Infinite non-simple C^{∗}-algebras: absorbing the Cuntz algebra O_{∞}

Eberhard Kirchberg^{∗} and Mikael Rørdam^{∗}

Abstract

The first named author has in [14] given a classification theorem for separable, nuclear, C^{∗}-algebras A that absorb the Cuntz algebra O_{∞}. (We say that A absorbs O_{∞} if A is isomorphic to A ⊗ O_{∞}.) Motivated by this theorem, we investigate here if one can intrinsically characterize which C^{∗}-algebras absorb O_{∞}. This leads us to three different notions of pure infiniteness of a C^{∗}-algebra, all given in terms of local, algebraic conditions on the C^{∗}-algebra.

The stronger of the three properties, called strongly purely infinite, is shown to be equivalent to absorbing O_{∞} for all separable, nuclear C^{∗}-algebras, that either are stable or have an approximate unit consisting of projections. In a previous paper, [16], we studied an intermediate, and perhaps more natural, condition: purely infinite, that extends the well-known property for simple C^{∗}-algebras. The weaker condition of the three, called weakly purely infinite, is shown to being equivalent to the absence of quasitraces in an ultrapower of the C^{∗}-algebra. The three conditions may be equivalent for all C^{∗}-algebras, and we prove that this is the case in the special cases where the C^{∗}-algebra is either simple, of real rank zero, or approximately divisible.

∗) Research at MSRI is supported in part by NSF grant DMS-9701755.

Contents

1 Introduction 2

2 Preliminaries 4

3 Purely infinite C^{∗}-algebras 9

4 Weakly purely infinite C^{∗}-algebras 11

5 Strongly purely infinite C^{∗}-algebras 23

6 Purely infinite C^{∗}-algebras of real rank zero 32

7 Approximately inner completely positive contraction on strongly purely infinite C^{∗}-algebras 37

8 Tensor products with O_{∞} 59

9 Summary and open problems 66

References 70

1 Introduction

It is well-known that each von Neumann algebra is the direct sum of two von Neumann algebras: one of which is finite and has a separating family of traces (the type I_{n}, n < ∞, and type II_{1} portions), and the other is properly infinite (the type I_{∞}, II_{∞}, and III portions). The properly infinite summand is again a direct sum of two von Neumann algebras: one of which has an essential ideal admitting a separating family of (unbounded, densely defined) traces (the type I_{∞} and II_{∞} portions), and one which is traceless and purely infinite (the type III portion). We investigate here an analogue to C^{∗}-algebras of the type III von Neumann algebras. More generally, we look at the C^{∗}-analogue of the quotient of a general von Neumann algebra by the ideal generated by its finite projections.

Our motivation for studying purely infinite C^{∗}-algebras stems primarily from the possibility of classifying these C^{∗}-algebras along the lines of Elliott’s classification program as described in [9]. More specifically, the first named author has recently proved that if A and B are nuclear, separable C^{∗}-algebras both with primitive ideal spectrum homeomorphic to some T_{0}-space X, then A ⊗ O_{∞} ⊗ K is isomorphic to B ⊗ O_{∞} ⊗ K if and only if A and B are KK_{X}-equivalent, where KK_{X} is a version of KK-theory that respects the primitive ideal spaces. As a step towards (and also a corollary of) this result, it is shown that two nuclear, separable C^{∗}-algebras A and B satisfy A ⊗ O_{2} ⊗ K ≅ B ⊗ O_{2} ⊗ K if and only if A and B have homeomorphic primitive ideal spaces.

Besides of answering questions, this classification also raises questions: Can one determine when A and B are KK_{X}-equivalent? Is there an intrinsic characterization of those C^{∗}-algebras that absorb O_{∞}? We shall not address the first question here except noting that one should be looking for a version of a universal coefficient theorem (UCT) for KK_{X}. An example of such a UCT was given in [21], where KK_{X}-equivalence was determined by isomorphism of 6-term exact sequences in the case where X consists of two points.

The first named author proved (in a paper published in [15]) that for a simple, nuclear, separable C^{∗}-algebra A one has A ≅ A ⊗ O_{∞} if and only if A is purely infinite (in the sense of J. Cuntz, [6]). The most optimistic generalization of this result to non-simple C^{∗}-algebras would be as follows: For any (separable, nuclear) C^{∗}-algebra A the following three conditions are equivalent:

(i) A ≅ A ⊗ O_{∞},

(ii) A is purely infinite (cf. Definition 3.4),

(iii) A is traceless, in the sense that no algebraic ideal in A admits a non-zero — possibly unbounded — quasitrace.

We shall here establish an equivalence similar to, but weaker than, this. Some of the technical difficulties are solved by inventing three different notions of being purely infinite. The stronger of the three, strongly purely infinite (defined in Section 5) is in Section 8 shown to be equivalent to (i) above for nuclear, separable C^{∗}-algebras that are either stable or have an approximate unit consisting of projections. In Section 4 we discuss weakly purely infinite C^{∗}-algebras, and it is shown that a C^{∗}-algebra A is weakly purely infinite if and only if its ultrapower A_{ω} is traceless. The intermediate condition was treated in detail in an earlier paper [16], and a brief survey of the properties of purely infinite C^{∗}-algebras is given in Section 3.

We show in Section 4 that every weakly purely infinite C^{∗}-algebra, that is either simple, approximately divisible, or has real rank zero, is purely infinite. In Section 6 we show that every purely infinite of real rank zero is strongly purely infinite. In particular, each simple purely infinite is strongly purely infinite. We also show that each approximately divisible, purely infinite C^{∗}-algebra is strongly purely infinite.

A more detailed summary of the main results of this paper is given in Section 9. This section also contains a list of open problems related to this article.

The main result on O_{∞}-absorbsion is obtained via a local Weyl–von Neumann theorem (Theorem 7.21) which says that an approximately inner completely positive map from a nuclear sub-C^{∗}-algebra of a strongly purely infinite C^{∗}-algebra can always be approximated by 1-step inner completely positive maps. Most of Section 7 is devoted to the proof of that result.

Both authors thank Etienne Blanchard for several useful discussions and for inviting us to Luminy, Marseilles, in the spring of 1999, where this work was initiated.

This article was completed in the fall of 2000 while both authors visited the Mathematical Sciences Research Institute (MSRI) in Berkeley, and we thank MSRI for its hospitality and support.

2 Preliminaries

This section has two subsections containing some background material that will be used frequently throughout this paper.

Cuntz comparison

The various notions of pure infiniteness we shall consider are defined in terms of comparison theory for positive elements in a C^{∗}-algebra. This theory, invented by Cuntz in [5], generalizes the comparison theory for projections in a von Neumann algebra. The reader is referred to [2], [20], and [16] for more reading about Cuntz comparison.

Definition 2.1 (Cuntz comparison) Let A be a C^{∗}-algebra, and let a, b be positive elements in A. Write a ≾ b if there is a sequence {x_{k}}_{k=1}^{∞} of elements in A such that x_{k}^{∗}bx_{k} → a. Write a ≈ b if a ≾ b and b ≾ a, and write a ∼ b if a = x^{∗}x and b = xx^{∗} for some x in A.

More generally, if a in M_{n}(A) and b in M_{m}(A) are positive matrices, then write a ≾ b if x_{k}^{∗}bx_{k} → a for some sequence {x_{k}}_{k=1}^{∞} of rectangular matrices in M_{m,n}(A), and let a ≈ b and a ∼ b have similar meanings as above.

With a in M_{n}(A) and b in M_{m}(A) one has a ≈ b if a ∼ b (but not conversely). Let a ⊕ b denote the element diag(a, b) in M_{n+m}(A), and let a ⊗ 1_{n} denote the n-fold direct sum a ⊕ a ⊕ ⋯ ⊕ a.

To each positive element a in a C^{∗}-algebra A and for each ε ≥ 0 define (a − ε)_{+} to be the positive part of the self-adjoint element a − ε·1 in the unitization of A. We remark that (a − ε)_{+} actually belongs to A and that (a − ε)_{+} = h_{ε}(a), where h_{ε}: ℝ^{+} → ℝ^{+} is the continuous function given by h_{ε}(t) = max{t − ε, 0}. We note the frequently used facts:

(a − ε_{1} − ε_{2})_{+} = ((a − ε_{1})_{+} − ε_{2})_{+},  ∥(a − ε)_{+} − a∥ ≤ ε,  (2.1)

that hold for all a in A^{+} and all ε, ε_{1}, ε_{2} ≥ 0.

Lemma 2.2 Let A be a C^{∗}-algebra, let a, b be positive elements in A, and let ε > ∥a − b∥ be given. Then there is a contraction d in A such that dbd^{∗} = (a − ε)_{+}.

Proof: For each r > 1 define g_{r}: ℝ^{+} → ℝ^{+} by g_{r}(t) = min{t, t^{r}}. Observe that g_{r}(b) → b as r → 1. Choose r > 1 such that (ε_{1} =) ∥a − g_{r}(b)∥ < ε and set b_{0} = g_{r}(b). Then b_{0} ≤ b, b_{0} ≤ b^{r}, and a − ε_{1} ≤ b_{0}. Find a positive contraction e in C^{∗}(a) with e(a − ε_{1})e = (a − ε)_{+}. Then (a − ε)_{+} ≤ eb_{0}e. Put x = b_{0}^{1/2}e and write x = v|x| for some partial isometry v in A^{∗∗}. As (a − ε)_{+} ≤ eb_{0}e = x^{∗}x, the element y = v(a − ε)_{+}^{1/2} belongs to A, y^{∗}y = (a − ε)_{+}, and

yy^{∗} = v(a − ε)_{+}v^{∗} ≤ vx^{∗}xv^{∗} = xx^{∗} = b_{0}^{1/2}e^{2}b_{0}^{1/2} ≤ b_{0}.

Now, following the proof of [18, Proposition 1.4.5], put d_{n} = y^{∗}(\frac{1}{n} +b^{r})^{−1/2}b^{(r−1)/2}. Because yy^{∗} ≤ b_{0} ≤ b^{r}, [18, Lemma 1.4.4] applies (with α = 1 and β = (r − 1)/r) and shows that {d_{n}}_{n=1}^{∞} is a Cauchy sequence in A. Let d be the limit of this Cauchy sequence. As in the proof of [18, Proposition 1.4.5], we have db^{1/2} = y^{∗}, so that dbd^{∗} = y^{∗}y = (a − ε)_{+}. Since yy^{∗} ≤ b_{0} ≤ b we get

d_{n}^{∗}d_{n} ≤ b^{(r−1)/2}(\frac{1}{n} + b^{r})^{−1/2}b(\frac{1}{n} + b^{r})^{−1/2}b^{(r−1)/2} ≤ 1.

Hence ∥d_{n}∥ ≤ 1 for each n which entails that d is a contraction. □

Lemma 2.3 ([20, Proposition 2.4]) Let A be a C^{∗}-algebra and let a, b be positive elements in A. The following conditions are equivalent:

(i) a ≾ b,

(ii) (a − ε)_{+} ≾ b for all ε > 0,

(iii) for every ε > 0 there is δ > 0 and x in A such that x^{∗}(b − δ)_{+}x = (a − ε)_{+}.

(iv) for every ε > 0 there is x in A such that x^{∗}x = (a − ε)_{+} and xx^{∗} belongs to \overline{bAb}.

In particular, if a is a positive element in the hereditary sub-C^{∗}-algebra \overline{bAb}, then a ≾ b.

Lemma 2.4 Let a, b be positive elements in a C^{∗}-algebra A and let δ > 0.

(i) If a = x^{∗}(b − δ)_{+}x for some x in A, then a = y^{∗}by for some y in A with ∥y∥ ≤ δ^{−1/2}∥a∥^{1/2}.

(ii) If a ≾ (b − δ)_{+}, then for each r > 1 there exists y in A with a = y^{∗}by and with ∥y∥ ≤ rδ^{−1/2}∥a∥^{1/2}.

(iii) If a ≤ b, then for each ε > 0 there is a contraction d in A with d^{∗}bd = (a − ε)_{+}. Proof: We shall need some functions in the proof. For δ > 0 and for α in the interval [0, 1/2] define f_{δ,α}, g_{δ}: ℝ^{+} → ℝ^{+} by

f_{δ,α}(t) = \begin{cases}\sqrt{\frac{(t−δ)^{2α}}{t}} , & t ≥ δ \\ 0, & t < δ \end{cases} ,  g_{δ}(t) = \begin{cases}1/t, & t ≥ δ \\ δ^{−2}t, & t < δ \end{cases} ,  (2.2)

and put f_{δ} = f_{δ,1/2}. Then

tf_{δ,α}(t)^{2} = (t − δ)_{+}^{2α}, f_{δ}(t)^{2} = (t − δ)_{+}g_{δ}(t).

(i). Put y = f_{δ}(b)x. Then

y^{∗}by = x^{∗}bf_{δ}(b)^{2}x = x^{∗}(b − δ)_{+}x = a, y^{∗}y = x^{∗}f_{δ}(b)^{2}x ≤ ∥g_{δ}(b)∥x^{∗}(b − δ)_{+}x ≤ δ^{−1}a.

The latter inequality yields the desired norm estimate for y.

(ii). Choose δ_{0} > 0 such that δ_{0} < δ and δ_{0}^{−1/2} ≤ rδ^{1/2}. If a ≾ (b − δ)_{+}, then a = x^{∗}(b − δ_{0})_{+}x for some x in A by Lemma 2.3 (iii) and (2.1). Hence a = y^{∗}by for some y in A with ∥y∥ ≤ δ_{0}^{−1/2}∥a∥^{1/2} ≤ rδ^{−1/2}∥a∥^{1/2} by (i).

(iii). The system \{f_{δ}(b)\}_{δ>0} is an approximate unit for \overline{bAb}, and so we can choose δ > 0 with ∥f_{δ}(b)af_{δ}(b) − a∥ < ε. Observe that

lim_{α→1/2^{−}} ∥f_{δ,α}(b)∥ = ∥f_{δ}(b)∥ = ((∥b∥ − δ)/∥b∥)^{1/2} < 1.

We can therefore choose α in the interval [0, 1/2) such that ∥a∥^{1/2−α}∥f_{δ,α}(b)∥ ≤ 1. Put y = f_{δ,α}(b), so that y^{∗}by = (b − δ)_{+}^{2α}. By Lemma 2.2 there is a contraction t in A such that (a − ε)_{+} = t^{∗}f_{δ}(b)af_{δ}(b)t. We have

f_{δ}(b)af_{δ}(b) ≤ f_{δ}(b)bf_{δ}(b) = (b − δ)_{+}.

Use [18, Proposition 1.4.5] to find u in A with u^{∗}(b − δ)_{+}^{2α}u = f_{δ}(b)af_{δ}(b) and ∥u∥ ≤ ∥a∥^{1/2−α}. Put d = yut. Then d^{∗}bd = (a − ε)_{+} and ∥d∥ ≤ ∥y∥∥u∥∥t∥ ≤ 1. □

The proof of Lemma 2.4 (iii) actually yields an element d in A of norm slightly less than 1 with d^{∗}bd = (a − ε)_{+}.

Limit algebras

A filter on a set Ω is an upper directed collection of subsets of Ω that is closed under finite intersections. To each filter ω on ℕ and to each C^{∗}-algebra A one defines the C^{∗}-algebra A_{ω} to be the quotient ℓ^{∞}(A)/c_{ω}(A), where c_{ω}(A) is the closed two-sided ideal in ℓ^{∞}(A) consisting of those sequences {a_{n}}_{n=1}^{∞} for which lim_{ω} ∥a_{n}∥ = 0. Recall that lim_{ω} ∥α_{n}∥ = α if for each ε > 0 there is a subset X in ω such that |α − α_{n}| < ε for all n in X. We shall denote the quotient mapping ℓ^{∞}(A) → A_{ω} by π_{ω}.

There is a canonical embedding of A into A_{ω} given by a ↦ π_{ω}(a, a, …,). We shall often view A as a sub-C^{∗}-algebra of A_{ω} using this embedding implicitly.

A filter ω on ℕ is called free if it contains all co-finite subsets of ℕ, and ω is called an ultrafilter if it is a maximal filter. Each filter is contained in an ultrafilter. The set of all cofinite subsets of ℕ is a free filter, and any ultrafilter containing this filter is a free ultra filter. If ω contains a finite set, then it is not free and there is a finite subset X_{0} = {n_{1}, …, n_{k}} of ℕ such that ω is the collection of all subsets of ℕ containing X_{0}. In this case, A_{ω} = A ⊕ A ⊕ ⋯ ⊕ A (with k summands), and π_{ω}(a_{1}, a_{2}, …) = (a_{n_{1}}, …, a_{n_{k}}). There are filters on ℕ that are neither free nor contain a finite set.

Lemma 2.5 Let A be a C^{∗}-algebra and let ω be a free filter on ℕ. Let {p_{i}}_{i∈𝕀} be a finite or countably infinite family of polynomials over A_{ω} in two non-commuting variables (cf. the examples below). Suppose for some finite constant C there is a sequence {d_{n}}_{n=1}^{∞} of elements in A_{ω} such that ∥d_{n}∥ ≤ C for all n in ℕ and

lim_{n→∞} ∥p_{i}(d_{n}, d_{n}^{∗})∥ = 0

for all i ∈ 𝕀. Then there is d in A_{ω} such that ∥d∥ ≤ C and p_{i}(d, d^{∗}) = 0 for all i ∈ 𝕀.

We shall typically apply the lemma in situations where the polynomials p_{i} are of the form:

p_{1}(d, d^{∗}) = d^{∗}ad − b,  p_{2}(d, d^{∗}) = [d^{∗}ad, b],  p_{3}(d, d^{∗}) = [d^{∗}ad, d^{∗}bd]

for some a, b in A_{ω}.

Proof: Each coefficient of each p_{i} is an element in A_{ω}. Upon lifting each such coefficient to an element (of the same norm) in ∏_{n=1}^{∞} A we obtain a sequence {p_{i,n}}_{n=1}^{∞} of polynomials over A in two non-commuting variables such that {p_{i,n}(e_{n}, e_{n}^{∗})}_{n=1}^{∞} is a bounded sequence for each bounded sequence {e_{n}}_{n=1}^{∞} in A, and such that

p_{i}(e, e^{∗}) = π_{ω}(p_{i,1}(e_{1}, e_{1}^{∗}), p_{i,2}(e_{2}, e_{2}^{∗}), p_{i,3}(e_{3}, e_{3}^{∗}), …) when e = π_{ω}(e_{1}, e_{2}, e_{3}, …).

Recall that

∥p_{i}(e, e^{∗})∥ = inf_{X∈ω} sup_{n∈X} ∥p_{i,n}(e_{n}, e_{n}^{∗})∥.

Write I as an increasing union of finite subsets {I_{k}}_{k=1}^{∞} of I. For each k find d_{k} in A_{ω} such that ∥d_{k}∥ ≤ C and ∥p_{i}(d_{k}, d_{k}^{∗})∥ < 1/k for all i in I_{k}. Write d_{k} = π_{ω}(d_{k,1}, d_{k,2}, d_{k,3}, …), where each d_{k,n} is an element in A with ∥d_{k,n}∥ ≤ C.

For each k find Y_{k} in ω such that

∥p_{i,n}(d_{k,n}, d_{k,n}^{∗})∥ < 1/k,  n ∈ Y_{k}, i ∈ I_{k}.  (2.3)

Define X_{k} ∈ ω inductively by setting X_{1} = Y_{1}, and

X_{k} = Y_{k} ∩ X_{k−1} ∩ (ℕ \ {1, 2, …, k})

for k ≥ 2. (The set ℕ \ {1, 2, …, k} belongs to ω by the assumption that ω is free.) Then (2.3) holds for all n in X_{k} and for all i in I_{k}, the sequence {X_{k}}_{k=1}^{∞} is decreasing and ∩_{k=1}^{∞}X_{k} = ∅. The last assumption implies that

ℕ = (ℕ \ X_{1}) ∪ (X_{1} \ X_{2}) ∪ (X_{2} \ X_{3}) ∪ · · · .

Let {e_{n}}_{n=1}^{∞} in ℓ^{∞}(A) be given by

e_{n} = \begin{cases}0, & if n ∈ ℕ \ X_{1}, \\ d_{1,n}, & if n ∈ X_{1} \ X_{2}, \\ d_{2,n}, & if n ∈ X_{2} \ X_{3}, \\ ⋮ & ⋮ \end{cases}

and put d = π_{ω}(e_{1}, e_{2}, e_{3}, …) in A_{ω}. Then ∥p_{i,n}(e_{n}, e_{n}^{∗})∥ ≤ 1/k for all n in X_{k} and for all i in I_{k}. Hence ∥p_{i}(d, d^{∗})∥ ≤ 1/k for all i in I_{k}. This holds for all k, and so p_{i}(d, d^{∗}) = 0 for all i in I as desired. □

3 Purely infinite C^{∗}-algebras

We give here a brief review of some of the results on purely infinite C^{∗}-algebras from [16].

Definition 3.1 (Properly infinite elements) A positive element a in a C^{∗}-algebra A is said to be properly infinite if a is non-zero and a ⊕ a ≾ a.

The condition a ⊕ a ≾ a means by definition that there is a sequence {d_{n}}_{n=1}^{∞} of elements in M_{2}(A) such that

∥d_{n}^{∗} (\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) d_{n} − (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) ∥ → 0.  (3.1)

We note as a side remark that if there is a sequence {d_{n}} such that (3.1) holds for all a in A, and if A is separable, then A is isomorphic to A ⊗ O_{∞} by Proposition 8.4. (See also Proposition 7.8.)

Some properties of properly infinite elements, established in [16, Proposition 3.3], include:

Lemma 3.2 The following conditions are equivalent when a is a positive element in a C^{∗}-algebra A:

(i) a is properly infinite.

(ii) For each ε > 0 there are positive elements a_{1}, a_{2} in \overline{aAa} such that a_{1}a_{2} = 0 and (a − ε)_{+} ≾ a_{j} for j = 1, 2.

(iii) For each natural number n and for each ε > 0 there are elements d_{1}, …, d_{n} in \overline{aAa} such that d_{j}^{∗}ad_{i} = δ_{ij}(a − ε)_{+}.

The property that an element in a C^{∗}-algebra is properly infinite depends on the C^{∗}-alge-bra to which the element belongs. However, as follows readily from Lemma 3.2 (iii), if a is a positive element in a hereditary sub-C^{∗}-algebra B of a C^{∗}-algebra A, and if a is properly infinite relatively to A, then a is also properly infinite relatively to B.

In the spirit of comparison theory for projections, a positive element a is called infinite if a ⊕ b ≾ a for some non-zero positive element b in A, cf. Definition 2.1.

If φ: A → B is a ^{∗}-homomorphism between C^{∗}-algebras A and B, and if a is a properly infinite element in A, then φ(a) is properly infinite if non-zero. Moreover, a positive element a is properly infinite if and only if φ(a) is either infinite or zero for every ^{∗}-homomorphism φ on A, cf. [16, Proposition 3.14].

If a is a properly infinite element in A, then b ≾ a for each positive element b in the closed two-sided ideal, \overline{AaA}, generated by a, cf. [16, Proposition 3.5].

The set of properly infinite positive elements in a C^{∗}-algebra is not dense. For example, no finite rank projections on a Hilbert space H is properly infinite, but they can be approximated in norm by properly infinite positive elements in B(H). We have the following (weaker) approximation lemma:

Lemma 3.3 Let a be a positive element in a C^{∗}-algebra A and suppose that for each ε > 0 there is a properly infinite element b in A with b ≾ a and ∥a − b∥ < ε. Then a is properly infinite.

Proof: By Lemma 2.2 we have

(a − ε)_{+} ⊕ (a − ε)_{+} ≾ b ⊗ b ≾ b ≾ a,

and if (a − ε)_{+} ⊕ (a − ε)_{+} ≾ a for all ε > 0, then a is properly infinite, cf. Lemma 2.3. □

Definition 3.4 (Purely infinite C^{∗}-algebras) A C^{∗}-algebra A is said to be purely infinite if A has no Abelian quotients and if for each pair of positive elements a, b in A such that a belongs to \overline{AbA}, the closed two-sided ideal generated by b, we have a ≾ b

It is shown in [16, Theorem 4.16] that A is purely infinite if and only if each non-zero positive element in A is properly infinite. Other facts about purely infinite C^{∗}-algebras, proved in [16], include:

Proposition 3.5 (Permanence properties)

(i) For each short exact sequence 0 → I → A → B → 0 one has that A is purely infinite if and only if I and B are purely infinite.

(ii) If A and B are stably isomorphic and if A is purely infinite, then so is B.

(iii) Each hereditary sub-C^{∗}-algebra of a purely infinite C^{∗}-algebra is purely infinite.

(iv) If A is a purely infinite C^{∗}-algebra and if ω is a free filter on ℕ, then A_{ω} is purely infinite.

(v) Any inductive limit of a system of purely infinite C^{∗}-algebras is purely infinite.

(vi) A ⊗ O_{∞} is purely infinite for every C^{∗}-algebra A.

A simple C^{∗}-algebra is purely infinite if and only if each of its non-zero hereditary sub-C^{∗}-algebras contain an infinite projection, in agreement with Cuntz’ original definition in [6].

4 Weakly purely infinite C^{∗}-algebras

One motivation for introducing the notion of weakly purely infinite C^{∗}-algebras is found in [16, Theorem 5.9] which says that an approximate divisible C^{∗}-algebra is purely infinite if and only if it is traceless. (The notions of being approximate divisible and traceless are defined below.) We shall here characterize tracelessness in terms of being weakly purely infinite (without assuming approximate divisibility).

Definition 4.1 (Approximate divisibility) A C^{∗}-algebra A is said to be approximately divisible if for every natural number n, for every finite subset F of A, and for every ε > 0 there is a unital ^{∗}-homomorphism φ: M_{n}(ℂ) ⊕ M_{n+1}(ℂ) → M(A), where M(A) denotes the multiplier algebra of A, such that

∥φ(x)a − aφ(x)∥ ≤ ε∥x∥

for all a in F and all x in M_{n}(ℂ) ⊕ M_{n+1}(ℂ).

As remarked in [16, Lemma 5.6], if A is approximately divisible and if B is any C^{∗}-alge-bra, then A ⊗ B is approximately divisible (where ⊗ is any tensor product). As O_{∞} is approximately divisible (a consequence of [15], see also Corollary 8.3) we find that A ⊗ O_{∞} is approximately divisible for every C^{∗}-algebra A.

Definition 4.2 (Traceless C^{∗}-algebras) A C^{∗}-algebra A will be called traceless if no algebraic ideal of A admits a non-zero quasitrace ^{1}.

There is a one-to-one correspondence between quasitraces and lower semi-continuous dimension functions (established by Blackadar and Handelman in [3]), and so being traceless is the same as being absent of non-zero lower semi-continuous dimension functions, cf. [16, Theorem 5.9].

1By a quasitrace we shall always mean a lower semi-continuous, positive 2-quasitrace. A 2-quasitrace on a C^{∗}-algebra A is a quasitrace that extends (not necessarily in the obvious way) to a quasitrace on M_{2}(A). Haagerup proved in [10] that each quasitrace on a unital, exact C^{∗}-algebra extends to a trace (and the first named author has extended this result to non-unital, exact C^{∗}-algebras in [13]). Thus an exact C^{∗}-algebra A is traceless if and only if no algebraic ideal of A admits a non-zero trace.

The C^{∗}-algebra B(H) of all bounded operators on an infinite dimensional Hilbert space H is not traceless, since it has a non-zero trace defined on the trace class operators on H. We do not require our traces (or quasitraces) to be bounded.

Definition 4.3 (Weakly purely infinite C^{∗}-algebras) A C^{∗}-algebra A will be said to have property pi-n if the n-fold direct sum a ⊕ a ⊕ ⋯ ⊕ a = a ⊗ 1_{n} is properly infinite (cf. Definition 3.1) for every non-zero positive element a in A. If A is pi-n for some n, then we shall call A weakly purely infinite.

By [16, Theorem 4.16], a C^{∗}-algebra is pi-1 if and only if it is purely infinite. By definition, a ⊗ 1_{n} is properly infinite if and only if a ⊗ 1_{2n} ≾ a ⊗ 1_{n}.

Lemma 4.4 Let a be a positive element in a C^{∗}-algebra A and let n be a natural number. The following conditions are equivalent:

(i) a ⊗ 1_{n} is properly infinite,

(ii) a ⊗ 1_{m} ≾ a ⊗ 1_{n} for all natural numbers m,

(iii) a ⊗ 1_{m} ≾ a ⊗ 1_{r} for all natural numbers r, m with r ≥ n,

(iv) a ⊗ 1_{n+1} ≾ a ⊗ 1_{n},

(v) for each ε > 0 and for each m in ℕ there is x in M_{m,n}(A) such that x^{∗}x belongs to M_{n}(\overline{aAa}) and xx^{∗} = (a − ε)_{+} ⊗ 1_{m}.

Proof: The implications (iii) ⇒ (ii) ⇒ (i) ⇒ (iv) are trivial, and (ii) and (v) are equivalent by Lemma 2.3. Assume that (iv) holds. Then

a ⊗ 1_{k+1} = (a ⊗ 1_{n+1}) ⊕ (a ⊗ 1_{k−n}) ≾ (a ⊗ 1_{n}) ⊕ (a ⊗ 1_{k−n}) = a ⊗ 1_{k}

for all k ≥ n. By transitivity of the relation ≾ and by induction we obtain (iii). □

Proposition 4.5 (Permanence properties)

(i) If A is pi-n, then A is pi-m for every m ≥ n.

(ii) Every hereditary sub-C^{∗}-algebra of a pi-n C^{∗}-algebra is pi-n.

(iii) Every non-zero quotient of a pi-n C^{∗}-algebra is again pi-n.

(iv) If 0 → I → A → B → 0 is a short exact sequence, if I is pi-n, and B is pi-m, then A is pi-(n + m).

(v) If A is an inductive limit of a system of C^{∗}-algebras, each of which is pi-n for some n, then A is pi-n.

(vi) If A and B are stably isomorphic C^{∗}-algebras, and if A is pi-n, then B is pi-n^{2}.

By (iii), an extension of two weakly purely infinite C^{∗}-algebras is again weakly purely infinite. The given estimate on the degree of pure infiniteness is not optimal (cf. Proposition 3.5 (i)), but it suffices for our purposes.

Proof: (i) follows from Lemma 4.4.

(ii). This follows from the remark below Lemma 3.2 that being properly infinite is preserved when passing to hereditary sub-C^{∗}-algebras.

(iii). This follows from the fact, remarked below Lemma 3.2, that a non-zero image under a ^{∗}-homomorphism of a properly infinite element is again properly infinite.

(iv). Let a be a non-zero positive element in A and let ε > 0 be given. View I as an ideal of A and let π denote the quotient mapping A → B. If a belongs to I, then a ⊗ 1_{n} is properly infinite because I is pi-n.

Assume now that π(a) is non-zero. Then π(a) ⊗ 1_{m} is properly infinite in M_{m}(B). By Lemma 3.2 we can find positive elements b_{1}, b_{2} in M_{m}(\overline{π(a)Bπ(a)}) such that b_{1}b_{2} = 0 and π((a − ε/3)_{+}) ⊗ 1_{m} ≾ b_{j} for j = 1, 2. Since π((a − ε/3)_{+}) ⊗ 1_{m} is properly infinite we also have

π((a − ε/3)_{+}) ⊗ 1_{m+n} ≾ b_{j}, j = 1, 2.

Lift b_{1}, b_{2} to mutually orthogonal, positive elements a_{1}, a_{2} in M_{m}(\overline{aAa}). There are positive elements c′_{1}, c′_{2} in M_{m+n}(\overline{aIa}) such that (a − ε/2)_{+} ⊗ 1_{m+n} ≾ a_{j} ⊕ c′_{j} for j = 1, 2, cf. [16, Lemma 4.12]. There is δ > 0 such that

(a − ε)_{+} ⊗ 1_{m+n} ≾ a_{j} ⊕ (c′_{j} − δ)_{+}, j = 1, 2,  (4.1)

cf. Lemma 2.3. We show next that there are positive elements c_{1}, c_{2} in M_{n}(\overline{aIa}) such that c_{1}c_{2} = 0 and (c′_{j} −δ)_{+} ≾ c_{j}. Find a positive element c_{0} in \overline{aIa} and η > 0 such that (c′_{1} −δ)_{+} and (c′_{2} − δ)_{+} belong to M_{m+n}(\overline{I(c_{0} − η)_{+}I}). Since I is pi-n, (c_{0} − η)_{+} ⊗ 1_{n} is properly infinite, and so (c′_{j} − δ)_{+} ≾ (c_{0} − η)_{+} ⊗ 1_{n}. Because c_{0} ⊗ 1_{n} also is properly infinite we can use Lemma 3.2 to find c_{1}, c_{2} in the hereditary sub-C^{∗}-algebra of M_{n}(I) generated by c_{0} ⊗ 1_{n} such that c_{1}c_{2} = 0 and (c_{0} − η)_{+} ⊗ 1_{n} ≾ c_{j} for j = 1, 2. Notice that c_{1}, c_{2} belong to M_{n}(\overline{aAa}). Hence d_{j} = diag(a_{j}, c_{j}) belongs to M_{m+n}(\overline{aAa}), d_{1}d_{2} = 0, and

(a − ε)_{+} ⊗ 1_{m+n} ≾ a_{j} ⊕ (c′_{j} − δ)_{+} ≾ a_{j} ⊕ c_{j} ≾ d_{j}.

It follows from Lemma 3.2 that a ⊗ 1_{m+n} is properly infinite, and this proves (iv).

(v). Assume that A is the inductive limit of a system {A_{i}}_{i∈𝕀} of C^{∗}-algebras each of which is pi-n. By (iii) we can assume that each map A_{i} → A is an inclusion mapping (so that A is the closure of the directed union of the algebras A_{i}). Let a be a non-zero positive element in A and let ε > 0 be given. Use [16, Lemma 2.5] to find i in 𝕀 and a non-zero positive element b in A_{i} such that (a − ε)_{+} ≾ b ≾ a. Then

(a − ε)_{+} ⊗ 1_{n+1} ≾ b ⊗ 1_{n+1} ≾ b ⊗ 1_{n} ≾ a ⊗ 1_{n}.

Since this holds for all ε > 0, we get a ⊗ 1_{n+1} ≾ a ⊗ 1_{n}, and so A is pi-n.

(vi). By (ii) and (v) it suffices to show that if A is pi-n, then M_{k}(A) is pi-n^{2} for all natural numbers k.

We show first that b ⊗ 1_{n} is properly infinite for every non-zero positive element b in M_{n}(A). Indeed, let b_{ij} be the matrix entries of b. Then b_{ii} ≾ b ≾ b_{11} ⊕ ⋯ ⊕ b_{nn} for i = 1, …, n (cf. Lemma 5.3 below) and by Lemma 4.4 this implies:

b ⊗ 1_{n^{2}+1} ≾ (b_{11} ⊕ ⋯ ⊕ b_{nn}) ⊗ 1_{n^{2}+1} ≾ (b_{11} ⊕ ⋯ ⊕ b_{nn}) ⊗ 1_{n} ≾ b ⊗ 1_{n^{2}}.

Take now a non-zero positive matrix a in M_{k}(A) and let ε > 0 be given. Choose ε_{1} > 0 such that if ∥a − c∥ < ε_{1}, then ∥a − c∥ < ε/2 and ∥(a − ε/2)_{+} − (c − ε/2)_{+}∥ < ε/2. By Lemma 2.2, this will entail that (a − ε)_{+} ≾ (c − ε/2)_{+} ≾ a. Find a positive contraction e in A and δ > 0 such that

∥a − a^{1/2}((e − δ)_{+} ⊗ 1_{k})a^{1/2}∥ < ε_{1}.

Since e ⊗ 1_{n} is properly infinite there is an element d in M_{n,k}(A) such that d^{∗}(e ⊗ 1_{n})d = (e − δ)_{+} ⊗ 1_{k}. With t = (e^{1/2} ⊗ 1_{n})da^{1/2} in M_{n,k}(A) we have ∥t^{∗}t − a∥ < ε_{1} and tt^{∗} (= b) is a positive element in M_{n}(A). Now (tt^{∗} − ε/2)_{+} ⊗ 1_{n^{2}}, and hence (t^{∗}t − ε/2)_{+} ⊗ 1_{n^{2}}, are properly infinite by the first part of the proof. By the choice of ε_{1} we therefore get

(a − ε)_{+} ⊗ 1_{n^{2}+1} ≾ (t^{∗}t − ε/2)_{+} ⊗ 1_{n^{2}+1} ≾ (t^{∗}t − ε/2)_{+} ⊗ 1_{n^{2}} ≾ a ⊗ 1_{n^{2}}.

As ε > 0 was arbitrary, this proves that a ⊗ 1_{n^{2}} is properly infinite. □

Proposition 4.6 The following conditions are equivalent for every C^{∗}-algebra A:

(i) A is pi-n.

(ii) ℓ^{∞}(A) is pi-n.

(iii) A_{ω} is pi-n for every filter ω on ℕ.

(iv) A_{ω} is pi-n for some filter ω on ℕ.

Proof: (i) ⇒ (ii). Let a = (a_{1}, a_{2}, …) be a non-zero positive element in ℓ^{∞}(A) and let ε > 0 be given. We show that a⊗1_{n} is properly infinite. It is no loss of generality to assume that ∥a∥ = 1. Since A is pi-n, (a_{k}−ε)_{+}⊗1_{n+1} ≾ (a_{k}−ε)_{+}⊗1_{n}, and hence (a_{k}−ε)_{+}⊗1_{n+1} = x_{k}^{∗}(a_{k} ⊗ 1_{n})x_{k} for some x_{k} in M_{n,n+1}(A) with ∥x_{k}∥ ≤ 2ε^{−1/2}, cf. Lemma 2.4 (ii). It follows that x = (x_{1}, x_{2}, …) belongs to ℓ^{∞}(A) and that (a − ε)_{+} ⊗ 1_{n+1} = x^{∗}(a ⊗ 1_{n})x. Since this holds for all ε > 0 we get a ⊗ 1_{n+1} ≾ a ⊗ 1_{n}. Hence a ⊗ 1_{n} is properly infinite, and (ii) must hold.

(ii) ⇒ (iii) follows from Proposition 4.5 (iii); and (iii) ⇒ (iv) is trivial.

(iv) ⇒ (i). Let a be a non-zero positive element in A and let ε > 0. Since A_{ω} is pi-n there is x = π_{ω}(x_{1}, x_{2}, …) in M_{n,n+1}(A_{ω}) such that x^{∗}(a ⊗ 1_{n})x = (a − ε/2)_{+} ⊗ 1_{n+1}. Each x_{k} belongs to M_{n,n+1}(A) and

lim_{ω} ∥x_{k}^{∗}(a ⊗ 1_{n})x_{k} − (a − ε/2)_{+} ⊗ 1_{n+1}∥ = 0.

Hence ∥x_{k}^{∗}(a⊗1_{n})x_{k} −(a−ε/2)_{+} ⊗1_{n+1}∥ < ε/2 for some k. By Lemma 2.2 this entails that (a − ε)_{+} ⊗ 1_{n+1} ≾ a ⊗ 1_{n}. Since ε > 0 was arbitrary we conclude that a ⊗ 1_{n+1} ≾ a ⊗ 1_{n}, and so A is pi-n. □

Lemma 4.7 Let A be a C^{∗}-algebra which is pi-n. Then for each pair of positive elements a, b in A, where a belongs to \overline{AbA}, and for each ε > 0 there are elements x_{1}, …, x_{n} in A such that

∑_{j=1}^{n} x_{j}^{∗}bx_{j} = (a − ε)_{+}.

Proof: Let ε > 0 be given. By [16, Proposition 2.7 (v)] there is a natural number k such that (a − ε/2)_{+} ≾ b ⊗ 1_{k}. By Lemma 4.4, b ⊗ 1_{k} ≾ b ⊗ 1_{n}, and so (a − ε/2)_{+} ≾ b ⊗ 1_{n}. It follows that (a − ε)_{+} = x^{∗}(b ⊗ 1_{n})x for some x in M_{n,1}(A). With x_{1}, …, x_{n} the entries of x, we obtain the desired identity. □

Theorem 4.8 Let A be a C^{∗}-algebra.

(i) For each free filter ω on ℕ the following three conditions are equivalent:

(a) A_{ω} is traceless,

(b) A_{ω} is weakly purely infinite,

(c) A is weakly purely infinite.

(ii) If A is weakly purely infinite, then A is traceless.

Proof: (i). (c) ⇒ (b) follows from Proposition 4.6, and (b) ⇒ (a) follows from (i). We proceed to prove (a) ⇒ (c). We do so indirectly by assuming that A is not weakly purely infinite. We construct below a positive element a in A_{ω} which satisfies

((a − 1/4)_{+} ⊕ (a − 1/4)_{+}) ⊗ 1_{k} ≾| a ⊗ 1_{k}  (4.2)

for all natural numbers k. It then follows from [16, Proposition 5.7] that A_{ω} admits a non-zero lower semi-continuous dimension functions, and hence that A is not traceless.

For each natural number k there is a positive contraction a_{k} in A such that

((a_{k} − 1/2)_{+} ⊕ (a_{k} − 1/2)_{+}) ⊗ 1_{m} ≾| a_{k} ⊗ 1_{m},  m = 1, 2, …, k.  (4.3)

To see this, find for each k a non-zero positive element b_{k} in A such that b_{k} ⊗ 1_{k} is not properly infinite. Then b_{k} ⊗ 1_{m} is not properly infinite for m = 1, 2, …, k, cf. Lemma 4.4. Hence b_{k} ⊗ 1_{2m} ≾| b_{k} ⊗ 1_{m} for m ≤ k. By Lemma 2.3, (b_{k} − ε_{m})_{+} ⊗ 1_{2m} ≾| b_{k} ⊗ 1_{m} for some ε_{m} > 0. Taking ε to be the minimum of ε_{1}, …, ε_{k} we can assume that ε_{m} = ε for each m. Define h: ℝ^{+} → ℝ^{+} to be h(t) = min{t/(2ε), 1}, and set a_{k} = h(b_{k}). Then a_{k} is a positive contraction in A, (b_{k} − ε)_{+} ≈ (a_{k} − 1/2)_{+}, and a_{k} ≈ b_{k}. This shows that (4.3) holds.

With π_{ω}: ℓ^{∞}(A) → A_{ω} denoting the quotient mapping, set a = π_{ω}(a_{1}, a_{2}, …). We proceed to show that (4.2) holds. Assume the contrary. Then (a − 1/4)_{+} ⊗ 1_{2k} ≾ a ⊗ 1_{k} for some natural number k, and so ∥x^{∗}(a ⊗ 1_{k})x − (a − 1/4)_{+} ⊗ 1_{2k}∥ < 1/4 for some x in M_{k,2k}(A_{ω}). Write x = π_{ω}(x_{1}, x_{2}, …). Then

lim_{n→ω} ∥x_{n}^{∗}(a_{n} ⊗ 1_{k})x_{n} − (a_{n} − 1/4)_{+} ⊗ 1_{2k}∥ < 1/4.

Because ω is a free filter there is an infinite subset X of ℕ such that

∥x_{n}^{∗}(a_{n} ⊗ 1_{k})x_{n} − (a_{n} − 1/4)_{+} ⊗ 1_{2k}∥ < 1/4, n ∈ X.

Use Lemma 2.2 to deduce that (a_{n} − 1/2)_{+} ⊗ 1_{2k} ≾ a_{n} ⊗ 1_{k} for all n in X. Because X is infinite it contains an element n ≥ k. But this is in contradiction with (4.3).

(ii). If A is weakly purely infinite, then A is pi-n for some n. Arguing as in the proof of [16, Proposition 5.1] we see that A admits no non-zero dimension function, and hence no quasitrace: Indeed, if d were a dimension function with domain I (an algebraic ideal of A) and if a is a non-zero positive element in I, then a ⊗ 1_{n} is properly infinite, and hence a ⊗ 1_{n+1} ≾ a ⊗ 1_{n}. This implies nd(a) ≤ (n + 1)d(a), and so d(a) = 0. □

It is not known if the sum of two properly infinite elements is again properly infinite. We do however have the following weaker result.

Lemma 4.9 Let a_{1}, …, a_{n} be properly infinite elements in a C^{∗}-algebra A, and put a = a_{1} + a_{2} + ⋯ + a_{n}. Then:

(i) a ⊗ 1_{n} is properly infinite,

(ii) and if a_{1}, …, a_{n} are mutually orthogonal, then a is properly infinite.

Proof: (i) follows from the relations

a ⊗ 1_{2n} ≾ (a_{1} ⊕ a_{2} ⊕ ⋯ ⊕ a_{n}) ⊗ 1_{2n} ≾ (a_{1} ⊕ a_{2} ⊕ ⋯ ⊕ a_{n}) ≾ a ⊗ 1_{n}.

(ii) follows from [16, Lemma 3.9]. □

The next lemma is used for our proof of Proposition 4.11 saying that the multiplier algebra of a weakly purely infinite C^{∗}-algebra is weakly purely infinite. Its proof uses a technique of Elliott’s from [8].

Lemma 4.10 Let A be a σ-unital weakly purely infinite C^{∗}-algebra, let T be a positive element in M(A), and let ε > 0. Then there is an increasing, countable, approximate unit {e_{n}}_{n=1}^{∞} for A consisting of positive contractions satisfying e_{n+1}e_{n} = e_{n} and e_{0} = 0, such that

T = a + ∑_{n=1}^{∞} f_{2n−1}^{1/2}T f_{2n−1}^{1/2} + ∑_{n=1}^{∞} f_{2n}^{1/2}T f_{2n}^{1/2},  f_{n} = e_{n} − e_{n−1},  (4.4)

where a is a (not necessarily positive) element in A with ∥a∥ ≤ ε.

We have f_{n} ⊥ f_{m} whenever |n − m| ≥ 2, so that the summands in the two (strictly convergent) sums in (4.4) are mutually orthogonal elements in A.

Proof: Take a countable approximate unit {e_{n}}_{n=1}^{∞} for A consisting of positive contractions satisfying e_{n+1}e_{n} = e_{n} and ∥e_{n}T − T e_{n}∥ → 0, cf. [18, Theorem 3.12.14]. Upon passing to a subsequence of {e_{n}}_{n=1}^{∞} we may assume that

∑_{n=1}^{∞} ∥(e_{n} − e_{n−1})^{1/2}T − T (e_{n} − e_{n−1})^{1/2}∥ < ε,  (4.5)

(where e_{0} = 0.) Now, 1 = ∑_{n=1}^{∞} f_{n} (the sum is strictly convergent), and so

a = T − ∑_{n=1}^{∞} f_{2n−1}^{1/2}T f_{2n−1}^{1/2} + ∑_{n=1}^{∞} f_{2n}^{1/2}T f_{2n}^{1/2}
= ∑_{n=1}^{∞} T f_{n} − ∑_{n=1}^{∞} f_{n}^{1/2}T f_{n}^{1/2} = ∑_{n=1}^{∞}(T f_{n}^{1/2} − f_{n}^{1/2}T)f_{n}^{1/2}.

By (4.5) we have ∥a∥ ≤ ∑_{n=1}^{∞} ∥T f_{n}^{1/2} − f_{n}^{1/2}T ∥ < ε, and because this sum is norm convergent, we can also conclude that a belongs to A. □

Proposition 4.11 Let A be a σ-unital weakly purely infinite C^{∗}-algebra. Then its multiplier algebra M(A) is weakly purely infinite.

Proof: Assume that A is pi-r.

We show first that if {e_{n}}_{n=1}^{∞} is an approximate unit as in Lemma 4.10 and if R is non-zero and given by a (strictly convergent) sum R = ∑_{k=1}^{∞} a_{k}, where a_{1}, a_{2}, … is a bounded sequence of positive, mutually orthogonal elements of A such that a_{k}e_{k−1} = 0 for all k, then R ⊗ 1_{r} is properly infinite. Let ε > 0 be given. Since A is pi-r there are elements x_{k} in M_{r,2r}(A) such that x_{k}^{∗}x_{k} = (a_{k} − ε)_{+} ⊗ 1_{2r} and x_{k}x_{k}^{∗} ∈ M_{r}(\overline{a_{k}Aa_{k}}), cf. Lemma 4.4. Now, ∥x_{k}∥ = ∥(a_{k} − ε)_{+}∥^{1/2} which shows that the sequence {x_{k}}_{k=1}^{∞} is normbounded. The (i, j)th entry, x_{k}(i, j), of x_{k} belongs to \overline{a_{k}Aa_{k}}, and so x_{k}(i, j)e_{k−1} = 0. It follows that X(i, j) = ∑_{k=1}^{∞} x_{k}(i, j) is strictly convergent in M(A). The resulting matrix X in M_{r,2r}(M(A)), whose (i, j)th entry is X(i, j), satisfies X^{∗}X = (R − ε)_{+} ⊗ 1_{2r} and XX^{∗} ∈ M_{r}(\overline{RM(A)R}). Since ε > 0 was arbitrary, we conclude that R ⊗ 1_{2r} ≾ R ⊗ 1_{r}, cf. Lemma 2.3.

The argument above and Lemma 4.10 show that each positive element T in M(A) can be written as T = a+T_{1} +T_{2}, where T_{1}, T_{2} are positive elements in M(A) for which T_{j} ⊗1_{r} is properly infinite and a belongs to A. With π: M(A) → M(A)/A denoting the quotient mapping we get π(T) = π(T_{1}) + π(T_{2}), and π(T_{j}) ⊗ 1_{r} is properly infinite (cf. the remarks below Lemma 3.2). Hence π(T) ⊗ 1_{2r} is properly infinite by Lemma 4.9 (i). This proves that M(A)/A is pi-2r. By Proposition 4.5 we conclude that M(A) is pi-3r and hence weakly purely infinite. □

We do not know if the multiplier algebra of a purely infinite C^{∗}-algebra is again purely infinite.

We end this section by discussing a sufficient condition under which one can deduce pure infiniteness from weak pure infiniteness.

It is a consequence of a lemma by Glimm that if n is a natural number and if A is a C^{∗}-algebra that admits an irreducible representation on a Hilbert space of dimension at least n, then there is a non-zero ^{∗}-homomorphism from M_{n}(C_{0}((0, 1])) into A, cf. [16, Proposition 4.10].

Definition 4.12 (The global Glimm property) A C^{∗}-algebra A is said to have the global Glimm property if for each natural number n, for each positive element a in A, and for each ε > 0 there is a ^{∗}-homomorphism φ: M_{n}(C_{0}((0, 1])) → \overline{aAa} such that (a − ε)_{+} belongs to the closed two-sided ideal of A generated by the image of φ.

Equivalently, A has the global Glimm property if for each positive element a in A, for each ε > 0, and for each natural number n there are mutually orthogonal elements t_{1}, …, t_{n} in \overline{aAa} such that t_{1} ∼ t_{2} ∼ ⋯ ∼ t_{n}, cf. Definition 2.1, and such that (a − ε)_{+} belongs to the closed two-sided ideal generated by t = t_{1} + ⋯ + t_{n}.

The latter condition is illustrated in our next result:

Lemma 4.13 Let a be a positive element in a C^{∗}-algebra A and suppose that there is a full ^{2} ^{∗}-homomorphism φ: M_{n}(ℂ) → M(\overline{aAa}). Then, for each ε > 0, there are mutually orthogonal and mutually equivalent positive elements t_{1}, …, t_{n} in \overline{aAa} such that (a − ε)_{+} belongs to the closed two-sided ideal generated by t = t_{1} + ⋯ + t_{n}.

Proof: Observe first that if p is a full projection in M(\overline{aAa}), then p\overline{aAa}p is a full hereditary sub-C^{∗}-algebra of \overline{aAa}. Let \{e_{ij}\} denote the matrix units of M_{n}(ℂ). Take a positive contraction h in φ(e_{11})\overline{aAa}φ(e_{11}) such that (a − ε)_{+} belongs to \overline{AhA}. Put t_{j} = φ(e_{j1})hφ(e_{1j}). Then t_{1}, …, t_{n} are mutually orthogonal and mutually equivalent positive elements in \overline{aAa} and (a − ε)_{+} belongs to the ideal generated by t = t_{1} + ⋯ + t_{n}. □

Lemma 4.14 Let A be a C^{∗}-algebra such that no non-zero hereditary sub-C^{∗}-algebra of A a finite dimensional representation. Then A has the global Glimm property if A is either simple, approximately divisible, or purely infinite.

2A ^{∗}-homomorphism A → B is called full if its image is not contained in a proper closed two-sided ideal in B.

Proof: The local version of Glimm’s lemma (see [16, Proposition 4.10]) implies the global Glimm property when A is simple.

Assume next that A is approximately divisible. Let a be a positive element in A and let ε > 0 be given. Let B denote the algebra M_{n}(ℂ)⊕M_{n+1}(ℂ) and find an asymptotically central sequence of unital ^{∗}-homomorphisms φ_{k}: B → M(A). Let μ denote the Haar measure on the compact unitary group of B, and define a conditional expectation E_{k}: A → A ∩ φ_{k}(B)′ by

E_{k}(a) = ∫_{U(B)} φ_{k}(u)aφ_{k}(u)^{∗} dμ(u).

Then ∥a − E_{k}(a)∥ → 0. Choose k large enough so that ∥a − E_{k}(a)∥ < ε/2, and put a_{0} = (E_{k}(a) − ε/2)_{+}. Then a_{0} is a positive element in A ∩ φ_{k}(B)′ and ∥a − a_{0}∥ < ε. By Lemma 2.2 there are elements d, e in A such that a_{0} = d^{∗}ad and (a − ε)_{+} = e^{∗}a_{0}e. Put a_{1} = a^{1/2}dd^{∗}a^{1/2}. There is an isomorphism

α: \overline{a_{0}Aa_{0}} → \overline{a_{1}Aa_{1}} ⊆ \overline{aAa}

such that α(a_{0}) = a_{1}, cf. [16, Lemma 2.4]. Being equivalent, a_{0} and a_{1} generate the same closed two-sided ideal of A, and (a − ε)_{+} belongs to this ideal.

The canonical isomorphism β: B ⊗ C(sp(a_{0})) → C^{∗}(φ_{k}(B), a_{0}) restricts to a ^{∗}-homo-morphism

β_{0} : B ⊗ C_{0}(sp(a_{0}) \ \{0\}) → \overline{a_{0}Aa_{0}},

and a_{0} = β_{0}(1 ⊗ ι), where ι in C_{0}(sp(a_{0}) \ {0}) is given by ι(t) = t. Take a full ^{∗}-homo-morphism M_{n}(ℂ) → B, a surjective ^{∗}-homomorphism C_{0}((0, 1]) → C_{0}(sp(a_{0}) \ {0}), and obtain in this way a ^{∗}-homomorphism

M_{n}(C_{0}((0, 1])) = M_{n}(ℂ) ⊗ C_{0}((0, 1]) → B ⊗ C_{0}(sp(a_{0}) \ \{0\}) → \overline{a_{0}Aa_{0}},

whose image generates an ideal of A which contains (a − ε)_{+}.

Suppose finally that A is purely infinite. Let a be a positive element in A and let ε > 0 and n in ℕ be given. By Lemma 2.3 there is d in M_{1,n}(A) such that d^{∗}ad = (a − ε)_{+} ⊗ 1_{n}. Write d = (d_{1}, d_{2}, …, d_{n}). Then d_{i}^{∗}ad_{j} = δ_{ij}(a − ε)_{+} for all i, j. Put t_{j} = a^{1/2}d_{j}d_{j}^{∗}a^{1/2}. Then t_{1}, …, t_{n} are mutually orthogonal elements in \overline{aAa} each equivalent to (a − ε)_{+}, and this shows that A has the global Glimm property. □

Proposition 4.15 A weakly purely infinite is purely infinite if and only if it has the global Glimm property.

Proof: The “only if” part follows from Proposition 4.14. Assume next that A is weakly purely infinite, say pi-n, with the global Glimm property. Let a be a non-zero positive element in A and let ε > 0 be given. Then \overline{aAa} contains pairwise orthogonal and equivalent positive elements t_{1}, t_{2}, …, t_{n} such that (a − ε)_{+} belongs to the closed two-sided ideal generated by t = t_{1} + t_{2} + ⋯ + t_{n}. Now,

t_{1} ⊗ 1_{n} ∼ t_{1} ⊕ t_{2} ⊕ ⋯ ⊕ t_{n} ∼ t_{1} + t_{2} + ⋯ + t_{n} = t.

Hence t is properly infinite. Since (a − ε)_{+} belongs to the ideal generated by t we conclude that (a − ε)_{+} ≾ t. Conversely, t ≾ a because t belongs to \overline{aAa}, cf. [16, Proposition 2.7 (i)]. Using again that t is properly infinite, we get

(a − ε)_{+} ⊕ (a − ε)_{+} ≾ t ⊕ t ≾ t ≾ a.

Since this holds for all ε > 0 we conclude that a is properly infinite. This shows that A is purely infinite because a was arbitrary. □

Every hereditary sub-C^{∗}-algebra and every quotient of a weakly purely infinite is again weakly purely infinite, and no weakly purely infinite C^{∗}-algebra can be finite dimensional (e.g. by Theorem 4.8). Hence no hereditary sub-C^{∗}-algebra of a weakly purely infinite C^{∗}-algebra admits a finite dimensional representation. Together with Proposition 4.14 and Lemma 4.15 this proves:

Corollary 4.16 Any weakly purely infinite C^{∗}-algebra, which is either simple or approximately divisible, is purely infinite.

A C^{∗}-algebra A is said to have property (SP) (“small projections”) if every non-zero hereditary sub-C^{∗}-algebra of A contains a non-zero projection.

Proposition 4.17 Each non-zero projection in a weakly purely infinite C^{∗}-algebra with property (SP) is infinite.

Proof: Let A be a weakly purely infinite C^{∗}-algebra with property (SP), and let p be a non-zero projection in A. Upon replacing A with pAp (which again is weakly purely infinite by Proposition 4.5 (ii) and has property (SP)), it suffices to show that the unit is infinite in a unital weakly purely infinite C^{∗}-algebra A with property (SP).

Since A is weakly purely infinite it admits no finite dimensional representations. Hence Glimm’s lemma applies (cf. [16, Proposition 4.10]), and there is a non-zero ^{∗}-homomor-phism from M_{n}(C_{0}((0, 1])) into A. It follows that A contains mutually orthogonal positive elements t_{1}, t_{2}, …, t_{n} and elements x_{2}, …, x_{n} such that x_{j}x_{j}^{∗} = t_{j} and x_{j}^{∗}x_{j} = t_{1}. Find a non-zero projection p_{1} in \overline{t_{1}At_{1}}. Write x_{j} = v_{j}|x_{j}| for some partial isometry v_{j} in A^{∗∗}. Then u_{j} = v_{j}p_{1} belongs to A, u_{j}^{∗}u_{j} = p_{1}, and p_{j} = u_{j}u_{j}^{∗} belongs to \overline{t_{j}At_{j}}. In particular, p_{1}, p_{2}, …, p_{n} are mutually orthogonal and mutually equivalent projections. Consequently,

p = p_{1} + p_{2} + ⋯ + p_{n} ∼ p_{1} ⊕ p_{2} ⊕ ⋯ ⊕ p_{n} ∼ p_{1} ⊗ 1_{n}

is properly infinite. As A contains a non-zero properly infinite projection, the unit of A must be infinite. □

Proposition 4.18 Every weakly purely infinite C^{∗}-algebra of real rank zero is purely infinite.

Proof: We check that if B is a non-zero hereditary sub-C^{∗}-algebra of a quotient of a weakly purely infinite C^{∗}-algebra A, then B contains an infinite projection. By [16, Proposition 4.7], this will ensure that A is purely infinite. By Proposition 4.5 (ii), B is weakly purely infinite. Any C^{∗}-algebra of real rank zero has property (SP). Proposition 4.17 therefore yields that every non-zero projection in B is infinite. □

One can relax the real rank zero condition in Proposition 4.18 to the weaker condition that every quotient of the C^{∗}-algebra has property (SP).

We conclude this section with further reformulations of the open problem, if weak pure infiniteness implies pure infiniteness:

Proposition 4.19 The following four conditions are equivalent:

(i) all weakly purely infinite C^{∗}-algebras are purely infinite,

(ii) the (unit of the) multiplier algebra of any σ-unital weakly purely infinite C^{∗}-algebra is properly infinite,

(iii) all non-zero projections in any weakly purely infinite C^{∗}-algebra are properly infinite,

(iv) all non-zero projections in any weakly purely infinite C^{∗}-algebra are infinite.

Proof: All non-zero projections in a purely infinite C^{∗}-algebra are properly infinite by [16, Theorem 4.16], and therefore (i) ⇒ (ii) and (i) ⇒ (iii).

(iv) ⇒ (iii). Suppose that p is a non-zero, not properly infinite projection in a weakly purely infinite A. Then there is a closed two-sided ideal I of A such that p+I is a non-zero finite projection in A/I, cf. [16, Corollary 3.15]. But A/I is weakly purely infinite by Lemma 4.5 (iii), thus contradicting (iv).

(ii) ⇒ (i). If M(\overline{aAa}) is properly infinite, then there is a full (possibly non-unital) embedding of M_{n}(ℂ) into M(\overline{aAa}) for every natural number n. Lemma 4.13 therefore implies that A has the global Glimm property, and by Proposition 4.15 we conclude that A is purely infinite if A is weakly purely infinite. □

5 Strongly purely infinite C^{∗}-algebras

Our third notion of pure infiniteness is perhaps not very intuitive, but the definition is still local and algebraic like the definitions of being purely infinite and weakly purely infinite. It turns out that this notion is precisely what is required to obtain an O_{∞}-absorbsion theorem (Theorem 8.6) and a Weyl–von Neumann type result such as Theorem 7.21.

Definition 5.1 A C^{∗}-algebra A is said to have be strongly purely infinite if for every

(\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) ∈ M_{2}(A)^{+}

and for every ε > 0 there exist d_{1}, d_{2} in A such that

∥ (\begin{matrix}d_{1}^{∗} & 0 \\ 0 & d_{2}^{∗} \end{matrix}) (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) (\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}) − (\begin{matrix}a & 0 \\ 0 & b \end{matrix}) ∥ ≤ ε.

The matrix diagonalization appearing in this definition can be rephrased in a number of ways (see also Remark 5.10 below):

Lemma 5.2 For each element ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) in M_{2}(A)^{+}, the following conditions are equivalent:

(i) For each ε > 0 there exist d_{1}, d_{2} in M(A) such that ∥d_{1}^{∗}ad_{1} −a∥ ≤ ε, ∥d_{2}^{∗}bd_{2} −b∥ ≤ ε, and ∥d_{2}^{∗}xd_{1}∥ ≤ ε.

(ii) For each ε > 0 there exist d_{1}, d_{2} in A such that

∥ (\begin{matrix}d_{1}^{∗} & 0 \\ 0 & d_{2}^{∗} \end{matrix}) (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) (\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}) − (\begin{matrix}a & 0 \\ 0 & b \end{matrix}) ∥ ≤ ε.

(iii) For each ε > 0 and δ > 0 there exist d_{1} in \overline{aAa} and d_{2} in \overline{bAb} such that d_{1}^{∗}ad_{1} = (a − ε)_{+}, d_{2}^{∗}bd_{2} = (b − ε)_{+}, and ∥d_{2}^{∗}xd_{1}∥ ≤ δ.

Proof: The implications (iii) ⇒ (ii) ⇒ (i) are obvious. Assume that (i) holds. Let ε > 0 and δ > 0 be given, and find e_{1}, e_{2} in M(A) such that ∥e_{1}^{∗}ae_{1} − a∥ ≤ ε/2, ∥e_{2}^{∗}be_{2} − b∥ ≤ ε/2, and ∥e_{2}^{∗}xe_{1}∥ ≤ δ/2. Set f_{j} = g_{j}^{1/2}e_{j}g_{j}^{1/2}, j = 1, 2, where g_{1} and g_{2} are positive contractions in \overline{aAa}, respectively \overline{bAb}, chosen such that ∥f_{1}^{∗}af_{1} − a∥ < ε, ∥f_{2}^{∗}bf_{2} − b∥ < ε, and ∥f_{2}^{∗}xf_{1}∥ ≤ δ. By Lemma 2.2 there are contractions h_{1} and h_{2} in \overline{aAa}, respectively \overline{bAb}, such that h_{1}^{∗}f_{1}^{∗}af_{1}h_{1} = (a − ε)_{+} and h_{2}^{∗}f_{2}^{∗}bf_{2}h_{2} = (b − ε)_{+}. We can therefore take d_{j} to be f_{j}h_{j}, j = 1, 2. □

In a strongly purely infinite (or in a purely infinite) C^{∗}-algebra A, if a = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) is a positive element in M_{2}(A), then diag(a_{11}, a_{22}) ≾ a. The converse holds in any C^{∗}-algebra:

Lemma 5.3 Let A be any C^{∗}-algebra, let n be a positive integer, and let a = (a_{ij}) be a positive element in M_{n}(A). Then a ≾ diag(a_{11}, …, a_{nn}).

Proof: This follows from the fact that a belongs to the hereditary sub-C^{∗}-algebra generated by diag(a_{11}, …, a_{nn}). □

Proposition 5.4 Every strongly purely infinite C^{∗}-algebra is purely infinite.

Proof: If A is strongly purely infinite and if a is a positive element in A, then

a ⊕ a = (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) ≾ (\begin{matrix}a & a \\ a & a \end{matrix}) = rr^{∗} ∼ r^{∗}r = 2a ≈ a,

cf. Definition 2.1, where r is the column matrix in M_{2,1}(A) with both entries equal to a^{1/2}. □

The following definition is convenient for the formulation of some of our next lemmas.

Definition 5.5 (The matrix diagonalization property) An n-tuple (a_{1}, …, a_{n}) in a C^{∗}-algebra A is said to have the matrix diagonalization property if for every positive matrix a = (a_{ij}) in M_{n}(A) with a_{jj} = a_{j}, and for every ε_{1} > 0, …, ε_{n} > 0 and every δ > 0 there are elements d_{1}, …, d_{n} in A with

d_{j}^{∗}a_{jj}d_{j} = (a_{jj} − ε_{j})_{+},  ∥d_{i}^{∗}a_{ij}d_{j}∥ ≤ δ for i ≠ j.  (5.1)

Lemma 5.6 Let a_{1}, …, a_{n} be positive elements in a C^{∗}-algebra A, let ε_{1} > 0, …, ε_{n} > 0, and suppose that the n-tuple ((a_{1} − rε_{1})_{+}, …, (a_{n} − rε_{n})_{+}) has the matrix diagonalization property for some r in (1/2, 1). Then for each positive matrix a = (a_{ij}) in M_{n}(A), with a_{jj} = a_{j}, and for each δ > 0 there are elements d_{1}, …, d_{n} in A such that (5.1) holds and such that ∥d_{j}∥^{2} ≤ 2ε_{j}^{−1}∥a_{j}∥.

Proof: Let f_{ε}: ℝ^{+} → [0, 1] be a continuous function satisfying f_{ε}(t)^{2} = (t − rε)/t for t ≥ rε, and f_{ε}(t) = 0 for t ≤ ε/2. Then

(a − rε)_{+} = f_{ε}(a)^{2}a ≥ \frac{ε}{2}f_{ε}(a)^{2}

for all positive a in A. Put b_{ij} = f_{ε_{i}}(a_{i})a_{ij}f_{ε_{j}}(a_{j}). Then b = (b_{ij}) is a positive matrix in M_{n}(A) and b_{jj} = (a_{j} − rε_{j})_{+}. By assumption, and because rε_{j} < ε_{j}, there are elements e_{1}, …, e_{n} in A such that e_{j}^{∗}b_{jj}e_{j} = (a_{j} −ε_{j})_{+} and ∥e_{i}^{∗}c_{ij}e_{j}∥ ≤ δ for i ≠ j. Put d_{j} = f_{ε_{j}}(a_{j})e_{j}, so that d_{i}^{∗}a_{ij}d_{j} = e_{i}^{∗}b_{ij}e_{j}. Then (5.1) holds, and

∥d_{j}∥^{2} = ∥e_{j}^{∗}f_{ε_{j}}(a_{j})^{2}e_{j}∥ ≤ \frac{2}{ε_{j}} ∥e_{j}^{∗}f_{ε_{j}}(a_{j})a_{j}f_{ε_{j}}(a_{j})e_{j}∥ = \frac{2}{ε_{j}} ∥(a_{j} − ε_{j})_{+}∥ ≤ \frac{2}{ε_{j}} ∥a_{j}∥

as desired. □

The norm estimate on the elements d_{j} in the next lemma will be improved later in Corollary 7.22, where it is shown that we can take d_{1}, …, d_{n} to be contractions.

Lemma 5.7 Let a_{1}, …, a_{n} be positive elements in a C^{∗}-algebras A, and suppose that each pair ((a_{i} − η_{i})_{+}, (a_{j} − η_{j})_{+}), i ≠ j, has the matrix diagonalization property for every choice of η_{k} ≥ 0. Then the n-tuple (a_{1}, …, a_{n}) has the matrix diagonalization property.

Proof: The lemma is proved by induction on n. For n = 2 there is nothing to prove. Suppose that n ≥ 3 and that the lemma has been verified for n − 1. Let a = (a_{ij}) be a positive matrix in M_{n}(A) with a_{jj} = a_{j}, and let ε_{j} > 0 and δ > 0 be given. We find d_{1}, …, d_{n} in A with d_{j}^{∗}a_{jj}d_{j} = (a_{jj} − ε_{j})_{+} and ∥d_{i}^{∗}a_{ij}d_{i}∥ ≤ δ for i ≠ j. The proof has three steps: First we diagonalize the lower n − 1 by n − 1 sub-matrix of a, then we diagonalize the resulting upper n − 1 by n − 1 sub-matrix, and at the end we take care of the entries at the positions (1, n) and (n, 1).

Take δ_{0} > 0 and δ_{1} > 0 (to be determined later). By the induction hypothesis there are elements f_{2}, …, f_{n} in A such that f_{j}^{∗}a_{jj}f_{j} = (a_{jj} − ε_{j}/2)_{+} and ∥f_{i}^{∗}a_{ij}f_{j}∥ ≤ δ_{0} for i ≠ j. Set f_{1} = 1 and put b_{ij} = f_{i}^{∗}a_{ij}f_{j}, making b = (b_{ij}) a positive matrix in M_{n}(A). Use the induction hypothesis and Lemma 5.6 to find elements g_{1}, …, g_{n−1} in A such that

∥g_{j}∥ ≤ 2ε_{j}^{−1}∥a_{jj}∥,  g_{j}^{∗}b_{jj}g_{j} = (b_{jj} − ε_{j}/2)_{+},  ∥g_{i}^{∗}b_{ij}g_{j}∥ ≤ δ_{1} for i ≠ j.

Set g_{n} = 1 and set c_{ij} = g_{i}^{∗}b_{ij}g_{j}. Then

c_{jj} = \begin{cases}(a_{jj} − ε_{j}/2)_{+}, & j = 1, j = n, \\ (a_{jj} − ε_{j})_{+}, & 2 ≤ j ≤ n − 1, \end{cases} ∥c_{ij}∥ ≤ \begin{cases}δ_{1}, & i ≠ j, 1 ≤ i, j ≤ n − 1, \\ ∥g_{j}∥δ_{0}, & i = n, 2 ≤ j ≤ n − 1 \\ ∥g_{i}∥δ_{0}, & j = n, 2 ≤ i ≤ n − 1 \end{cases}

Use finally the assumption (and Lemma 5.6) to find h_{1} and h_{n} in A with ∥h_{j}∥ ≤ 2ε_{j}^{−1}∥a_{jj}∥ and

∥h_{1}^{∗}c_{1n}h_{n}∥ = ∥h_{n}^{∗}c_{n1}h_{1}∥ ≤ δ, h_{j}^{∗}c_{jj}h_{j} = (c_{jj} − ε_{j}/2)_{+} = (a_{jj} − ε_{j})_{+} for j = 1, n,

and put h_{j} = 1 for j = 2, …, n. Then d_{j} = f_{j}g_{j}h_{j} satisfies d_{j}^{∗}a_{jj}d_{j} = (a_{jj} − ε_{j})_{+}, and if δ_{0} and δ_{1} have been chosen small enough, then also ∥d_{i}^{∗}a_{ij}d_{j}∥ ≤ δ whenever i ≠ j. □

As each pair of positive elements in a strongly purely infinite has the matrix diagonalization property, Lemma 5.7 and Lemma 5.6 imply:

Lemma 5.8 Let A be a strongly purely infinite C^{∗}-algebra. Then for each positive matrix a = (a_{ij}) in M_{n}(A), for each choice of ε_{j} > 0, j = 1, …, n, and for each δ > 0 there are elements d_{1}, …, d_{n} in A such that

d_{j}^{∗}a_{jj}d_{j} = (a_{jj} − ε_{j})_{+},  ∥d_{i}^{∗}a_{ij}d_{j}∥ ≤ δ for i ≠ j,  ∥d_{j}∥^{2} ≤ 2ε_{j}^{−1}∥a_{jj}∥.  (5.2)

Lemma 5.9 Let A be a C^{∗}-algebra, and let a_{1}, a_{2}, …, a_{n} and b_{1}, b_{2}, …, b_{m} be two families of positive elements in A such that the (n + m)-tuple (a_{1}, …, a_{n}, b_{1}, …, b_{m}) has the matrix diagonalization property. Then the pair ( ∑_{i=1}^{n} a_{i}, ∑_{j=1}^{m} b_{j}) has the matrix diagonalization property.

Proof: We may assume that m = n (otherwise take for example b_{m+1} = ⋯ = b_{n} = 0). Put a = ∑_{i=1}^{n} a_{i} and b = ∑_{i=1}^{n} b_{i}, and let x in A be such that ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) is a positive matrix in M_{2}(A). Let ε > 0 be given. Put

s = (a_{1}^{1/2} ; ⋮ ; a_{n}^{1/2}) , t = (b_{1}^{1/2} ; ⋮ ; b_{n}^{1/2}) .

Then s^{∗}s = a and t^{∗}t = b. Write s = ua^{1/2} and t = vb^{1/2} for some partial isometries u, v in M_{n,1}(A^{∗∗}). Observe that uc, respectively, vc, belongs to M_{n,1}(A) for every c in \overline{aAa}, respectively, in \overline{bAb}. Also,

uau^{∗} = ss^{∗} = (\begin{matrix}a_{1} & ⋯ & a_{1}^{1/2}a_{n}^{1/2} \\ ⋮ & & ⋮ \\ a_{n}^{1/2}a_{1}^{1/2} & ⋯ & a_{n} \end{matrix}) , vbv^{∗} = tt^{∗} = (\begin{matrix}b_{1} & ⋯ & b_{1}^{1/2}b_{n}^{1/2} \\ ⋮ & & ⋮ \\ b_{n}^{1/2}b_{1}^{1/2} & ⋯ & b_{n} \end{matrix}) .

It follows that

(\begin{matrix}uau^{∗} & ux^{∗}v^{∗} \\ vxu^{∗} & vbv^{∗} \end{matrix}) ∈ M_{2n}(A)^{+},

and that the diagonal entries of this matrix are (a_{1}, …, a_{n}, b_{1}, …, b_{n}), and this 2n-tuple has the matrix diagonalization property by assumption. Arguing as in the proof of Proposition 5.11 (iv) we find matrices f_{1}, f_{2} in M_{n}(A) such that

∥f_{1}^{∗}uau^{∗}f_{1} − uau^{∗}∥ < ε,  ∥f_{2}^{∗}vbv^{∗}f_{2} − vbv^{∗}∥ < ε,  ∥f_{2}^{∗}vxu^{∗}f_{1}∥ < ε.

Replacing f_{j} by e_{j}f_{j}e_{j} where e_{1} and e_{2} are suitable (approximate units) in hereditary sub-C^{∗}-algebras generated by uau^{∗}, respectively, vbv^{∗}, we can assume that f_{1} and f_{2} belong to these respective hereditary sub-C^{∗}-algebras. This will ensure that d_{1} = u^{∗}f_{1}u and d_{2} = v^{∗}f_{2}v belong to A. Finally,

∥d_{1}^{∗}ad_{1} − a∥ = ∥ud_{1}^{∗}ad_{1}u^{∗} − uau^{∗}∥ = ∥f_{1}^{∗}uau^{∗}f_{1} − uau^{∗}∥ < ε,
∥d_{2}^{∗}bd_{2} − b∥ = ∥vd_{2}^{∗}bd_{2}v^{∗} − vbv^{∗}∥ = ∥f_{2}^{∗}vbv^{∗}f_{2} − vbv^{∗}∥ < ε,
∥d_{2}^{∗}xd_{1}∥ = ∥vd_{2}^{∗}xd_{1}u^{∗}∥ = ∥f_{2}^{∗}vxu^{∗}f_{1}∥ < ε,

as desired. □

Remark 5.10 (Matrix diagonalization revisited) A C^{∗}-algebra A is strongly purely infinite if and only if for each pair of positive element a, b in A and for each ε > 0 there are elements d_{1}, d_{2} in A such that

∥d_{1}^{∗}ad_{1} − a∥ ≤ ε,  ∥d_{2}^{∗}bd_{2} − b∥ ≤ ε,  ∥d_{2}^{∗}b^{1/2}a^{1/2}d_{1}∥ ≤ ε.  (5.3)

To see this, note first that

(\begin{matrix}a & a^{1/2}b^{1/2} \\ b^{1/2}a^{1/2} & b \end{matrix}) = (\begin{matrix}a^{1/2} & 0 \\ 0 & b^{1/2} \end{matrix}) (\begin{matrix}1 & 1 \\ 1 & 1 \end{matrix}) (\begin{matrix}a^{1/2} & 0 \\ 0 & b^{1/2} \end{matrix}) ∈ M_{2}(A)^{+}.

The “only if” part of the claim now follows immediately from Definition 5.1.

To prove the “if” part it suffices to show that ((a − η)_{+}, (b − η)_{+}) has the matrix diagonalization property for every pair of positive elements a, b in A and for every η > 0, cf. Lemma 5.6. Let x in A be such that ( \begin{matrix}(a−η)_{+} & x^{∗} \\ x & (b−η)_{+} \end{matrix} ) is positive. Put c = a + b. It follows from (5.3), as in the proof of Proposition 5.4, that A must be strongly purely infinite. Hence, for some δ > 0,

(\begin{matrix}(a − η)_{+} & x^{∗} \\ x & (b − η)_{+} \end{matrix}) ≾_{0} (\begin{matrix}(a − η/2)_{+} & 0 \\ 0 & (b − η/2)_{+} \end{matrix}) ≾_{0} (\begin{matrix}(c − δ)_{+} & 0 \\ 0 & (c − δ)_{+} \end{matrix}) ≾_{0} c,

where e ≾_{0} f means that e = z^{∗}fz for some z in (a rectangular matrix over) A. We need therefore only show that each positive matrix in M_{2}(A) of the form y^{∗}cy, where y belongs to M_{1,2}(A) and c is a positive element in A, can be approximately matrix diagonalized.

Let ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) = y^{∗}cy be given, so that a = y_{1}^{∗}cy_{1}, b = y_{2}^{∗}cy_{2}, and x = y_{2}^{∗}cy_{1}. Write y = (y_{1}, y_{2}), write c^{1/2}y_{1} = ua^{1/2} and c^{1/2}y_{2} = vb^{1/2}, where u, v are partial isometries in A^{∗∗}, and put a_{0} = uau^{∗} and b_{0} = vbv^{∗}. Then a_{0} and b_{0} belong to A, and

(\begin{matrix}u^{∗} & 0 \\ 0 & v^{∗} \end{matrix}) (\begin{matrix}a_{0} & a_{0}^{1/2}b_{0}^{1/2} \\ b_{0}^{1/2}a_{0}^{1/2} & b_{0} \end{matrix}) (\begin{matrix}u & 0 \\ 0 & v \end{matrix}) = (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) .

Find e_{1}, e_{2} in A such that

∥e_{1}^{∗}a_{0}e_{1} − a_{0}∥ < ε,  ∥e_{2}^{∗}b_{0}e_{2} − b_{0}∥ < ε,  ∥e_{2}^{∗}b_{0}^{1/2}a_{0}^{1/2}e_{1}∥ < ε.

Upon replacing e_{1} by ge_{1}g for a suitable positive contraction in the hereditary sub-C^{∗}-al-gebra \overline{a_{0}Aa_{0}} we may assume that e_{1} belongs to this sub-C^{∗}-algebra. Similarly, we may assume that e_{2} belongs to \overline{b_{0}Ab_{0}}. Then d_{1} = ue_{1} and d_{2} = ve_{2} belong to A, and

∥d_{1}^{∗}ad_{1} − a∥ < ε,  ∥d_{2}^{∗}bd_{2} − b∥ < ε,  ∥d_{2}^{∗}xd_{1}∥ < ε.

Proposition 5.11 (Permanence properties)

(i) If A is strongly purely infinite, then so is every non-zero quotient of A.

(ii) If A is strongly purely infinite, then so are all its non-zero hereditary sub-C^{∗}-algebras.

(iii) If A and B are stably isomorphic and if A is strongly purely infinite, then so is B.

(iv) Any inductive limit of a system of strongly purely infinite C^{∗}-algebras is again strongly purely infinite.

Proof: (i). We must show that A/I is strongly purely infinite whenever I is a closed two-sided ideal in A. To see this take a positive element b in M_{2}(A/I). Lift b to a positive element a in M_{2}(A). Find d_{1}, d_{2} in A that approximately matrix diagonalize a, cf. Definition 5.1. The image under the quotient mapping A → A/I of d_{1}, d_{2} will then approximately matrix diagonalize b.

(ii). This follows from Lemma 5.2.

(iv). In the light of (i) it suffices to show that if A is a C^{∗}-algebra with an upper directed family {A_{i}}_{i∈𝕀} of strongly purely infinite sub-C^{∗}-algebras A_{i} such that ⋃_{i∈𝕀} A_{i} is dense in A, then A is strongly purely infinite.

We must show that to each positive matrix ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) in M_{2}(A) and to each ε > 0, there are d_{1}, d_{2} in A such that ∥d_{1}^{∗}ad_{1} − a∥ ≤ ε, ∥d_{2}^{∗}bd_{2} − b∥ ≤ ε, and ∥d_{2}^{∗}xd_{1}∥ ≤ ε. It is no loss of generality to assume that the given positive matrix is a contraction.

Choose δ_{1} > 0 and δ_{2} > 0 such that δ_{2} ≤ ε/2 and (2δ_{2}^{−1} + 1)δ_{1} ≤ ε/2. Find i in 𝕀 and a positive element ( \begin{matrix}a_{0} & x_{0}^{∗} \\ x_{0} & b_{0} \end{matrix} ) in M_{2}(A_{i}) such that

∥a − a_{0}∥ < δ_{1},  ∥b − b_{0}∥ < δ_{1},  ∥x − x_{0}∥ < δ_{1}

Find next d_{1}, d_{2} in A_{i} with ∥d_{i}∥^{2} ≤ 2δ_{2}^{−1} and

d_{1}^{∗}a_{0}d_{1} = (a_{0} − δ_{2})_{+},  d_{2}^{∗}b_{0}d_{2} = (b_{0} − δ_{2})_{+},  ∥d_{2}^{∗}x_{0}d_{1}∥ < δ_{2},

cf. Lemma 5.6. Then

∥d_{1}^{∗}ad_{1} − a∥ ≤ ∥d_{1}∥^{2}∥a − a_{0}∥ + ∥a − a_{0}∥ + δ_{2} ≤ (2δ_{2}^{−1} + 1)δ_{1} + δ_{2} ≤ ε,

and, similarly, ∥d_{2}^{∗}bd_{2} − b∥ < ε. Also,

∥d_{2}^{∗}xd_{1}∥ ≤ ∥d_{1}∥∥d_{2}∥∥x − x_{0}∥ + ∥d_{2}^{∗}x_{0}d_{1}∥ ≤ 2δ_{2}^{−1}δ_{1} + δ_{2} ≤ ε.

This shows that A is strongly purely infinite.

(iii). By (i) it suffices to show that A ⊗ K is strongly purely infinite when A is strongly purely infinite, and using (iv) it suffices to show that M_{n}(A) is strongly purely infinite for all natural numbers n when A is strongly purely infinite. Let ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) be a positive element in M_{2}(M_{n}(A)) and let ε > 0. Then a and b are positive elements of M_{n}(A). Denote by â and b̂ the diagonal parts of a and b, i.e., â is the diagonal matrix in M_{n}(A) whose diagonal entries are equal to the diagonal entries of a, and similarly for b̂. Then diag(â, b̂) is the diagonal part of ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}). As remarked in Lemma 5.3, a ≾ â and b ≾ b̂. Hence there is δ > 0 and e_{1}, e_{2} in M_{n}(A) such that e_{1}^{∗}(â − δ)_{+}e_{1} = (a − ε)_{+} and e_{2}^{∗}(b̂ − δ)_{+}e_{2} = (b − ε)_{+}.

From Lemma 5.8 we can find (diagonal) matrices f_{1}, f_{2} in M_{n}(A) such that

f_{1}^{∗}af_{1} = (â − δ)_{+},  f_{2}^{∗}bf_{2} = (b̂ − δ)_{+},  ∥f_{2}^{∗}xf_{1}∥ ≤ ∥e_{1}∥^{−1}∥e_{2}∥^{−1}ε.

Put d_{1} = f_{1}e_{1} and d_{2} = f_{2}e_{2}. Then d_{1}^{∗}ad_{1} = (a − ε)_{+}, d_{2}^{∗}bd_{2} = (b − ε)_{+}, and ∥d_{2}^{∗}xd_{1}∥ ≤ ∥e_{1}∥∥e_{2}∥∥f_{2}^{∗}xf_{1}∥ ≤ ε. This shows that M_{n}(A) is strongly purely infinite. □

Proposition 5.12 The following conditions are equivalent for every C^{∗}-algebra A:

(i) A is strongly purely infinite.

(ii) ℓ^{∞}(A) is strongly purely infinite.

(iii) A_{ω} is strongly purely infinite for every filter ω on ℕ.

(iv) A_{ω} is strongly purely infinite for some filter ω on ℕ.

Proof: (i) ⇒ (ii). Assume that A is strongly purely infinite. To show that ℓ^{∞}(A) is strongly purely infinite take a positive matrix ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) in ℓ^{∞}(A). Upon scaling this matrix, we can assume that a and b are contractions. Write

a = (a_{1}, a_{2}, …),  b = (b_{1}, b_{2}, …),  x = (x_{1}, x_{2}, …),

where \{a_{n}\}, \{b_{n}\}, and \{x_{n}\} are sequences of contractions in A and ( \begin{matrix}a_{n} & x_{n}^{∗} \\ x_{n} & b_{n} \end{matrix} ) is positive in M_{2}(A) for each n. Let ε > 0 be given. Use Lemma 5.6 to find elements d_{1,n}, d_{2,n} in A with

d_{1,n}^{∗}a_{n}d_{1,n} = (a_{n} − ε)_{+}, d_{2,n}^{∗}b_{n}d_{2,n} = (b_{n} − ε)_{+}, ∥d_{2,n}^{∗}x_{n}d_{1,n}∥ ≤ ε, ∥d_{j,n}∥^{2} ≤ 2ε^{−1}.

Then d_{j} = (d_{j,1}, d_{j,2}, …) belongs to ℓ^{∞}(A) for j = 1, 2, and

d_{1}^{∗}ad_{1} = (a − ε)_{+},  d_{2}^{∗}bd_{2} = (b − ε)_{+},  ∥d_{2}^{∗}xd_{1}∥ ≤ ε.

This shows that ℓ^{∞}(A) is strongly purely infinite.

(ii) ⇒ (iii). This follows from Proposition 5.11 (i).

(iii) ⇒ (iv). Trivial!

(iv) ⇒ (i). Assume that A_{ω} is strongly purely infinite for some free filter ω on ℕ. Let ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) be a positive element in M_{2}(A) and let ε > 0 be given. Let π_{ω}: ℓ^{∞}(A) → A_{ω} denote the quotient mapping. Identify x in A with π_{ω}(x, x, …) (thus viewing A as a sub-C^{∗}-alge-bra of A_{ω}). Then ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) is a positive element in M_{2}(A_{ω}). We can therefore find d_{1}, d_{2} in A_{ω} such that ∥d_{1}^{∗}ad_{1} − a∥ < ε, ∥d_{2}^{∗}bd_{2} − b∥ < ε, and ∥d_{2}^{∗}xd_{2}∥ < ε. Write d_{j} = π_{ω}(d_{j,1}, d_{j,2}, …). Then

lim sup_{ω} ∥d_{1,n}^{∗}ad_{1,n} − a∥ < ε,  lim sup_{ω} ∥d_{2,n}^{∗}bd_{2,n} − b∥ < ε,  lim sup_{ω} ∥d_{2,n}^{∗}xd_{1,n}∥ < ε.

Hence ∥d_{1,n}^{∗}ad_{1,n} − a∥ < ε, ∥d_{2,n}^{∗}bd_{2,n} − b∥ < ε, and ∥d_{2,n}^{∗}xd_{1,n}∥ < ε for each n in some subset belonging to ω, and hence for at least one n. This completes the proof. □

If ω is a free filter on ℕ and if A is a strongly purely infinite C^{∗}-algebra, then A_{ω} is strongly purely infinite. Combining Lemma 5.6 and Lemma 2.5 we get:

Lemma 5.13 In the notation above, Lemma 5.8 holds with δ = 0 when A is replaced with the C^{∗}-algebra A_{ω}.

Proposition 5.14 Every approximately divisible purely infinite C^{∗}-algebra is strongly purely infinite.

Proof: Let A be an approximately divisible purely infinite C^{∗}-algebra, and let ω be a free filter on ℕ. We show that A_{ω} is strongly purely infinite, and it will then follow from Proposition 5.12 that A is strongly purely infinite.

Let T = ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) be a positive matrix in M_{2}(A_{ω}). Lift T to a positive matrix (T_{1}, T_{2}, …) in M_{2}(ℓ^{∞}(A)) and write T_{n} = ( \begin{matrix}a_{n} & x_{n}^{∗} \\ x_{n} & b_{n} \end{matrix} ). With E = M_{2}(ℂ) ⊕ M_{3}(ℂ) we can find unital ∗-homomorphisms φ_{n}: E → M(A) such that

lim_{n→∞} ∥φ_{n}(e)a_{n} − a_{n}φ_{n}(e)∥ = lim_{n→∞} ∥φ_{n}(e)b_{n} − b_{n}φ_{n}(e)∥ = lim_{n→∞} ∥φ_{n}(e)x_{n} − x_{n}φ_{n}(e)∥ = 0

for all e in E.

With π_{ω}: ℓ^{∞}(M(A)) → M(A)_{ω} the quotient mapping, define

φ: E → M(A)_{ω} ⊆ M(A_{ω}) by φ(e) = π_{ω}(φ_{1}(e), φ_{2}(e), …).

Then a, b, x commute with the image of φ.

Choose a full (non-unital) ^{∗}-homomorphism ι: M_{2}(ℂ) → E, let {e_{ij}} be the matrix units for M_{2}(ℂ), and put f_{ij} = ι(e_{ij}). There are elements e_{1}, e_{2}, e_{3} in E such that 1 = ∑_{j=1}^{3} e_{j}^{∗}f_{11}e_{j}. Hence

a = ∑_{j=1}^{3} φ(e_{j}^{∗})φ(f_{11})aφ(e_{j})

and so a belongs to the ideal generated by φ(f_{11})a (= φ(f_{11})aφ(f_{11})). Similarly, b belongs to the ideal generated by φ(f_{22})bφ(f_{22}). Let ε > 0 be given. Since A_{ω} is purely infinite, cf. Proposition 3.5, there are elements c_{1}, c_{2} in A_{ω} such that

c_{1}^{∗}φ(f_{11})aφ(f_{11})c_{1} = (a − ε)_{+}, c_{2}^{∗}φ(f_{22})bφ(f_{22})c_{2} = (b − ε)_{+}.

Put d_{1} = φ(f_{11})c_{1} and d_{2} = φ(f_{22})c_{2}. Then d_{2}^{∗}xd_{1} = 0 because x commutes with the image of φ, d_{1}^{∗}ad_{1} = (a − ε)_{+}, and d_{2}^{∗}bd_{2} = (b − ε)_{+}. Hence T can be matrix diagonalized, and this proves that A_{ω} is strongly purely infinite. □

6 Purely infinite C^{∗}-algebras of real rank zero

It is shown in this section that every purely infinite C^{∗}-algebra of real rank zero is strongly purely infinite.

Call an element a in a C^{∗}-algebra A locally central if it belongs to the center of the hereditary sub-C^{∗}-algebra \overline{aAa}. Every projection and every multiple of a projection are locally central. If a is locally central, then so is (a − ρ)_{+} for every ρ ≥ 0.

Definition 6.1 A C^{∗}-algebra A is said to have the locally central decomposition property if for every a in A^{+} and for every ε > 0 there exist e, f in A and locally central elements a_{1}, a_{2}, …, a_{n} in A^{+} such that

(i) e^{∗}ae = ∑_{j=1}^{n} a_{j},

(ii) f^{∗}(e^{∗}ae)f = (a − ε)_{+}.

Remark 6.2 A purely infinite C^{∗}-algebra A satisfies the locally central decomposition property if it has the following property: for every a in A^{+} and for every ε > 0 there exist locally central elements a_{1}, a_{2}, …, a_{n} in A^{+} such that

(i) a_{1}, …, a_{n} belong to \overline{AaA},

(ii) (a − ε)_{+} belongs to \overline{A(∑_{j=1}^{n} a_{j})A}.

To see that (i) and (ii) above imply the locally central decomposition property, take a in A^{+} and ε > 0. Applying the conditions above to (a − ε/3)_{+} and to ε/3, and using the assumption that A is purely infinite, we find locally central elements a_{1}, a_{2}, …, a_{n} in A^{+} such that

∑_{j=1}^{n} a_{j} ≾ (a − ε/3)_{+}, (a − 2ε/3)_{+} ≾ ∑_{j=1}^{n} a_{j}.

We can now use Lemma 2.4 to find e, f in A such that (i) and (ii) in Definition 6.1 hold.

Proposition 6.3 Every purely infinite C^{∗}-algebra of real rank zero has the locally central decomposition property.

Proof: Let a be a non-zero positive element in a purely infinite C^{∗}-algebra A of real rank zero. Since \overline{aAa} has an approximate unit consisting of projections, it contains for each ε > 0 a projection p such that (a − ε)_{+} belongs to the ideal generated by p. Hence the two conditions of Remark 6.2 are satisfied. □

It is shown in [4] that continuous field C^{∗}-algebras, whose fibers are purely infinite and simple, have the locally central decomposition property.

Lemma 6.4 Let A be a purely infinite C^{∗}-algebra. Let δ > 0, let x, f in A and e in \overline{x^{∗}Ax} be given such that x is a contraction and

∥f^{∗}xe∥ ≤ δ^{2}, (|x| − δ)_{+} ≤ |e|.

Let D be a hereditary sub-C^{∗}-algebra of A which contains x^{∗}x. Then for each positive element b in D and for each ε > 0 there exists d in D with d^{∗}d = (b − ε)_{+} and

∥f^{∗}xd∥ ≤ (2∥b∥∥f∥(1 + 5∥f∥))^{1/2}δ^{1/2}.

Proof: We may assume that δ < 1. Put b_{0} = (b − ε)_{+} and find positive contractions u_{1}, u_{2} in D such that u_{1}u_{2} = u_{2}u_{1} = u_{2} and u_{2}b_{0} = b_{0}u_{2} = b_{0}.

Let J be the closed two-sided ideal in A generated by (|e^{∗}| − δ)_{+} and let π_{J}: A → A/J be the quotient mapping. Then ∥π_{J}(e)∥ ≤ δ, and hence ∥π_{J}(|x|)∥ ≤ 2δ. Accordingly, we can find a positive contraction g in J such that ∥xu_{1}^{1/2}(1 − g)∥ ≤ \sqrt{5}δ.

Put a = u_{1}^{1/2}(1 − g)^{2}u_{1}^{1/2} + |e^{∗}|. Then a belongs to D and (u_{1} − δ)_{+} belongs to the closed two-sided ideal I in A generated by (a − δ)_{+}. To see the latter, observe first that ∥π_{I}(|e^{∗}|)∥ ≤ ∥π_{I}(a)∥ ≤ δ, which shows that (|e^{∗}| − δ)_{+} belongs to I. This entails that I contains J. Hence

π_{I}(u_{1}) = π_{I}(u_{1}^{1/2}(1 − g)^{2}u_{1}^{1/2}) ≤ π_{I}(a) ≤ δ,

whence (u_{1} − δ)_{+} belongs to I.

Since (a − δ)_{+} is properly infinite, we have (u_{1} − δ)_{+} ≾ (a − δ)_{+} (in A and hence in D), and u_{2} ≾ (u_{1} − δ)_{+}. By Lemma 2.4 (ii) we can find y in D such that ∥y∥^{2} ≤ 2/δ and y^{∗}ay = u_{2}. Put d = a^{1/2}yb_{0}^{1/2}. Then d belongs to D, d^{∗}d = b_{0}, and

∥f^{∗}xd∥^{2} = ∥f^{∗}xdd^{∗}x^{∗}f∥ ≤ ∥b_{0}∥∥y∥^{2}∥f^{∗}xax^{∗}f∥
≤ 2δ^{−1}∥b∥(∥f^{∗}xu^{1/2}(1 − g)^{2}u^{1/2}x^{∗}f∥ + ∥f^{∗}x|e^{∗}|x^{∗}f∥)
≤ 2δ^{−1}∥b∥(∥f∥^{2}∥xu^{1/2}(1 − g)∥^{2} + ∥f^{∗}x|e^{∗}|∥∥x^{∗}∥∥f∥)
≤ 2δ^{−1}∥b∥(5δ^{2}∥f∥^{2} + δ^{2}∥f∥)
= 2δ∥b∥∥f∥(5∥f∥ + 1).
□

Proposition 6.5 Let A be a purely infinite C^{∗}-algebra, let D_{1} and D_{2} be hereditary sub-C^{∗}-algebras of A, and let x be an element in A such that x^{∗}x ∈ D_{1} and xx^{∗} ∈ D_{2}. Let a be a positive element in D_{1}, let b be a positive element in D_{2}, and let ε_{1} > 0 and ε_{2} > 0 be given. It follows that for each δ > 0 there exist d_{1} ∈ D_{1} and d_{2} ∈ D_{2} such that d_{1}^{∗}d_{1} = (a − ε_{1})_{+}, d_{2}^{∗}d_{2} = (b − ε_{2})_{+}, and ∥d_{2}^{∗}xd_{1}∥ < δ.

Proof: Since δ > 0 is arbitrary, we may assume that a, b, and x are contractions. Choose η > 0 such that 12^{3}η ≤ δ^{4}. Using that A is purely infinite we can find h_{1}, h_{2} in \overline{x^{∗}Ax} such that

(\begin{matrix}(|x| − η)_{+}^{2} & 0 \\ 0 & (|x|^{1/2} − η)_{+}^{2} \end{matrix}) = (\begin{matrix}h_{1}^{∗} & 0 \\ h_{2}^{∗} & 0 \end{matrix}) (\begin{matrix}|x| & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}h_{1} & h_{2} \\ 0 & 0 \end{matrix}) ,

or, equivalently, such that

h_{1}^{∗}|x|h_{1} = (|x| − η)_{+}^{2},  h_{2}^{∗}|x|h_{2} = (|x|^{1/2} − η)_{+}^{2},  h_{1}^{∗}|x|h_{2} = 0.

Let x = v|x| = |x^{∗}|v be the polar decomposition for x, where v is a partial isometry in A^{∗∗}. Put x_{1} = v|x|^{1/2}, f_{1} = x_{1}h_{1}v^{∗}, and e_{1} = h_{2}. Then e_{1} belongs to \overline{x^{∗}Ax} = \overline{x_{1}^{∗}Ax_{1}}, f_{1}^{∗}x_{1}e_{1} = vh_{1}^{∗}x_{1}^{∗}x_{1}h_{2} = vh_{1}^{∗}|x|h_{2} = 0,

|e_{1}| = (h_{2}^{∗}h_{2})^{1/2} ≥ (h_{2}^{∗}|x|h_{2})^{1/2} = (|x|^{1/2} − η)_{+} = (|x_{1}| − η)_{+},

and f_{1} is a contraction. It follows from Lemma 6.4 that there exists d_{1} in D_{1} with d_{1}^{∗}d_{1} = (a − ε_{1})_{+} and ∥f_{1}^{∗}x_{1}d_{1}∥ ≤ (12η)^{1/2} (= η_{1}).

Put x_{2} = x^{∗}, f_{2} = d_{1}, and e_{2} = vh_{1}v^{∗}. Then

f_{1}^{∗}x_{1}d_{1} = vh_{1}x_{1}^{∗}x_{1}f_{2} = vh_{1}|x|f_{2}
= (f_{2}^{∗}|x|h_{1}^{∗}v^{∗})^{∗} = (f_{2}^{∗}x^{∗}vh_{1}v^{∗})^{∗} = (f_{2}^{∗}x_{2}e_{2})^{∗},

and so ∥f_{2}^{∗}x_{2}e_{2}∥ ≤ η_{1}. Moreover, e_{2} belongs to v(\overline{x^{∗}Ax})v^{∗} = \overline{xAx^{∗}} = \overline{x_{2}^{∗}Ax_{2}}, x_{2}^{∗}x_{2} = xx^{∗} belongs to D_{2}, and

|e_{2}| = v(h_{1}^{∗}h_{1})^{1/2}v^{∗} ≥ v(h_{1}^{∗}|x|h_{1})^{1/2}v^{∗} = v(|x| − η)_{+}v^{∗}
= (|x^{∗}| − η)_{+} = (|x_{2}| − η)_{+} ≤ (|x_{2}| − η_{1})_{+}.

Apply Lemma 6.4 once more to get an element d_{2} in D_{2} satisfying d_{2}^{∗}d_{2} = (b − ε_{2})_{+}, and

∥f_{2}^{∗}x_{2}d_{2}∥ ≤ (12η_{1}∥b∥)^{1/2} ≤ (12η_{1})^{1/2} ≤ (12(12η)^{1/2})^{1/2} ≤ δ.

This proves the lemma because ∥d_{2}^{∗}xd_{1}∥ = ∥d_{1}^{∗}x^{∗}d_{2}∥ = ∥f_{2}^{∗}x_{2}d_{2}∥. □

Remark 6.6 (Pure versus strong pure infiniteness) At a first glance it would seem that Proposition 6.5 proves that every purely infinite C^{∗}-algebra is strongly purely infinite. Here is what actually follows from this proposition: If a, b are positive elements in a purely infinite C^{∗}-algebra A and if x = b^{1/2}x_{0}a^{1/2}, where x_{0} belongs to \overline{(b − ρ)_{+}A(a − ρ)_{+}} for some ρ > 0, then for each ε > 0 and δ > 0 there are elements d_{1}, d_{2} in A such that d_{1}^{∗}ad_{1} = (a − ε)_{+}, d_{2}^{∗}bd_{2} = (b − ε)_{+}, and ∥d_{2}^{∗}xd_{1}∥ ≤ δ.

Indeed, assume as we may that ρ < ε, and put

a_{0} = (a − ρ)_{+}, b_{0} = (b − ρ)_{+}, D_{1} = \overline{a_{0}Aa_{0}}, D_{2} = \overline{b_{0}Ab_{0}},

so that x_{0}^{∗}x_{0} belongs to D_{1}, x_{0}x_{0}^{∗} belongs to D_{2}, (a − ε)_{+} = (a_{0} − (ε − ρ))_{+}, and (b − ε)_{+} = (b_{0} − (ε − ρ))_{+}. We can apply Proposition 6.5 to get e_{1} in D_{1} and e_{2} in D_{2} satisfying e_{1}^{∗}e_{1} = (a − ε)_{+}, e_{2}^{∗}e_{2} = (b − ε)_{+}, and ∥e_{2}^{∗}x_{0}e_{1}∥ ≤ δ. Now, find d_{1}, d_{2} in A such that e_{1} = a^{1/2}d_{1} and e_{2} = b^{1/2}d_{2}. Then d_{1}^{∗}ad_{1} = e_{1}^{∗}e_{1} = (a − ε)_{+}, d_{2}^{∗}bd_{2} = e_{2}^{∗}e_{2} = (b − ε)_{+}, and

∥d_{2}^{∗}xd_{1}∥ = ∥d_{2}^{∗}b^{1/2}x_{0}a^{1/2}d_{1}∥ = ∥e_{2}^{∗}x_{0}e_{1}∥ ≤ δ.

In general we cannot take x to be of the form b^{1/2}x_{0}a^{1/2}, with x_{0} as above, although, by Remark 5.10, to prove that A is strongly purely infinite, it would suffice to find d_{1}, d_{2} as above for x = b^{1/2}a^{1/2}.

The two next lemmas concern the matrix diagonalization property of n-tuples as in Definition 5.5 and locally central elements (defined above Definition 6.1).

Lemma 6.7 Each n-tuple (a_{1}, …, a_{n}) of locally central positive elements has the matrix diagonalization property.

Proof: By Lemma 5.7 it suffices to prove the lemma for n = 2. Recall that (a_{j} − η)_{+} is locally central for each η ≥ 0.

Let x in A be such that

(\begin{matrix}a_{1} & x^{∗} \\ x & a_{2} \end{matrix}) ∈ M_{2}(A)^{+}.

Choose a continuous function f: ℝ^{+} → [0, 1] satisfying that f(t) = 0 when t ≤ δ/2 and |f(t)t − t| < δ for all t ≥ 0. Put D_{j} = \overline{a_{j}Aa_{j}}, j = 1, 2, so that x^{∗}x belongs to D_{1} and xx^{∗} belongs to D_{2}. It follows from Proposition 6.5 that there exist d_{j} in D_{j}, j = 1, 2, satisfying d_{j}^{∗}d_{j} = f(a_{j}) and ∥d_{2}^{∗}xd_{1}∥ < δ. Notice that ∥d_{j}∥^{2} = ∥f(a_{j})∥ ≤ 1. Using that a_{1} and a_{2} are locally central, we get d_{j}^{∗}a_{j}d_{j} = d_{j}^{∗}d_{j}a_{j} = f(a_{j})a_{j}, and hence ∥d_{j}^{∗}a_{j}d_{j} − a_{j}∥ ≤ δ and ∥d_{2}^{∗}xd_{1}∥ ≤ δ. Since δ > 0 was arbitrary we can use Lemma 2.2 to conclude that (a_{1}, a_{2}) has the matrix diagonalization property. □

Theorem 6.8 Every purely infinite C^{∗}-algebra with the locally central decomposition property is strongly purely infinite.

Proof: Let A be a C^{∗}-algebra with the locally central decomposition property, and let

(\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) ∈ M_{2}(A)^{+}

and ε > 0 be given. By the assumption that A satisfies the locally central decomposition property and by Lemmas 6.7 and 5.9 there are elements e_{1}, f_{1}, e_{2}, f_{2} in A such that (e_{1}^{∗}ae_{1}, e_{2}^{∗}be_{2}) satisfies the matrix diagonalization property, and such that

∥f_{1}^{∗}(e_{1}^{∗}ae_{1})f_{1} − a∥ ≤ ε/2, ∥f_{2}^{∗}(e_{2}^{∗}be_{2})f_{2} − b∥ ≤ ε/2.

Now,

(\begin{matrix}e_{1}^{∗} & 0 \\ 0 & e_{2}^{∗} \end{matrix}) (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) (\begin{matrix}e_{1} & 0 \\ 0 & e_{2} \end{matrix}) = (\begin{matrix}e_{1}^{∗}ae_{1} & x_{1}^{∗} \\ x_{1} & e_{2}^{∗}be_{2} \end{matrix}) ,

where x_{1} = e_{2}^{∗}xe_{1}. Find g_{1}, g_{2} in A such that

∥g_{1}^{∗}(e_{1}^{∗}ae_{1})g_{1} −e_{1}^{∗}ae_{1}∥ ≤ \frac{ε}{2∥f_{1}∥^{2}} , ∥g_{2}^{∗}(e_{2}^{∗}be_{2})g_{2} −e_{2}^{∗}be_{2}∥ ≤ \frac{ε}{2∥f_{2}∥^{2}} , ∥g_{2}^{∗}x_{1}g_{1}∥ ≤ \frac{ε}{∥f_{1}∥∥f_{2}∥}.

Put d_{j} = e_{j}g_{j}f_{j} for j = 1, 2. Then

∥d_{1}^{∗}ad_{1} − a∥ = ∥f_{1}^{∗}g_{1}^{∗}(e_{1}^{∗}ae_{1})g_{1}f_{1} − a∥
≤ ∥f_{1}∥^{2}∥g_{1}^{∗}(e_{1}^{∗}ae_{1})g_{1} − e_{1}^{∗}ae_{1}∥ + ∥f_{1}^{∗}(e_{1}^{∗}ae_{1})f_{1} − a∥ ≤ ε,

and, similarly, ∥d_{2}^{∗}bd_{2} − b∥ ≤ ε. Finally,

∥d_{2}^{∗}xd_{1}∥ = ∥f_{2}^{∗}g_{2}^{∗}e_{2}^{∗}xe_{1}g_{1}f_{1}∥ = ∥f_{2}^{∗}g_{2}^{∗}x_{1}g_{1}f_{1}∥ ≤ ∥f_{1}∥∥f_{2}∥∥g_{2}^{∗}x_{1}g_{1}∥ ≤ ε,

as desired. □

Combining Theorem 6.8 with Proposition 6.3 yields:

Corollary 6.9 Every purely infinite C^{∗}-algebra of real rank zero is strongly purely infinite.

7 Approximately inner completely positive contraction on strongly purely infinite C^{∗}-algebras

The main result of this section says that all approximately inner completely positive contractions from a sub-C^{∗}-algebra of a nuclear strongly purely infinite A into A are approximately 1-step inner. This result will in Section 8 be used to showing that A is isomorphic to A ⊗ O_{∞}, if A is also separable. The section is divided into three, the last of which contains a digression that will not be used in the rest of the paper.

Some preliminary results

We begin by defining what it means for a completely positive mapping to be (approximately) inner:

Definition 7.1 (Inner and approximately inner maps) Let A and B be C^{∗}-algebras both contained in a C^{∗}-algebra E, and let T: B → A be a completely positive map. The map T is said to be n-step inner (relatively to E) if there are elements e_{1}, …, e_{n} in E such that

T (b) = ∑_{j=1}^{n} e_{j}^{∗}be_{j}

for all b in B. We say that T is inner if T is n-step inner for some natural number n.

If for each finite subset F of B and for each ε > 0 there is an n-step inner, respectively, an inner completely positive map S: B → A such that ∥T (b) − S(b)∥ ≤ ε for all b in F, then T is called approximately n-step inner, respectively, approximately inner.

The C^{∗}-algebra E in the definition above will usually be either A, the multiplier algebra of A, or a limit algebra, A_{ω}, of A corresponding to some free filter ω on ℕ. If E = A or if it is clear from the context which ambient C^{∗}-algebra we are considering, then we might omit the reference “relatively to E”. Lemma 7.3 below says that (approximate) innerness is not really dependent of the ambient C^{∗}-algebra. First we need the following lemma:

Lemma 7.2 Let T: B → A be a completely positive contraction that is approximately n-step inner relatively to a C^{∗}-algebra E containing both A and B. Then for each finite subset F of B and for each ε > 0 there are elements e_{1}, …, e_{n} in E such that

∥T (b) − ∑_{j=1}^{n} e_{j}^{∗}be_{j}∥ ≤ ε, b ∈ F, ∥ ∑_{j=1}^{n} e_{j}^{∗}e_{j}∥ ≤ 1.

Proof: Take an n-step inner completely positive contraction S with ∥T (b) − S(b)∥ ≤ ε/2 for all b in F. Write S(b) = ∑_{j=1}^{n} d_{j}^{∗}bd_{j}, where d_{j} belong to E. Find a positive contraction f in B such that ∥f^{1/2}bf^{1/2}−b∥ ≤ ε/2 for all b in F, and set e_{j} = f^{1/2}d_{j}. Then ∑_{j=1}^{n} e_{j}^{∗}e_{j} = S(f), and ∥S(f)∥ ≤ ∥f∥ ≤ 1. Also,

∑_{j=1}^{n} e_{j}^{∗}be_{j} − T (b) = (S(f^{1/2}bf^{1/2}) − S(b)) + (S(b) − T (b)), b ∈ B,

and this proves the lemma. □

Lemma 7.3 Let A and B be C^{∗}-algebras with B separable, and let T: B → A be a completely positive contraction.

(i) Suppose that B is a sub-C^{∗}-algebra of A_{ω} for some free filter ω on ℕ, and that T is approximately n-step inner relatively to A_{ω}. Then T is (exactly) n-step inner relatively to A_{ω} and T is approximately n-step inner relatively to A.

(ii) Suppose that B is a sub-C^{∗}-algebra of M(A) and that T is approximately n step inner relatively to M(A). Then T is approximately n step inner relatively to A.

Proof: (i). Assume that T is approximately n-step inner relatively to A_{ω}. Then for each natural number k there are contractions d_{1,k}, …, d_{n,k} in A_{ω} such that

lim_{k→∞} ∑_{j=1}^{n} d_{j,k}^{∗}bd_{j,k} = T (b)

for all b in B. By (an obvious modification of) Lemma 2.5 we find contractions d_{1}, …, d_{n} in A_{ω} satisfying ∑_{j=1}^{n} d_{j}^{∗}bd_{j} = T (b) for all b in B.

Write d_{j} = π_{ω}(d_{j}^{(1)}, d_{j}^{(2)}, …) where each d_{j}^{(k)} is a contraction in A. Then

lim_{ω} ∥ ∑_{j=1}^{n}(d_{j}^{(k)})^{∗}bd_{j}^{(k)} − T (b)∥ = 0, b ∈ B.

For each finite subset F of B and for each ε > 0 we can therefore find a natural number k such that

∥ ∑_{j=1}^{n}(d_{j}^{(k)})^{∗}bd_{j}^{(k)} − T (b)∥ ≤ ε, b ∈ F.

Hence T is approximately inner relatively to A.

(ii). Assume that T is approximately n-step inner relatively to M(A). Take a finite subset F of B and ε > 0. Find an n-step inner completely positive contraction S: B → M(A) such that ∥T (b) − S(b)∥ ≤ ε/2 for all b in F. Write S(b) = ∑_{j=1}^{n} d_{j}^{∗}bd_{j} for suitable d_{j} in M(A). Find a positive contraction f in A with ∥f^{1/2}T (b)f^{1/2} − T (b)∥ ≤ ε/2 for all b in F, and put e_{j} = d_{j}f^{1/2}. Then each e_{j} belongs to A, and

∥ ∑_{j=1}^{n} e_{j}^{∗}be_{j} − T (b)∥ ≤ ∥f^{1/2}(S(b) − T (b))f^{1/2}∥ + ∥f^{1/2}T (b)f^{1/2} − T (b)∥ ≤ ε

for all b in F. □

Part (i) of our next preliminary result describes (approximately) 1-step inner maps, and part (ii) is a related result that shall be used in Section 8.

Lemma 7.4 Let A be a stable C^{∗}-algebra.

(i) Let B be a separable sub-C^{∗}-algebra of A, and let V : B → A be an approximately 1-step inner completely positive contraction. Then there is a sequence {t_{n}}_{n=1}^{∞} of isometries in M(A) such that ∥t_{n}^{∗}bt_{n} − V (b)∥ → 0 for all b in B.

(ii) If there is a sequence {d_{n}} of elements in M_{2}(A) such that

lim_{n→∞} ∥d_{n}^{∗} (\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) d_{n} − (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) ∥ = 0  (7.1)

for all a in A, then there are isometries u_{n}, v_{n} in M(A) such that u_{n}u_{n}^{∗} + v_{n}v_{n}^{∗} ≤ 1 and

lim_{n→∞} ∥ (\begin{matrix}u_{n}^{∗} & 0 \\ v_{n}^{∗} & 0 \end{matrix}) (\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}u_{n} & v_{n} \\ 0 & 0 \end{matrix}) − (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) ∥ = 0  (7.2)

for all a in A.

Proof: Because A is stable, we can write A = A_{0} ⊗ K(ℓ^{2}(ℕ)). View 1 ⊗ B(ℓ^{2}(ℕ)) as a sub-C^{∗}-algebra of M(A). Choose an increasing approximate unit {e_{n}}_{n=1}^{∞} consisting of projections for K(ℓ^{2}(ℕ)). Then ∥a(1 − 1 ⊗ e_{n})∥ → 0 for every a in A. Take isometries s_{n,1}, s_{n,2} in 1 ⊗ B(ℓ^{2}(ℕ)) satisfying s_{n,1}s_{n,1}^{∗} + s_{n,2}s_{n,2}^{∗} = 1 and s_{n,2}s_{n,2}^{∗} ≤ 1 − 1 ⊗ e_{n}. Then ∥as_{n,2}s_{n,2}^{∗}∥ → 0 for all a in A.

(i). Use Lemma 7.2 to find a sequence {d_{n}}_{n=1}^{∞} of contractions in A such that d_{n}^{∗}bd_{n} → V (b) for all b in B, and put

t_{n} = s_{n,1}s_{n,1}^{∗}d_{n} + s_{n,2}(1 − d_{n}^{∗}s_{n,1}s_{n,1}^{∗}d_{n})^{1/2}.  (7.3)

Then each t_{n} is an isometry, and t_{n}^{∗}bt_{n} → V (b) as desired.

(ii). Observe first that M(M_{2}(A)) = M_{2}(M(A)). Let e in M_{2}(M(A)) be diag(1, 0). By Lemma 7.2 we can assume that each d_{n} is a contraction. Upon replacing d_{n} by ed_{n} we may also assume that ed_{n} = d_{n}. Let s_{1,n}, s_{2,n} be as above with s_{1,n}s_{1,n}^{∗} + s_{2,n}s_{2,n}^{∗} = e = s_{j,n}^{∗}s_{j,n}, and let t_{n} in M_{2}(M(A)) be given as in (7.3). Then t_{n} is an isometry satisfying (7.1) (with t_{n} in the place of d_{n}), t_{n} = et_{n}, and hence

t_{n} = (\begin{matrix}u_{n} & v_{n} \\ 0 & 0 \end{matrix}) .

for some u_{n}, v_{n} that necessarily are isometries with u_{n}u_{n}^{∗} orthogonal to v_{n}v_{n}^{∗} because t_{n} is an isometry. □

A local Weyl–von Neumann theorem

The main result of this subsection is Theorem 7.21 which will be proved in a series of lemmas. Two of these lemmas (Lemma 7.7 and Lemma 7.12) will use the following:

Remark 7.5 (A construction with commuting elements) Let a_{1}, a_{2}, …, a_{n} be commuting, positive contractions in a strongly purely infinite C^{∗}-algebra A, and let η_{1} > 0, η_{2} > 0, and η_{3} > 0 be given.

Let X denote the primitive ideal spectrum of D = C^{∗}(a_{1}, a_{2}, …, a_{n}), so that D is isomorphic to C_{0}(X). For each x in X let ρ_{x}: D → ℂ denote the corresponding character on D. Identifying X with the image of the (injective) map

x ↦ (ρ_{x}(a_{1}), ρ_{x}(a_{2}), …, ρ_{x}(a_{n})),

X becomes a bounded subset of ℝ^{n}. Put

Ω = {x ∈ X : max{a_{1}(x), …, a_{r}(x)} ≥ η_{2}},

so that Ω is a compact subset of X. It is a standard fact from dimension theory that every open cover of Ω ⊆ ℝ^{n} has an open sub-cover such that each point in ℝ^{n} belongs to at most n + 1 of the open sets in the sub-cover. Use this to find open sets U_{1}, …, U_{r} and V_{1}, …, V_{r} such that

(i) Ω ⊆ V_{1} ∪ V_{2} ∪ ⋯ ∪ V_{r} ⊆ U_{1} ∪ U_{2} ∪ ⋯ ∪ U_{r} ⊆ X, and \overline{V}_{j} ⊆ U_{j},

(ii) |ρ_{x}(a_{k}) − ρ_{x′}(a_{k})| ≤ η_{2} for all j = 1, 2, …, r, for all x, x′ in U_{j}, and for all k,

(iii) for each x in X is contained in at most n + 1 of the open sets U_{1}, …, U_{r}.

Choose x_{j} in V_{j} for each j. Let g_{1}, …, g_{r} be positive contractions in D such that g_{1} + ⋯ + g_{r} ≤ 1, ρ_{x}(g_{1} + ⋯ + g_{r}) = 1 for x in Ω, and ρ_{x}(g_{j}) = 0 when x ∉ V_{j}. Choose next positive contractions f_{1}, …, f_{r} in D such that f_{j}g_{j} = g_{j} and ρ_{x}(f_{j}) = 0 when x ∉ U_{j}. Then

∥ca_{k}c − ρ_{x_{j}}(a_{k})c^{2}∥ ≤ η_{2},  ∥a_{k} − ∑_{j=1}^{r} ρ_{x_{j}}(a_{k})g_{j}∥ ≤ η_{2}  (7.4)

for each contraction c in \overline{f_{j}Df_{j}} and for k = 1, …, n. Put

f = (f_{1}^{1/2} ⋯ f_{r}^{1/2}) ∈ M_{1,r}(A), g = (g_{1}^{1/2} ; ⋮ ; g_{r}^{1/2}) ∈ M_{r,1}(A).

For each δ ≥ 0 define the map S_{δ}: D → M_{r}(D) by

S_{δ}(c) = (\begin{matrix}ρ_{x_{1}}(c)(f_{1} − δ)_{+} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & ρ_{x_{r}}(c)(f_{r} − δ)_{+} \end{matrix})

Applying Lemma 5.8 (with ε_{j} = η_{1} and δ small enough) to the positive matrix

f^{∗}f = (\begin{matrix}f_{1} & ⋯ & f_{1}^{1/2}f_{r}^{1/2} \\ ⋮ & & ⋮ \\ f_{r}^{1/2}f_{1}^{1/2} & ⋯ & f_{r} \end{matrix})

we obtain elements d_{1}, d_{2}, …, d_{r} in A such that d_{j}^{∗}f_{j}d_{j} = (f_{j} − η_{1})_{+}, ∥d_{j}∥^{2} ≤ 2/η_{1}, and

∥ (\begin{matrix}d_{1}^{∗} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & d_{r}^{∗} \end{matrix}) f^{∗}f (\begin{matrix}d_{1} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & d_{r} \end{matrix}) − (\begin{matrix}(f_{1} − η_{1})_{+} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & (f_{r} − η_{1})_{+} \end{matrix}) ∥ ≤ η_{3}. (7.5)

Set h_{j} = f_{j}^{1/2}d_{j}, and set h = (h_{1}, …, h_{r}) in M_{1,r}(A), so that h_{j}^{∗}h_{j} = d_{j}^{∗}f_{j}d_{j} = (f_{j} − η_{1})_{+}.

Lemma 7.6 In the notation of Remark 7.5,

∥h^{∗}a_{k}h − S_{η_{1}}(a_{k})∥ ≤ 2(n + 1)η_{2}η_{1}^{−1} + η_{3}, ∥g^{∗}S_{2η_{1}}(a_{k})g − a_{k}∥ ≤ 2η_{1} + η_{2},

g is a contraction, and ∥f∥^{2} ≤ n + 1.

Proof: Since ff^{∗} = ∑_{j=1}^{n} f_{j} ≤ n+1 by condition (iii) in Remark 7.5, we have ∥f∥^{2} ≤ n+1.

For each j = 1, …, r let p_{j} in A^{∗∗} correspond to the indicator function 1_{U_{j}} so that f_{j}p_{j} = f_{j} and ∥p_{j}a_{k} − ρ_{x_{j}}(a_{k})p_{j}∥ ≤ η_{2}, cf. (7.4). Then, using (7.5), we get

∥h^{∗}a_{k}h − S_{η_{1}}(a_{k})∥
= ∥ (\begin{matrix}d_{1}^{∗} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & d_{r}^{∗} \end{matrix}) (f^{∗}a_{k}f − f^{∗}f (\begin{matrix}ρ_{x_{1}}(a_{k}) & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & ρ_{x_{r}}(a_{k}) \end{matrix}) ) (\begin{matrix}d_{1} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & d_{r} \end{matrix}) ∥ + η_{3}

≤ 2η_{1}^{−1}∥f^{∗}f (\begin{matrix}a_{k} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & a_{k} \end{matrix}) − f^{∗}f (\begin{matrix}ρ_{x_{1}}(a_{k}) & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & ρ_{x_{r}}(a_{k}) \end{matrix}) ∥ + η_{3}

= 2η_{1}^{−1}∥f^{∗}f (\begin{matrix}p_{1}a_{k} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & p_{r}a_{k} \end{matrix}) − f^{∗}f (\begin{matrix}ρ_{x_{1}}(a_{k})p_{1} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & ρ_{x_{r}}(a_{k})p_{r} \end{matrix}) ∥ + η_{3}
≤ 2η_{1}^{−1}∥f^{∗}f∥η_{2} + η_{3} ≤ 2(n + 1)η_{2}η_{1}^{−1} + η_{3}.

Next, g^{∗}g = ∑_{j=1}^{r} g_{j}, so that g is a contraction, and

g^{∗}S_{2η_{1}}(a_{k})g = ∑_{j=1}^{r} ρ_{x_{j}}(a_{k})g_{j}^{1/2}(f_{j} − 2η_{1})_{+}g_{j}^{1/2} = (1 − 2η_{1}) ∑_{j=1}^{r} ρ_{x_{j}}(a_{k})g_{j}.

This together with the second identity in (7.4) yield ∥g^{∗}S_{2η_{1}}(a_{k})g − a_{k}∥ ≤ 2η_{1} + η_{2}. □

Lemma 7.7 Let a_{1}, a_{2}, …, a_{n} be commuting, positive elements in a strongly purely infinite C^{∗}-algebra A. Then for each ε > 0 and for each m in ℕ there exists b in M_{m}(A) such that

∥b^{∗} (\begin{matrix}a_{k} & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) b − (\begin{matrix}a_{k} & 0 & ⋯ & 0 \\ 0 & a_{k} & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & a_{k} \end{matrix}) ∥ ≤ ε.

for k = 1, …, n.

Proof: Upon replacing the elements a_{1}, …, a_{n} with ta_{1}, …, ta_{n} for some positive real number t we may assume that each a_{k} is a contraction.

Apply Remark 7.5 and Lemma 7.6 with

η_{1} = ε/4,  η_{2} = min \{\frac{ε^{3}}{320(n + 1)}, \frac{ε}{4}\},  η_{3} = ε/40.

Then by Lemma 7.6,

∥h^{∗}a_{k}h − S_{ε/4}(a_{k})∥ ≤ \frac{ε^{2}}{20},  ∥g^{∗}S_{ε/2}(a_{k})g − a_{k}∥ ≤ \frac{3ε}{4}, for k = 1, …, n.  (7.6)

By Lemma 2.4 (i) there are t_{1}, …, t_{r} in M_{m,1}(A) with

∥t_{j}∥^{2} ≤ 5/ε, t_{j}^{∗}(f_{j} − ε/4)_{+}t_{j} = (f_{j} − ε/2)_{+} ⊗ 1_{m}.

Write t_{j} = (t_{j,i}) and put

t = (\begin{matrix}t_{1,1} & … & 0 & t_{1,2} & … & 0 & … & t_{1,m} & … & 0 \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ & … & ⋮ & ... & ⋮ \\ 0 & … & t_{r,1} & 0 & … & t_{r,2} & … & 0 & … & t_{r,m} \end{matrix}) ∈ M_{r,rm}(A).

Then ∥t∥^{2} ≤ 5/ε and t^{∗}S_{ε/4}(c)t = S_{ε/2}(c) ⊗ 1_{m} for all c in D, and hence

∥t^{∗}h^{∗}a_{k}ht − S_{ε/2}(a_{k}) ⊗ 1_{m}∥ ≤ ε/4 for k = 1, …, n,  (7.7)

by the first inequality in (7.6). Put b_{0} = ht(g ⊗ 1_{m}) ∈ M_{1,m}(A). By (7.7) and the second inequality in (7.6) we get the estimate:

∥b_{0}^{∗}a_{k}b_{0} − a_{k} ⊗ 1_{m}∥ ≤ ∥(g^{∗} ⊗ 1_{m})(t^{∗}h^{∗}a_{k}ht − (S_{ε/2}(a_{k}) ⊗ 1_{m}))(g ⊗ 1_{m})∥
+∥(g^{∗} ⊗ 1_{m})(S_{ε/2}(a_{k}) ⊗ 1_{m})(g ⊗ 1_{m}) − a_{k} ⊗ 1_{m}∥
≤ ∥t^{∗}h^{∗}a_{k}ht − (S_{ε/2}(a_{k}) ⊗ 1_{m})∥ + ∥g^{∗}S_{ε/2}(a_{k})g − a_{k}∥ ≤ ε.

Taking b in M_{m}(A) to be the matrix whose first row is b_{0} and whose other rows are zero, the lemma is proved. □

Proposition 7.8 Let A be a strongly purely infinite C^{∗}-algebra.

(i) Let a_{1}, a_{2}, …, a_{n} be commuting, positive elements in A. For each natural number m and for each ε > 0 there is a contraction b such that

∥b^{∗} (\begin{matrix}a_{k} & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) b − (\begin{matrix}a_{k} & 0 & ⋯ & 0 \\ 0 & a_{k} & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & a_{k} \end{matrix}) ∥ ≤ ε

for k = 1, …, n.

(ii) Let ω be a free filter on ℕ. Let a_{1}, a_{2}, …, a_{n} be commuting, positive elements in A_{ω}, and let m be a natural number. Then there exists a contraction b in M_{m}(A_{ω}) such that

b^{∗} (\begin{matrix}a_{k} & 0 & ⋯ & 0 \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) b = (\begin{matrix}a_{k} & 0 & ⋯ & 0 \\ 0 & a_{k} & ⋯ & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & ⋯ & a_{k} \end{matrix})

for k = 1, …, n.

Proof: (i). As in the proof of Lemma 7.7 we may assume that each a_{k} is a contraction. Choose a positive contraction e in the Abelian C^{∗}-algebra generated by a_{1}, …, a_{n} with ∥ea_{k}e − a_{k}∥ ≤ ε/3. Use Lemma 7.7 to find d in M_{1,m}(A) satisfying ∥d^{∗}cd − (c ⊗ 1_{m})∥ ≤ ε/3 for all c in the set {ea_{1}e, …, ea_{k}e, e^{2}}. Put b_{0} = (1 + ε/3)^{−1/2}ed. Then

b_{0}^{∗}b_{0} = (1 + ε/3)^{−1}d^{∗}e^{2}d ≤ 1,

so b_{0} is a contraction. Also,

∥b_{0}^{∗}a_{k}b_{0} − a_{k} ⊗ 1_{m}∥ = ∥(1 + ε/3)^{−1}d^{∗}ea_{k}ed − a_{k} ⊗ 1_{m}∥
≤ ∥(1 + ε/3)^{−1}d^{∗}ea_{k}ed − ea_{k}e ⊗ 1_{m}∥ + ε/3
≤ (1 + ε/3)^{−1}∥d^{∗}ea_{k}ed − ea_{k}e ⊗ 1_{m}∥ + ε/3 + ε/3 ≤ ε.

The element b in M_{m}(A) whose first row is b_{0} and whose other rows are zero will therefore be as desired.

(ii). This follows from (i) and from Lemma 2.5. □

For any positive element a in a purely infinite C^{∗}-algebra A, for any m in ℕ, and for every ε > 0, one can find d in M_{1,m}(A) such that ∥d^{∗}ad−a⊗1_{m}∥ ≤ ε. It is, however, not know to us if one always can choose d to be a contraction. One of the offsprings of Proposition 7.8 above is that d can be chosen to be a contraction in a strongly purely infinite C^{∗}-algebra.

Lemma 7.9 Let D be a C^{∗}-algebra, let a be a positive element in D, and let d be a contraction in D. Then the following two conditions are equivalent:

(i) d commutes with a, and d^{∗}da = a,

(ii) d^{∗}ad = a and d^{∗}a^{2}d = a^{2}.

Proof: (ii) ⇒ (i). Set x = ad − da. Then

x^{∗}x = d^{∗}a^{2}d − d^{∗}ada − ad^{∗}ad + ad^{∗}da = ad^{∗}da − a^{2} ≤ 0

(because d is a contraction). This shows that x must be zero. Hence d commutes with a, and d^{∗}da = d^{∗}ad = a. (i) ⇒ (ii) is trivial. □

Lemma 7.10 Suppose that A is a strongly purely infinite C^{∗}-algebra. Let ω be a free filter on ℕ, and let a_{1}, a_{2}, …, a_{n} be a set of commuting normal elements in A_{ω}. Then for each natural number m there are contractions r_{1}, r_{2}, …, r_{m} in A_{ω} ∩ {a_{1}, a_{2}, …, a_{n}}′ such that

r_{i}^{∗}r_{j} = δ_{ij}r_{1}^{∗}r_{1}, r_{1}^{∗}r_{1}a_{k} = a_{k}, for all i, j, k.

Proof: We can without loss of generality assume that each a_{j} is positive. Use Lemma 2.5 to find a positive contraction e in A_{ω} such that ea_{k} = a_{k}e = a_{k}. Given m, take a contraction b in M_{m}(A_{ω}) as in Proposition 7.8 (ii) with respect to the set

{a_{1}, a_{2}, …, a_{n}, a_{1}^{2}, a_{2}^{2}, …, a_{n}^{2}, e, e^{2}}.

We can assume that b is a row matrix, i.e., that

b = (\begin{matrix}b_{1} & b_{2} & ⋯ & b_{m} \\ 0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & 0 \end{matrix}) .

Then each b_{j} is a contraction, b_{i}^{∗}a_{k}b_{j} = δ_{ij}a_{k}, and b_{i}^{∗}a_{k}^{2}b_{j} = δ_{ij}a_{k}^{2}.

We conclude from Lemma 7.9 that each b_{i} belongs to A_{ω} ∩ {a_{1}, a_{2}, …, a_{n}}′, and that b_{i}^{∗}b_{j}a_{k} = δ_{ij}a_{k}. Similarly we find that each b_{i} commutes with e and that b_{i}^{∗}b_{i}e = e. Set r_{i} = b_{i}e^{1/2}. Then each r_{i} is a contraction in A_{ω} ∩ {a_{1}, a_{2}, …, a_{n}}′ and r_{i}^{∗}r_{j} = δ_{ij}e. This proves the lemma. □

Lemma 7.11 Let D, E be C^{∗}-algebras, and let a be a positive element in D.

(i) Let S: D → E be a completely positive contraction. If ∥S(a^{2}) − S(a)^{2}∥ ≤ ε, then ∥S(ab) − S(a)S(b)∥ ≤ ε^{1/2} and ∥S(ba) − S(b)S(a)∥ ≤ ε^{1/2} for all contraction b in D.

(ii) If T: E → D is a completely positive contraction and if ∥S(T (a)) − a∥ ≤ ε and ∥S(T (a^{2})) − a^{2}∥ ≤ ε, then ∥S(T (a)^{2}) − S(T (a))^{2})∥ ≤ (1 + 2∥a∥)ε.

Proof: (i). By Stinespring’s theorem we can find a representation of E on a Hilbert space H, a ^{∗}-homomorphism π: D → B(H), and a projection p on H such that S(d) = pπ(d)p (when viewing E as a sub-C^{∗}-algebra of B(H)). Calculate

S(a^{2}) − S(a)^{2} = pπ(a)π(a)p − pπ(a)pπ(a)p = (pπ(a)(1 − p))(pπ(a)(1 − p))^{∗}.

We deduce that ∥pπ(a)(1 − p)∥ ≤ ε^{1/2}, and hence that ∥pπ(a) − π(a)p∥ ≤ ε^{1/2}. This leads to the estimate

∥S(ab) − S(a)S(b)∥ = ∥pπ(ab)p − pπ(a)pπ(b)∥ = ∥p(pπ(a) − π(a)p)π(b)p∥
≤ ∥b∥∥π(a)p − pπ(a)∥ ≤ ∥b∥ε^{1/2}.

The other inequality follows by taking adjoints.

(ii). A completely positive contraction V between two C^{∗}-algebras satisfies the inequality V (x)^{∗}V (x) ≤ V (x^{∗}x). (This can for example be proved using Stinespring’s theorem as in the proof of (i) above.) Using this fact twice together with the estimate ∥S(T (a))^{2} − a^{2}∥ ≤ 2∥a∥ε yields:

0 ≤ S(T (a)^{2}) − S(T (a))^{2} ≤ S(T (a^{2})) − S(T (a))^{2}
≤ (a^{2} + ε·1) − (a^{2} − 2∥a∥ε·1) = (1 + 2∥a∥)ε.

This proves (ii). □

Lemma 7.12 Suppose that A is a strongly purely infinite C^{∗}-algebra. Let ω be a free filter on ℕ, let a_{1}, a_{2}, …, a_{n} be a set of commuting normal elements in A_{ω}, and let b_{1}, b_{2}, …, b_{m} be any finite subset of A_{ω}. Then there exists a contraction d in A_{ω} with

d^{∗}da_{k} = a_{k}, [d, a_{k}] = 0, [d^{∗}b_{j}d, a_{k}] = 0, [d^{∗}b_{j}d, d^{∗}b_{i}d] = 0

for all i, j, and k.

Proof: It is no loss of generality to assume that each a_{k} and each b_{j} is a positive contraction. Next, it suffices to prove the lemma in the case where m = 1. Indeed, for m ≥ 2 use the case m = 1 to find a contraction d_{1} satisfying

d_{1}^{∗}d_{1}a_{k} = a_{k},  [d_{1}, a_{k}] = 0,  [d_{1}^{∗}b_{1}d, a_{k}] = 0.

Repeat the process to the n + 1 commuting elements

ã_{1} = d_{1}^{∗}a_{1}d_{1}, ã_{2} = d_{2}^{∗}a_{2}d_{2}, …, ã_{n} = d_{1}^{∗}a_{n}d_{1}, ã_{n+1} = d_{1}^{∗}b_{1}d_{1}

and to b̃_{1} = d_{1}^{∗}b_{2}d_{1}. The lemma is proved after m such steps.

Assume accordingly that m = 1 and that b = b_{1} is a positive contraction. By Lemma 7.9 it suffices to find a contraction d in A_{ω} such that d^{∗}a_{k}d = a_{k}, d^{∗}a_{k}^{2}d = a_{k}^{2}, and [d^{∗}bd, a_{k}] = 0 for all k. To find d replace the set {a_{1}, …, a_{n}} with the set {a_{1}, …, a_{n}, a_{1}^{2}, …, a_{n}^{2}}, show that for each ε > 0 there is a contraction d in A_{ω} such that

∥d^{∗}a_{k}d − a_{k}∥ ≤ ε,  ∥[d^{∗}bd, a_{k}]∥ ≤ ε,  (7.8)

and invoke Lemma 2.5 to complete the proof.

We apply Remark 7.5 and Lemma 7.6 with A_{ω} in the place of A, with η_{1} = ε/4, with η_{2} > 0, η_{3} > 0 chosen such that the right-hand estimates in (7.9) and (7.10) hold, and with {a_{1}, …, a_{n}, a_{1}^{2}, …, a_{n}^{2}} in the place of the set {a_{1}, …, a_{n}}. Hence n (from Remark 7.5) is replaced with 2n.

In addition to the positive contractions f_{1}, …, f_{r}, g_{1}, … g_{r}, and the elements h_{1}, …, h_{r} from Remark 7.5 we shall need positive contractions k_{1}, …, k_{r} in D such that f_{j}k_{j} = k_{j} and k_{j}g_{j} = g_{j}.

Observe that h_{j}^{∗}bh_{j} ≤ h_{j}^{∗}h_{j} = (f_{j} − η_{1})_{+} because b is a contraction. Apply Lemma 5.13 to the positive matrix

h^{∗}bh + (\begin{matrix}(f_{1} − η_{1})_{+} − h_{1}^{∗}bh_{1} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & (f_{r} − η_{1})_{+} − h_{r}^{∗}bh_{r} \end{matrix})

to obtain positive elements e_{1}, …, e_{r} in A_{ω} such that

∥e_{j}∥^{2} ≤ 2/η_{1},  e_{j}^{∗}(f_{j} − η_{1})_{+}e_{j} = (f_{j} − 2η_{1})_{+},  e_{i}^{∗}h_{i}^{∗}bh_{j}e_{j} = 0 when i ≠ j.

Put d = ∑_{j=1}^{r} h_{j}e_{j}g_{j}^{1/2}. Then

d^{∗}d = ∑_{j=1}^{r} g_{j}^{1/2}(f_{j} − 2η_{1})_{+}g_{j}^{1/4} = (1 − 2η_{1}) ∑_{j=1}^{r} g_{j},

and so d is a contraction. We proceed to verify (7.8).

Let g be as in Remark 7.5 and define R: M_{r}(A_{ω}) → A_{ω} and T: A_{ω} → M_{r}(A_{ω}) by R(c) = g^{∗}cg and

T (c) = (\begin{matrix}k_{1}e_{1}^{∗} & ⋯ & 0 \\ . & . & . \\ 0 & ⋯ & k_{r}e_{r}^{∗} \end{matrix}) h^{∗}ch (\begin{matrix}e_{1}k_{1} & ⋯ & 0 \\ . & . & . \\ 0 & ⋯ & e_{r}k_{r} \end{matrix}) .

Use the relation g_{j}^{1/2}k_{j} = g_{j}^{1/2} to see that d^{∗}bd = R(T (b)). The (ij)th entry of T (b) is given by

(T (b))_{ij} = k_{i}e_{i}^{∗}h_{i}^{∗}bh_{j}e_{j}k_{j} = \begin{cases}k_{j}e_{j}^{∗}h_{j}^{∗}bh_{j}e_{j}k_{j}, & if i = j \\ 0, & if i ≠ j \end{cases} ,

so that S(a_{k})T (b) = T (b)S(a_{k}) for all k, where S = S_{0} : D → M_{r}(D) is as in Remark 7.5.

Because g_{j}^{1/2}f_{j}g_{j}^{1/2} = g_{j} we get R(S(a_{k}^{ν})) = ∑_{j=1}^{r} ρ_{x_{j}}(a_{k}^{ν})g_{j}. Hence ∥R(S(a_{k}^{ν})) − a_{k}^{ν}∥ ≤ η_{2} for all k and for ν = 1, 2 by (7.4). Lemma 7.11 (ii) now yields

∥R(S(a_{k})^{2}) − R(S(a_{k}))^{2}∥ ≤ (1 + 2∥a_{k}∥)η_{2} ≤ 3η_{2},

whence

∥R(S(a_{k})T (b)) − R(S(a_{k}))R(T (b))∥ ≤ \sqrt{3η_{2}},
∥R(T (b)S(a_{k})) − R(T (b))R(S(a_{k}))∥ ≤ \sqrt{3η_{2}},

by Lemma 7.11 (i). Recalling that S(a_{k}) commutes with T (b) and that R(T (b)) = d^{∗}bd, this leads to the estimate ∥R(S(a_{k}))d^{∗}bd − d^{∗}bdR(S(a_{k}))∥ ≤ 2\sqrt{3η_{2}}, which in turns implies that

∥[d^{∗}bd, a_{k}]∥ ≤ 2\sqrt{3η_{2}} + 2η_{2} ≤ ε.  (7.9)

With S_{δ}: D → M_{r}(D) as in Remark 7.5, we have

d^{∗}a_{k}d = g^{∗} (\begin{matrix}e_{1}^{∗} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & e_{r}^{∗} \end{matrix}) h^{∗}a_{k}h (\begin{matrix}e_{1} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & e_{r} \end{matrix}) g,

g^{∗}S_{2η_{1}}(a_{k})g = g^{∗} (\begin{matrix}e_{1}^{∗} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & e_{r}^{∗} \end{matrix}) S_{η_{1}}(a_{k}) (\begin{matrix}e_{1} & ⋯ & 0 \\ ⋮ & ... & ⋮ \\ 0 & ⋯ & e_{r} \end{matrix}) g.

From Lemma 7.6 we know that

∥h^{∗}a_{k}h − S_{η_{1}}(a_{k})∥ ≤ 2(2n + 1)η_{2}η_{1}^{−1} + η_{3}, ∥g^{∗}S_{2η_{1}}(a_{k})g − a_{k}∥ ≤ 2η_{1} + η_{2}.

By the choice of η_{2} and η_{3}, this shows that

∥d^{∗}a_{k}d − a_{k}∥ ≤ 2η_{1}^{−1}(2(2n + 1)η_{2}η_{1}^{−1} + η_{3}) + 2η_{1} + η_{2} ≤ ε.  (7.10)

as desired. □

Proposition 7.13 (Extension) Let A be a strongly purely infinite C^{∗}-algebra, let ω be a free filter on ℕ, let B be a separable sub-C^{∗}-algebra of A_{ω}, and let C be an Abelian sub-C^{∗}-algebra of B. It follows that there is a 1-step inner completely positive contraction T: B → A_{ω} such that the image of T is contained in an Abelian sub-C^{∗}-algebra of A_{ω} and such that T (c) = c for all c in C.

Proof: Choose countable subsets {b_{1}, b_{2}, … } and {c_{1}, c_{2}, … } of positive contractions in B, respectively, C, which span dense subspaces of B and C, respectively.

By Lemma 7.12 there is for each natural number n a contraction d_{n} in A_{ω} such that

d_{n}^{∗}c_{k}d_{n} = c_{k}, [d_{n}^{∗}b_{j}d_{n}, c_{k}] = 0, [d_{n}^{∗}b_{j}d_{n}, d_{n}^{∗}b_{i}d_{n}] = 0 for all i, j, k = 1, 2, …, n. (7.11)

Lemma 2.5 then shows that there is a contraction d in A_{ω} such that (7.11) holds for all i, j, and k in ℕ. Now, the completely positive contraction given by T (b) = d^{∗}bd has the desired properties. □

Proposition 7.14 Let A be a strongly purely infinite C^{∗}-algebra.

(i) Let ω be a free filter on ℕ, let B be a separable sub-C^{∗}-algebra of A_{ω}, and let V : B → A_{ω} be an approximately inner completely positive contraction whose image is contained in an Abelian sub-C^{∗}-algebra of A_{ω}. It follows that V is 1-step inner relatively to A_{ω}.

(ii) Let B be a separable sub-C^{∗}-algebra of M(A), the multiplier algebra of A, and let V : B → A be an approximately inner completely positive contraction whose image is contained in an Abelian sub-C^{∗}-algebra of A. Then V is approximately 1-step inner relatively to A.

Proof: We prove (i) and (ii) simultaneously. It suffices to show that if B is either a separable subalgebra of A_{ω} or of M(A), if V : B → A_{ω} is approximately inner (relatively to A_{ω}), then V is 1-step inner relatively to A_{ω}. If V : B → A is 1-step inner relatively to A_{ω}, then V is approximately 1-step inner relatively to A by Lemma 7.3 (i).

Choose a set {b_{1}, b_{2}, b_{3}, … } of positive elements that span a dense subset of B. By Lemma 2.5 it suffices, for each n and for each ε > 0, to find a contraction f in A_{ω} such that ∥V (b_{k}) − f^{∗}b_{k}f∥ ≤ ε for k = 1, 2, …, n. By assumption there is an m-step inner completely positive contraction W: B → A_{ω} such that ∥V (b_{k}) − W (b_{k})∥ ≤ ε for all k = 1, …, n. Find d_{1}, d_{2}, …, d_{m} in A_{ω} such that W (b) = ∑_{i=1}^{m} d_{j}^{∗}bd_{j}.

By Lemma 7.12 there is a contraction d in A_{ω} such that each pair of elements in the finite set

{d^{∗}d_{i}^{∗}b_{k}d_{j}d : i, j = 1, …, m, k = 1, …, n}

commute, and such that d^{∗}V (b_{k})d = V (b_{k}) for k = 1, …, n, cf. Lemma 7.9.

In particular, each d^{∗}d_{i}^{∗}b_{k}d_{j}d is normal. Use Lemma 7.10 to find contractions r_{1}, …, r_{m} in A_{ω} commuting with each d^{∗}d_{i}^{∗}b_{k}d_{j}d, and such that r_{j}^{∗}d^{∗}d_{j}^{∗}b_{k}d_{i}dr_{i} = δ_{ij}d^{∗}d_{j}^{∗}b_{k}d_{i}d. Put f = ∑_{j=1}^{m} d_{j}dr_{j}. Then

f^{∗}b_{k}f = ∑_{i,j=1}^{m} r_{j}^{∗}d^{∗}d_{j}^{∗}b_{k}d_{i}dr_{i} = ∑_{j=1}^{m} d^{∗}d_{j}^{∗}b_{k}d_{j}d = d^{∗}W (b_{k})d, k = 1, …, n.

As ∥W (b_{k}) − V (b_{k})∥ ≤ ε, and d is a contraction satisfying d^{∗}V (b_{k})d = V (b_{k}), we conclude that ∥V (b_{k}) − f^{∗}b_{k}f∥ ≤ ε. □

Definition 7.15 For each pair of C^{∗}-algebras A and B define 𝒞_{0}(B, A) to be the set of all approximately inner completely positive maps from B to A whose image is contained in an Abelian sub-C^{∗}-algebra of A. Define 𝒞 (B, A) to be the set of all completely positive maps T: B → A such that

T (b) = ∑_{i,j=1}^{n} a_{j}^{∗}T_{0}(c_{j}^{∗}bc_{i})a_{i}, b ∈ B  (7.12)

for some natural number n, for some a_{1}, …, a_{n} in M(A), and for some c_{1}, …, c_{n} in M(B). Let \overline{𝒞} (B, A) denote the pointwise–norm closure of 𝒞 (B, A).

Let A be a C^{∗}-algebra such that its multiplier algebra M(A) contains a unital copy of the Cuntz algebra O_{2}. Each stable C^{∗}-algebra has this property, and more generally, if D is stable and if ω is any filter on ℕ, the A = D_{ω} has this property. Indeed, M(D) contains B(ℓ^{2}) and therefore O_{2} as a unital sub-C^{∗}-algebra, and there is a (canonical) unital embedding of M(D) into M(D_{ω}). (The C^{∗}-algebra D_{ω} need not be stable, even if D is stable.)

By assumption there are isometries s_{1}, s_{2} in M(A) satisfying the Cuntz relation: s_{1}s_{1}^{∗} + s_{2}s_{2}^{∗} = 1. Fixing two such isometries s_{1}, s_{2} we can for each C^{∗}-algebra B define the Cuntz sum of two maps T_{1}, T_{2} : B → A by

(T_{1} ⊕ T_{2})(b) = s_{1}T_{1}(b)s_{1}^{∗} + s_{2}T_{2}(b)s_{2}^{∗},  b ∈ B.  (7.13)

The operation ⊕ depends on the choice of s_{1}, s_{2}, but only up to unitary equivalence; ⊕ is not associative, but (T_{1} ⊕ T_{2}) ⊕ T_{3} is unitarily equivalent to T_{1} ⊕ (T_{2} ⊕ T_{3}). With this in mind, define inductively T_{1} ⊕ T_{2} ⊕ ⋯ ⊕ T_{n} to be (T_{1} ⊕ T_{2}) ⊕ ⋯ ⊕ T_{n}. Then

(T_{1} ⊕ T_{2} ⊕ ⋯ ⊕ T_{n})(b) = s_{1,n}T_{1}(b)s_{1,n}^{∗} + ⋯ + s_{n,n}T_{n}(b)s_{n,n}^{∗}  (7.14)

for some isometries s_{1,n}, …, s_{n,n} in A_{ω} satisfying 1 = s_{1,n}s_{1,n}^{∗} + ⋯ + s_{n,n}s_{n,n}^{∗}.

Lemma 7.16 Let A and B be C^{∗}-algebras, and assume that there is a unital embedding of O_{2} into M(A). Then the following holds:

(i) If T_{1}, T_{2} belong to 𝒞 (B, A), then so does their Cuntz sum T_{1} ⊕ T_{2}.

(ii) If T_{1}, …, T_{n} belong to 𝒞 (B, A), if a_{1}, …, a_{n} are elements in M(A), then T: B → A, given by T (b) = ∑_{j=1}^{n} a_{j}^{∗}T_{j}(b)a_{j}, belongs to 𝒞 (B, A). In particular, 𝒞 (B, A) is a cone.

(iii) If T belongs to 𝒞 (B, A), if a_{1}, …, a_{n} belong to M(A), c_{1}, …, c_{n} belong to M(B),

then S: B → A, given by S(b) = ∑_{i,j=1}^{n} a_{j}^{∗}T (c_{j}^{∗}bc_{i})a_{i}, belongs to 𝒞 (B, A). Straightforward continuity considerations show that Lemma 7.16 holds with 𝒞 (B, A) replaced by \overline{𝒞} (B, A).

Proof: (i). Suppose first that T_{1}, T_{2} belong to 𝒞_{0}(A, B). The image of T_{j} is then contained in an Abelian sub-C^{∗}-algebra D_{j} of A. Now, D = s_{1}D_{1}s_{1}^{∗} + s_{2}D_{2}s_{2}^{∗} is an Abelian sub-C^{∗}- algebra of A which contains the image of T_{1} ⊕ T_{2}. Hence T_{1} ⊕ T_{2} belongs to 𝒞_{0}(B, A). Assume now that T_{1}, T_{2} belong to 𝒞 (B, A). Then

T_{k}(b) = ∑_{i,j=1}^{n_{k}} a_{j,k}^{∗}S_{k}(c_{j,k}^{∗}bc_{i,k})a_{i,k}

for suitable a_{j,k} in M(A), c_{j,k} in M(B), and with S_{1}, S_{2} in 𝒞_{0}(B, A). We saw above that S = S_{1} ⊕ S_{2} belongs to 𝒞_{0}(B, A). The Cuntz relation for the isometries s_{1}, s_{2} implies that s_{k}^{∗}S(b)s_{l} = δ_{k,l}S_{k}(b). Hence

(T_{1} ⊕ T_{2})(b) = ∑_{k=1}^{2} ∑_{i,j=1}^{n_{k}} s_{k}a_{j,k}^{∗}S_{k}(c_{j,k}^{∗}bc_{i,k})a_{i,k}s_{k}^{∗}
= ∑_{k,l=1}^{2} ∑_{i,j=1}^{n_{k}} s_{k}a_{j,k}^{∗}s_{k}^{∗}S(c_{j,k}^{∗}bc_{i,l})s_{l}a_{i,l}s_{l}^{∗},

and it follows from the latter expression that T_{1} ⊕ T_{2} belongs to 𝒞 (B, A).

(ii). Let T_{1}, …, T_{n} be in 𝒞 (B, A), let a_{1}, …, a_{n} be elements in M(A), and assume that T given by T (b) = ∑_{j=1}^{n} a_{j}^{∗}T_{j}(b)a_{j} is a contraction. In the notation of (7.14), put a = ∑_{j=1}^{n} s_{j,n}a_{j} ∈ M(A). Then

T (b) = a^{∗}(T_{1} ⊕ ⋯ ⊕ T_{n})(b)a.

From (i) we know that T_{1} ⊕ ⋯ ⊕ T_{n} belongs to 𝒞. It is now clear from (7.12) that T belongs to 𝒞 (B, A).

(iii). Let S, T be as in (iii). Find R in 𝒞_{0}(B, A), elements e_{1}, …, e_{m} in M(A), and elements f_{1}, …, f_{m} in M(B) such that

T (b) = ∑_{k,l=1}^{m} e_{k}^{∗}R(f_{k}^{∗}bf_{l})e_{l}, b ∈ B.

Then

S(b) = ∑_{i,j=1}^{n} a_{j}^{∗}T (c_{j}^{∗}bc_{i})a_{i} = ∑_{i,j=1}^{n} ∑_{k,l=1}^{m} a_{j}^{∗}e_{k}^{∗}R(f_{k}^{∗}c_{j}^{∗}bc_{i}f_{l})e_{l}a_{i}.

This expression agrees with (7.12), and so S belongs to 𝒞 (B, A). □

The following two (well-known) facts, formulated as a lemma, are needed for the proof of Lemma 7.18 below. If ρ is a positive functional on a C^{∗}-algebra D and if d is an element in D, then d^{∗}ρd denotes the positive functional on D given by (d^{∗}ρd)(x) = ρ(d^{∗}xd).

Lemma 7.17 Let D be a C^{∗}-algebra.

(i) Let f_{1}, …, f_{n} be elements in the dual of D. Then there is a cyclic representation π of D on some Hilbert space H, say with cyclic vector ξ, and elements c_{1}, …, c_{n} in π(D)′ ∩ B(H) such that f_{j}(d) = ⟨π(d)ξ, c_{j}^{∗}ξ⟩.

(ii) Suppose that 𝒦 is a weak–^{∗} closed sub-cone of the cone of all positive linear functionals on D, and suppose that d^{∗}ρd belongs to 𝒦 for every d in D and for every ρ in 𝒦 . Let J be the set of all d in D such that ρ(d^{∗}d) = 0 for all ρ in 𝒦 . Then J is a closed two-sided ideal in D; and if ρ_{0} is a positive functional on D such that ρ_{0}(d^{∗}d) = 0 for all d in J, then ρ_{0} belongs to 𝒦 .

Proof: (i). Since each state on D is a linear combination of positive functionals on D it suffices to consider the case where each f_{j} is a positive. Put f = f_{1} + ⋯ + f_{n}, and let (π, H, ξ) be the GNS-representation with respect to f, so that f(d) = ⟨π(d)ξ, ξ⟩. Now f and each f_{j} extend to positive functionals g, respectively, g_{j} on the weak closure of π(D), i.e., g(π(d)) = f(d) and g_{j}(π(d)) = f_{j}(d), and g_{j} ≤ g. By (a linear version of) Sakai’s Radon– Nikodỳm theorem (see [12, Proposition 7.3.5]) there are elements c_{j} in B(H) ∩ π(D)′ such that g_{j}(d) = g(dc_{j}) for all d in the weak closure of π(D). It follows that

f_{j}(d) = g_{j}(π(d)) = g(π(d)c_{j}) = ⟨π(d)c_{j}ξ, ξ⟩ = ⟨π(d)ξ, c_{j}^{∗}ξ⟩, d ∈ D,

as desired.

(ii). The set J is a closed left-ideal of D (because the left-kernel of each ρ in 𝒦 is a closed left-ideal). Also, for each z in J and each d in D we have ρ((zd)^{∗}(zd)) = (d^{∗}ρd)(z^{∗}z) = 0 for each ρ in 𝒦 because d^{∗}ρd belongs to 𝒦 . This proves that zd belongs to J, and hence that J is a closed two-sided ideal in D.

To prove the second part of (ii) we may assume that J = 0. (Otherwise just replace A by A/J and view ρ_{0} and each functional in 𝒦 as a functional on A/J.) Suppose that ρ_{0} does not belong to 𝒦 . Then, by Hahn–Banach’s theorem and the characterization of weak-^{∗} continuous linear mappings on the dual space of A as being evaluation maps ρ ↦ ρ(a) for some a in A, there is a an element a in A such that ρ_{0}(a) < 0 and ρ(a) ≥ 0 for all ρ in 𝒦 . Upon replacing a by (a + a^{∗})/2 we may assume that a is self-adjoint. Write a = a_{+} − a_{−} where a_{+} and a_{−} are the positive and negative parts of a. Let {e_{n}} be an approximate unit consisting of positive contractions for the C^{∗}-algebra generated by a_{−}. Then e_{n}ae_{n} → −a_{−} and e_{n}ae_{n} ≤ 0 for all n. For each ρ in 𝒦 , e_{n}^{∗}ρe_{n} belongs to 𝒦 , and hence

0 ≤ (e_{n}^{∗}ρe_{n})(a) = ρ(e_{n}ae_{n}) ≤ 0,

so that ρ(−e_{n}ae_{n}) = 0 for ρ in 𝒦 and for all n. Since −e_{n}ae_{n} is positive, this entails that e_{n}ae_{n} = 0 for all n, and hence that a_{−} = 0. But this contradict the fact that ρ_{0}(a) < 0.

□

Lemma 7.18 Let B be a separable nuclear sub-C^{∗}-algebra of a stable C^{∗}-algebra A, and suppose that for each positive b in B there is V in \overline{𝒞} (B, A) such that V (b) = b. Then the inclusion mapping ι: B → A belongs to \overline{𝒞} (B, A).

Proof: For each finite set b_{1}, …, b_{n} in B and for each ε > 0 we must find V in \overline{𝒞} (B, A) such that ∥V (b_{j}) − b_{j}∥ ≤ ε for all j. Equivalently, we must show that

(b_{1}, …, b_{n}) ∈ \overline{\{(V (b_{1}), …, V (b_{n})) : V ∈ 𝒞 (B, A)\}}.  (7.15)

By a Hahn–Banach separation argument, and because the set displayed above is convex, cf. Lemma 7.16 (ii), it will suffice to show the following: For each finite set b_{1}, …, b_{n} in B, for each f_{1}, …, f_{n} in A^{∗}, and for each ε > 0 there is V in 𝒞 (B, A) such |f_{j}(b_{j})−f_{j}(V (b_{j}))| ≤ ε for all j. Choose a cyclic representation π: A → B(H), a cyclic vector ξ in H, and elements c_{1}, …, c_{n} in π(A)′ ∩ B(H) such that f_{j}(b) = ⟨π(b)ξ, c_{j}^{∗}ξ⟩ for all b in B and for all j, cf. Lemma 7.17 (i). Let C be sub-C^{∗}-algebra of B(H)∩π(A)′ generated by c_{1}, …, c_{n}. Keeping π and ξ fixed, let φ_{V} be the positive functional on B ⊗ C defined by

φ_{V} (b ⊗ c) = ⟨π(V (b))ξ, c^{∗}ξ⟩,  b ∈ B, c ∈ C.  (7.16)

(A priori, φ_{V} defines a functional on the maximal tensor product B⊗_{max}C, but the maximal and the minimal tensor products on B ⊙ C coincide because B is nuclear.) Let 𝒦 be the weak–^{∗} closure of the cone \{φ_{V} : V ∈ \overline{𝒞} (B, A)\}. Observe that

|f_{j}(b_{j}) − f_{j}(V (b_{j}))| = |⟨π(b_{j})ξ, c_{j}^{∗}ξ⟩ − ⟨π(V (b_{j}))ξ, c_{j}^{∗}ξ⟩| = |φ_{ι}(b_{j} ⊗ c_{j}) − φ_{V} (b_{j} ⊗ c_{j})|.

Hence it will suffice to show that φ_{ι} belongs to 𝒦 .

We proceed to check that d^{∗}ρd belongs to 𝒦 for all ρ in 𝒦 and for all d in B ⊗ C. By continuity of the maps ρ ↦ d^{∗}ρd and d ↦ d^{∗}ρd (with d, respectively, ρ fixed) it suffices to show that d^{∗}φ_{V} d belongs to 𝒦 for d = ∑_{j=1}^{n} x_{j} ⊗ y_{j} in the algebraic tensor product and for V in 𝒞 (B, A). Now,

(d^{∗}φ_{V} d)(b ⊗ c) = ∑_{i,j=1}^{n} φ_{V} (x_{i}^{∗}bx_{j} ⊗ y_{i}^{∗}cy_{j}) = ∑_{i,j=1}^{n} ⟨π(V (x_{i}^{∗}bx_{j}))ξ, (y_{i}^{∗}cy_{j})^{∗}ξ⟩
= ∑_{i,j=1}^{n} ⟨π(V (x_{i}^{∗}bx_{j}))y_{j}ξ, c^{∗}y_{i}ξ⟩.

The vectors y_{j}ξ can be approximated arbitrarily well by vectors of the form π(a_{j})ξ for suitable elements a_{j} in A. Let W: B → A be given by W (b) = ∑_{i,j=1}^{n} a_{i}^{∗}V (x_{i}^{∗}bx_{j})a_{j}. Then W belongs to 𝒞 (B, A) by Lemma 7.16 (iii), and

φ_{W} (b ⊗ c) = ∑_{i,j=1}^{n} ⟨π(V (x_{i}^{∗}bx_{j}))π(a_{j})ξ, cπ(a_{i})ξ⟩.

We conclude that d^{∗}φ_{V} d belongs to the weak-^{∗} closure of the set of elements of the form φ_{W} with W in 𝒞 (B, A).

As in Lemma 7.17 (ii), let J be the closed two-sided ideal in B ⊗ C consisting of those elements z such that φ_{V} (z^{∗}z) = 0 for all V in \overline{𝒞} (B, A). By Lemma 7.17 (ii) the present lemma is proved if we can show that φ_{ι}(z^{∗}z) = 0 for all z in J. It is a consequence of a theorem of Blackadar, [1, Theorem 3.3], see also [14, Proposition 2.13], that because B is nuclear, then J is the closed linear span of the set of elementary tensors x ⊗ y in J. The left kernel L of φ_{ι}, consisting of all z in B ⊗ C such that φ_{ι}(z^{∗}z) = 0, is a closed linear subspace of B ⊗ C, and so it suffices to show that φ_{ι}(x^{∗}x ⊗ y^{∗}y) = 0 whenever x ∈ B and y ∈ C are such that x ⊗ y belongs to J. Now, if x ⊗ y belongs to J, then π(x) vanishes on the range of y. Indeed, by assumption there is V_{x} in \overline{𝒞} (B, A) such that V_{x}(x^{∗}x) = x^{∗}x, and by Lemma 7.16 (ii) there is for each a in A an element V_{x,a} in \overline{𝒞} (B, A) such that V_{x,a}(x^{∗}x) = a^{∗}x^{∗}xa. Hence

0 = φ_{V_{x,a}}(x^{∗}x ⊗ y^{∗}y) = ⟨π(a^{∗}x^{∗}xa)ξ, y^{∗}yξ⟩ = ∥π(x)yπ(a)ξ∥^{2}.

Since ξ is cyclic for the representation A, we conclude that π(x)yη = 0 for all η in H. It follows that

φ_{ι}(x^{∗}x ⊗ y^{∗}y) = ⟨π(x^{∗}x)ξ, y^{∗}yξ⟩ = ⟨π(x)yξ, π(x)yξ⟩ = 0,

as desired. □

Lemma 7.19 Let B be a sub-C^{∗}-algebra of a stable C^{∗}-algebra A and assume that the inclusion mapping B → A belongs to \overline{𝒞} (B, A). Then \overline{𝒞} (B, A) contains every approximately inner completely positive map from B to A.

Proof: Because \overline{𝒞} (B, A) is closed in the pointwise–norm topology, it suffices to show that each inner completely positive contraction T: B → A belongs to \overline{𝒞} (B, A). Let ι: B → A denote the inclusion mapping (that belongs to \overline{𝒞} (B, A)), and find d_{1}, …, d_{n} in A such that

T (b) = ∑_{j=1}^{n} d_{j}^{∗}bd_{j} = ∑_{j=1}^{n} d_{j}^{∗}ι(b)d_{j}.

It now follows from Lemma 7.16 (ii) (and the remark below Lemma 7.16) that T belongs to \overline{𝒞} (B, A). □

Lemma 7.20 Let A be a C^{∗}-algebra, let B be a sub-C^{∗}-algebra of M(A), and assume that every map in 𝒞_{0}(B, A) is approximately 1-step inner. Then every map in \overline{𝒞} (B, A) is approximately 1-step inner.

Proof: By the definition of being approximately 1-step inner, it suffices to show that every map in 𝒞 (B, A) is approximately 1-step inner. Take T in 𝒞 (B, A) and find S in 𝒞_{0}(B, A), a_{1}, …, a_{n} in M(A) and c_{1}, …, c_{n} in M(B) such that (7.12) holds. Put

C = max{∥a_{1}∥, ∥a_{2}∥, …, ∥a_{n}∥}.

Let F be a finite subset of B and let ε > 0. Choose a positive contraction f in B such that ∥fbf − b∥ ≤ ε/2 for all b in F. By the assumption that S is approximately 1-step inner, there is d in M(A) such that

∥S(c_{j}^{∗}fbfc_{i}) − d^{∗}c_{j}^{∗}fbfc_{i}d∥ ≤ ε/(2C^{2}n^{2}), b ∈ F, i, j = 1, … n.

Put e = ∑_{i=1}^{n} fc_{i}da_{i} ∈ M(A). Then

∥T (b) − e^{∗}be∥ ≤ ∥T (b) − T (fbf)∥ + ∥T (fbf) − e^{∗}be∥
≤ ε/2 + ∑_{i,j=1}^{n} ∥a_{j}^{∗}S(c_{j}^{∗}fbfc_{i})a_{i} − a_{j}^{∗}d^{∗}c_{j}^{∗}fbfc_{i}da_{i}∥
≤ ε/2 + ∑_{i,j=1}^{n} ∥a_{j}∥∥a_{i}∥ε(2C^{2}n^{2})^{−1} ≤ ε.

This shows that T is approximately 1-step inner. □

Theorem 7.21 Let A be a strongly purely infinite C^{∗}-algebra and let B be a nuclear, separable sub-C^{∗}-algebra of A. Then each approximately inner completely positive map from B to A is approximately 1-step inner.

Proof: It suffices to prove the theorem in the case where A is stable. Indeed, we may view any C^{∗}-algebra A as being a sub-C^{∗}-algebra of A ⊗ K. If V : B → A is approximately inner, then clearly so is V : B → A⊗K. Next, if {d_{n}}_{n=1}^{∞} is a sequence of elements in A⊗K satisfying d_{n}^{∗}bd_{n} → V (b), then we will also have g_{n}^{∗}bg_{n} → V (b), when g_{n} = e_{n}d_{n}e_{n} ∈ A for a suitable approximate unit {e_{n}}_{n=1}^{∞} for A.

Let ω be a free filter on ℕ, and view A as a sub-C^{∗}-algebra of the limit algebra A_{ω}. By Lemma 7.3 (i) it suffices to show that each approximately inner completely positive map from B to A_{ω} is approximately 1-step inner. The inclusion mapping ι: B → A_{ω} belongs to \overline{𝒞} (B, A_{ω}) by Lemma 7.18 and Proposition 7.13, and \overline{𝒞} (B, A_{ω}) therefore contains all approximately inner completely positive maps by Lemma 7.19. Proposition 7.14 (i) says that each map in 𝒞_{0}(B, A_{ω}) is 1-step inner. Lemma 7.20 then shows that each map in \overline{𝒞} (B, A_{ω}), and hence each approximately inner completely positive contraction from B to A_{ω}, is approximately 1-step inner. □

Other applications

In the definition of strongly purely infinite there is no assumptions made on the norm of the elements d_{1}, d_{2}. A norm estimate for d_{1} and d_{2} was proved in Lemma 5.6. Now we can show that d_{1} and d_{2} can be chosen to be contractions:

Corollary 7.22 Let A be a strongly purely infinite C^{∗}-algebra. For each

(\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) ∈ M_{2}(A)^{+}

and for each ε > 0 there are contractions d_{1}, d_{2} in A such that

∥ (\begin{matrix}d_{1}^{∗} & 0 \\ 0 & d_{2}^{∗} \end{matrix}) (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) (\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}) − (\begin{matrix}a & 0 \\ 0 & b \end{matrix}) ∥ ≤ ε.

If A = B_{ω} for some C^{∗}-algebra B and for some free filter ω on ℕ, then there are contractions d_{1}, d_{2} in A such that

(\begin{matrix}d_{1}^{∗} & 0 \\ 0 & d_{2}^{∗} \end{matrix}) (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) (\begin{matrix}d_{1} & 0 \\ 0 & d_{2} \end{matrix}) = (\begin{matrix}a & 0 \\ 0 & b \end{matrix})

Proof: The second statement (concerning the case A = B_{ω}) follows from the first statement and from Lemma 2.5. We proceed to prove the first statement of the corollary. Let ( \begin{matrix}a & x^{∗} \\ x & b \end{matrix}) in M_{2}(A)^{+} and let ε > 0 be given. Choose positive contractions e in C^{∗}(a) and f in C^{∗}(b) such that ∥a − eae∥ < ε/2 and ∥b − fbf∥ < ε/2. Put B_{0} = C^{∗}(a, b, x), put B = M_{2}(B_{0}), and put

C = \{ (\begin{matrix}c_{1} & 0 \\ 0 & c_{2} \end{matrix}) : c_{1} ∈ C^{∗}(a), c_{2} ∈ C^{∗}(b)\}.

Then C is an Abelian sub-C^{∗}-algebra of the separable C^{∗}-algebra B. Let ω be a free ultrafilter on ℕ. Apply Proposition 7.13 (and its proof) to find a contraction d in B_{ω} and an Abelian sub-C^{∗}-algebra D of B_{ω} such that d^{∗}bd belongs to D for all b in B and d^{∗}cd = c for all c in C. Write

d = (\begin{matrix}d_{11} & d_{12} \\ d_{21} & d_{22} \end{matrix}) ∈ B_{ω} = M_{2}((B_{0})_{ω}).

For each c_{1} in C^{∗}(a) and for each c_{2} in C^{∗}(b) we have

(\begin{matrix}c_{1} & 0 \\ 0 & 0 \end{matrix}) = d^{∗} (\begin{matrix}c_{1} & 0 \\ 0 & 0 \end{matrix}) d = (\begin{matrix}d_{11}^{∗}c_{1}d_{11} & d_{11}^{∗}c_{1}d_{12} \\ d_{12}^{∗}c_{1}d_{11} & d_{12}^{∗}c_{1}d_{12} \end{matrix})
(\begin{matrix}0 & 0 \\ 0 & c_{2} \end{matrix}) = d^{∗} (\begin{matrix}0 & 0 \\ 0 & c_{2} \end{matrix}) d = (\begin{matrix}d_{21}^{∗}c_{2}d_{21} & d_{21}^{∗}c_{2}d_{22} \\ d_{22}^{∗}c_{2}d_{21} & d_{22}^{∗}c_{2}d_{22} \end{matrix}) .

We conclude that

d_{11}^{∗}c_{1}d_{11} = c_{1},  d_{22}^{∗}c_{2}d_{22} = c_{2},  e^{1/2}d_{12} = 0,  f^{1/2}d_{21} = 0,  (7.17)

for all n. Using this, we get

d^{∗} (\begin{matrix}0 & 0 \\ f^{1/2}xe^{1/2} & 0 \end{matrix}) d = (\begin{matrix}0 & 0 \\ d_{22}^{∗}f^{1/2}xe^{1/2}d_{11}^{∗} & 0 \end{matrix}) ,

and this element commutes with diag(e^{1/2}, 0) and with diag(0, f^{1/2}). Hence

0 = d_{22}^{∗}f^{1/2}xe^{1/2}d_{11}^{∗}e^{1/2} = f^{1/2}d_{22}^{∗}f^{1/2}xe^{1/2}d_{11}^{∗}.  (7.18)

Put s = e^{1/2}d_{11}e^{1/2} and t = f^{1/2}d_{22}f^{1/2}. Then s and t are contractions, t^{∗}xs = 0 by (7.18), and

s^{∗}as = e^{1/2}d_{11}^{∗}(e^{1/2}ae^{1/2})d_{11}e^{1/2} = eae, t^{∗}bt = f^{1/2}d_{22}^{∗}(f^{1/2}bf^{1/2})d_{22}f^{1/2} = fbf

by (7.17). Write

s = π_{ω}(s_{1}, s_{2}, …), t = π_{ω}(t_{1}, t_{2}, …),

where s_{n} and t_{n} are contractions in B_{0} ⊆ A. Then

lim_{ω} ∥t_{n}^{∗}xs_{n}∥ = 0,  lim_{ω} ∥s_{n}^{∗}as_{n} − a∥ < ε/2,  lim_{ω} ∥t_{n}^{∗}bt_{n} − b∥ < ε/2.

We can therefore find n such that ∥t_{n}^{∗}xs_{n}∥ ≤ ε/2, ∥s_{n}^{∗}as_{n}−a∥ ≤ ε/2, and ∥t_{n}^{∗}bt_{n}−b∥ ≤ ε/2, and we can take d_{1} = s_{n} and d_{2} = t_{n}. □

8 Tensor products with O_{∞}

Corollary 8.3 below gives a McDuff type description of when a C^{∗}-algebra A satisfies A ≅ A⊗O_{∞}. This corollary is proved in [14] by the first named author. For the convenience of the reader, we give here the proof in a version taken almost vertabim from [19, Chapter 7] (a work in progress at the time where this is written).

Proposition 8.1 (An approximate intertwining) Let A and B be separable C^{∗}-alge-bras, and let φ: A → B be an injective ^{∗}-homomorphism. Suppose that there is a sequence {v_{n}}_{n=1}^{∞} of unitaries in M(B) such that

lim_{n→∞} ∥v_{n}φ(a) − φ(a)v_{n}∥ = 0,  lim_{n→∞} dist(v_{n}^{∗}bv_{n}, φ(A)) = 0,

for all a in A and all b in B. Then A and B are isomorphic, and there is an isomorphism ψ: A → B which is approximately unitarily equivalent to φ.

Proof: Let {a_{1}, a_{2}, a_{3}, … } and {b_{1}, b_{2}, b_{3}, … } be (countable) dense subsets of A, respectively, of B. Put v_{0} = 1, and find next, inductively, unitaries v_{n} in M(B) and elements a_{j,n} in A such that

∥v_{n}^{∗}(v_{n−1}^{∗} ⋯ v_{1}^{∗}v_{0}^{∗}b_{j}v_{0}v_{1} ⋯ v_{n−1})v_{n} − φ(a_{n,j})∥ ≤ 1/n,
∥v_{n}φ(a_{j}) − φ(a_{j})v_{n}∥ ≤ 2^{−n}, ∥v_{n}φ(a_{m,j}) − φ(a_{m,j})v_{n}∥ ≤ 2^{−n},

for j = 1, 2 …, n and m = 1, 2, …, n − 1. Being a Cauchy sequence the limit

ψ(a) = lim_{n→∞} v_{1}v_{2} ⋯ v_{n}φ(a)v_{n}^{∗} ⋯ v_{2}^{∗}v_{1}^{∗}

exists for all a in {a_{1}, a_{2}, a_{3}, … }, and hence for all a in A; and ψ: A → B is a ^{∗}-homo-morphism. Clearly ψ is injective, because ∥ψ(a_{j})∥ = ∥a_{j}∥ for all j, and approximately unitarily equivalent to φ. Observe that

∥ψ(a_{n,j}) − v_{1}v_{2} ⋯ v_{n}φ(a_{n,j})v_{n}^{∗} ⋯ v_{2}^{∗}v_{1}^{∗}∥ < 2^{−n},

and use this to deduce

∥b_{j} − ψ(a_{n,j})∥ ≤ 2^{−n} + ∥v_{n}^{∗}v_{n−1}^{∗} ⋯ v_{1}^{∗}v_{0}^{∗}b_{j}v_{0}v_{1} ⋯ v_{n−1}v_{n} − φ(a_{n,j})∥ ≤ 2^{−n} + 1/n.

Since ψ(A) is closed and {b_{1}, b_{2}, b_{3}, … } is dense in B we conclude that ψ(A) = B. □

Theorem 8.2 ( [14, Corollary 10.8]) Let A be a separable C^{∗}-algebra and let B be a unital and separable C^{∗}-algebra. Then A is isomorphic to A ⊗ B if

(i) there is a sequence {φ_{n}}_{n=1}^{∞} of unital injective ^{∗}-homomorphisms from B into M(A) satisfying ∥φ_{n}(x)a − aφ_{n}(x)∥ → 0 for all a in A and all b in B, and

(ii) the two ^{∗}-homomorphisms α, β: B → B ⊗ B given by α(b) = b ⊗ 1 and β(b) = 1 ⊗ b, b ∈ B, are approximately unitarily equivalent.

It is shown in [15], based on ideas from the paper [7] by Effros, that a C^{∗}-algebra B that satisfies (ii) must necessarily be simple and nuclear.

Proof: We shall show that conditions of Proposition 8.1 are satisfied with respect to the injective ^{∗}-homomorphism φ: A → A ⊗ B given by a ↦ a ⊗ 1_{B}. More specifically we shall for each finite subset F of A, for each finite subset G of B, and for each ε > 0 find a unitary v in M(A ⊗ B) such that

∥vφ(a) − φ(a)v∥ ≤ ε,  dist(v^{∗}(a ⊗ b)v, φ(A)) ≤ ε,  a ∈ F, b ∈ G.  (8.1)

Let ω be a free filter on ℕ. Let α_{n}: B → M(A ⊗ B) be given by α_{n}(b) = φ_{n}(b) ⊗ 1, and define α: B → M(A ⊗ B)_{ω} to be α(b) = π_{ω}(α_{1}(b), α_{2}(b), …). By the assumption on the ^{∗}-homomorphisms φ_{n} the image of α commutes with the image of φ (when viewing M(A ⊗ B) as a sub-C^{∗}-algebra of M(A ⊗ B)_{ω}). Let β: B → M(A ⊗ B) ⊆ M(A ⊗ B)_{ω} be given by β(b) = 1 ⊗ b. The images of α and β commute with each other and with the image of φ. The C^{∗}-algebra generated by α(B) and β(B) is isomorphic to B ⊗ B (because B is nuclear and simple). By (ii) there is a unitary w in C^{∗}(α(B), β(B)) with ∥w^{∗}β(b)w − α(b)∥ ≤ ε/(2C) for all b in G, where C = max{∥a∥ : a ∈ F }. Since w commutes with the image of φ we have

w^{∗}(a ⊗ b)w = w^{∗}φ(a)β(b)w = φ(a)w^{∗}β(b)w, a ∈ A, b ∈ B.

Write w = π_{ω}(w_{1}, w_{2}, …) where each w_{n} is a unitary in M(A ⊗ B). Then

lim_{ω} ∥w_{n}φ(a) − φ(a)w_{n}∥ = 0,  lim_{ω} ∥w_{n}^{∗}(a ⊗ b)w_{n} − φ(a)w_{n}^{∗}β(b)w_{n}∥ = 0,

lim_{ω} ∥w_{n}^{∗}β(b)w_{n} − α_{n}(b)∥ = ∥w^{∗}β(b)w − α(b)∥ ≤ ε/(2C).

for all a in A and all b in B. It follows that

∥w_{n}φ(a) − φ(a)w_{n}∥ ≤ ε,  ∥w_{n}^{∗}(a ⊗ b)w_{n} − φ(a)α_{n}(b)∥ ≤ ε,  a ∈ F, ∈ G,

for all n ∈ X for some X ∈ ω, and hence for at least one n. The element φ(a)α_{n}(b) belongs to A ⊗ 1 = φ(A) and therefore (8.1) is satisfied with v = w_{n}. □

Corollary 8.3 Let A be a separable C^{∗}-algebra. Then A is isomorphic to A ⊗ O_{∞} if and only if there is a sequence of unital ^{∗}-homomorphisms φ_{n}: O_{∞} → M(A) such that ∥φ_{n}(x)a − aφ_{n}(x)∥ → 0 for all x in O_{∞} and for all a in A.

Proof: The “if” part follows immediately from Theorem 8.2 together with the theorem of Lin and Phillips, [17], that any pair of unital ^{∗}-homomorphisms O_{∞} → O_{∞} ⊗ O_{∞} are approximately unitarily equivalent.

It is shown in Lin and Phillips paper, [17], that O_{∞} is isomorphic to ⊗_{n=1}^{∞}O_{∞}. As a consequence of this isomorphism we get a sequence ψ_{n}: O_{∞} → O_{∞} of unital ^{∗}-homo-morphisms satisfying ∥ψ_{n}(x)y − yψ_{n}(x)∥ → 0 for all x, y in O_{∞}. Let A be any C^{∗}-alge-bra. The C^{∗}-algebra M(A) ⊗ O_{∞} is a sub-C^{∗}-algebra of M(A ⊗ O_{∞}), and so φ_{n}(x) = 1⊗ψ_{n}(x) defines a sequence of^{∗}-homomorphisms from O_{∞} into M(A⊗O_{∞}) which satisfies ∥φ_{n}(x)a − aφ_{n}(x)∥ → 0 for all x in O_{∞} and for all a in A ⊗ O_{∞}. □

It follows easily from Corollary 8.3 that any C^{∗}-algebra that absorbs O_{∞} is approximately divisible.

Proposition 8.4 Let A be a separable C^{∗}-algebra which is either unital or stable. Then the following conditions are equivalent:

(i) A ≅ A ⊗ O_{∞},

(ii) for each natural number m, every approximately inner completely positive contraction from a sub-C^{∗}-algebra B of M_{m}(A) into M_{m}(A) is approximately 1-step inner,

(iii) there is sequence {d_{n}}_{n=1}^{∞} in M_{2}(A) such that

lim_{n→∞} ∥d_{n}^{∗} (\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) d_{n} − (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) ∥ = 0  (8.2)

for all a in A.

Proof: (i) ⇒ (ii). If A satisfies (i), then so does M_{m}(A) for all natural numbers m. We need therefore only consider the case m = 1. Let B be a sub-C^{∗}-algebra of A. It suffices to show that each n-step inner completely positive contraction V : B → A is approximately 1-step inner. Find d_{1}, …, d_{n} in M(A) with V (b) = ∑_{j=1}^{n} d_{j}^{∗}bd_{j}. By Corollary 8.3 we can find sequences of isometries {t_{j,k}}_{k=1}^{∞} in M(A) for j = 1, …, n satisfying

t_{1,k}t_{1,k}^{∗} + ⋯ + t_{n,k}t_{n,k}^{∗} ≤ 1,  lim_{k→∞} ∥t_{j,k}a − at_{j,k}∥ = 0,

for all a in A and all j. Put f_{k} = ∑_{j=1}^{n} t_{j,k}d_{j}. Then f_{k}^{∗}af_{k} → V (a), and this shows that V is approximately 1-step inner.

(ii) ⇒ (iii). Let B be the sub-C^{∗}-algebra of M_{2}(A) consisting of all elements of the form diag(a, 0) with a in A. Let V : B → M_{2}(A) be given by

V (\begin{matrix}a & 0 \\ 0 & 0 \end{matrix}) = (\begin{matrix}a & 0 \\ 0 & a \end{matrix}) .

The map V is approximately 2-step inner, and hence V is approximately 1-step inner if (ii) holds.

(iii) ⇒ (i). If A is stable, then by Lemma 7.4 (ii) we can take d_{n} in M_{2}(M(A)) satisfying (8.2) to be isometries of the form

d_{n} = (\begin{matrix}u_{n} & v_{n} \\ 0 & 0 \end{matrix}) .  (8.3)

Notice that (8.3) and (8.2) imply that u_{n} and v_{n} are isometries in M(A) with orthogonal range projections, that u_{n}^{∗}au_{n} → a and v_{n}^{∗}av_{n} → a for all a in A, and hence (by Lemma 7.9) that

lim_{n→∞} ∥u_{n}a − au_{n}∥ = 0,  lim_{n→∞} ∥v_{n}a − av_{n}∥ = 0  (8.4)

for all a in A.

We show next that we also can take the d_{n}’s to be of the form in (8.3) when A is unital. Put e = diag(1, 0) in M_{2}(A). Upon replacing d_{n} by ed_{n} we may assume that d_{n} = ed_{n}. Applying (8.2) to a = 1 we find that d_{n}^{∗}d_{n} → 1. Hence d_{n}^{∗}d_{n} is invertible (for n large enough), and d_{n} = v_{n}|d_{n}| for some isometry v_{n}. Since ∥v_{n} − d_{n}∥ → 0 and since v_{n} = ev_{n} we obtain (8.3) by replacing d_{n} by v_{n}.

Let E_{2} be the universal C^{∗}-algebra generated by two isometries with orthogonal range projections. The two isometries u_{n}, v_{n} have orthogonal range projections (for each fixed n), and so there is a unital ^{∗}-homomorphism ψ_{n}: E_{2} → M(A) mapping the two canonical generators of E_{2} onto u_{n} and v_{n}. By (8.4) we see that ψ_{n}(x)a − aψ_{n}(x) → 0 for all x in E_{2} and all a in A. The C^{∗}-algebra E_{2} has a unital sub-C^{∗}-algebra isomorphic to O_{∞}. Taking φ_{n}: O_{∞} → M(A) to be the restriction of ψ_{n}, an application of Corollary 8.3 yields that A is isomorphic to A ⊗ O_{∞}. □

Recall that a C^{∗}-algebra A is called O_{∞}-absorbing if A ⊗ O_{∞} is isomorphic to A.

Proposition 8.5 (Permanence properties)

(i) If A is an O_{∞}-absorbing C^{∗}-algebra, then so is every closed two-sided ideal in A.

(ii) If A is an O_{∞}-absorbing C^{∗}-algebra, then so is every quotient of A.

(iii) If A is a separable O_{∞}-absorbing C^{∗}-algebra, and if B is a hereditary sub-C^{∗}-algebra of A admitting an approximate unit consisting of projections, then B is O_{∞}-absorbing.

(iv) If A is an inductive limit of a sequence A_{1} → A_{2} → A_{3} → ⋯ of separable C^{∗}-al-gebras A_{n}, each of which absorbs O_{∞}, and if each connecting map A_{n} → A_{n+1} is non-degenerate ^{3}, then A absorbs O_{∞}.

Proof: (i) and (ii). Write A = A_{0}⊗O_{∞} for some C^{∗}-algebra A_{0}. Suppose that I is an ideal in A. Because O_{∞} is exact and simple, I = I_{0} ⊗ O_{∞} for some ideal I_{0} in I, cf. [1, Theorem 3.3] and [14, Proposition 2.13]. Hence I is O_{∞}-absorbing, and A/I ≅ (A_{0}/I_{0})⊗O_{∞} because O_{∞} is exact.

(iii). Let {p_{k}}_{k=1}^{∞} be an increasing approximate unit for B where each p_{k} is a projection. Observing that there are unital embeddings of E_{2} into O_{∞} and vice versa, we may use Corollary 8.3 — with E_{2} in the place of O_{∞} — to find an asymptotically central sequence of unital ^{∗}-homomorphisms φ_{n}: E_{2} → M(A). Let s, t be the two canonical generators of E_{2}, i.e., s, t are isometries with ss^{∗} ⊥ tt^{∗}. Put p_{0} = 0 and put q_{k} = p_{k} − p_{k−1} for all k in ℕ. Then 1 = ∑_{j=1}^{∞} q_{k} in M(B) (the sum is strictly convergent). Also,

lim_{n→∞} ∥φ_{n}(s)q_{k} − q_{k}φ_{n}(s)∥ = lim_{n→∞} ∥φ_{n}(t)q_{k} − q_{k}φ_{n}(t)∥ = 0.

The relations satisfied by s, t are stable, and we can therefore for each k in ℕ find sequences {s_{n,k}}_{n=1}^{∞} and {t_{n,k}}_{n=1}^{∞} of isometries in q_{k}Aq_{k} = q_{k}Bq_{k} such that s_{n,k}s_{n,k}^{∗} ⊥ t_{n,k}t_{n,k}^{∗}, and such that

lim_{n→∞} ∥q_{k}φ_{n}(s)q_{k} − s_{n,k}∥ = lim_{n→∞} ∥q_{k}φ_{n}(t)q_{k} − t_{n,k}∥ = 0.

For each n there is a (unique) unital ^{∗}-homomorphism ψ_{n}: E_{2} → M(B) that satisfies

ψ_{n}(s) = ∑_{k=1}^{∞} s_{n,k}, ψ_{n}(t) = ∑_{k=1}^{∞} t_{n,k},

3A ^{∗}-homomorphism φ: D → E is said to be non-degenerate if φ(D)Eφ(D) is dense in E.

(the sums are strictly convergent). We have ∥ψ_{n}(x)b − bψ_{n}(x)∥ → 0 for each x in E_{2} and for each b in B. (To see this, consider first b in p_{k}Bp_{k} for some k.) Hence the conditions of Corollary 8.3 are satisfied, and so B is isomorphic to B ⊗ O_{∞}.

(iv). Let μ_{n}: A_{n} → A be the inductive limit map. Then μ_{n} is non-degenerate for each n and therefore extends to a unital ^{∗}-homomorphism μ̂_{n}: M(A_{n}) → M(A). Use Corollary 8.3 to find a sequence of unital ^{∗}-homomorphisms φ_{k,n}: O_{∞} → M(A_{n}) such that

lim_{k→∞} ∥φ_{k,n}(x)a − aφ_{k,n}(x)∥ = 0, x ∈ O_{∞}, a ∈ A_{n}.

Let ψ_{k,n}: O_{∞} → M(A) be the composition μ̂_{n} ∘ φ_{k,n}. Then

lim_{k,n→∞} ∥ψ_{k,n}(x)a − aψ_{k,n}(x)∥ = 0, x ∈ O_{∞}, a ∈ A.

Corollary 8.3 now shows that A absorbs O_{∞}. □

Theorem 8.6 Let A be a separable C^{∗}-algebra. If A is strongly purely infinite and nuclear, and if A is either stable or has an approximate unit consisting of projections, then A is isomorphic to A ⊗ O_{∞}. Conversely, for any C^{∗}-algebra A, if A is isomorphic to A ⊗ O_{∞}, then A is strongly purely infinite.

Proof: Assume first that A is isomorphic to A ⊗ O_{∞}. Then, by Corollary 8.3, there are sequences {u_{n}}_{n=1}^{∞} and {v_{n}}_{n=1}^{∞} of isometries in M(A) such that u_{n}u_{n}^{∗} ⊥ v_{n}v_{n}^{∗} and

lim_{n→∞} ∥u_{n}a − au_{n}∥ = 0,  lim_{n→∞} ∥v_{n}a − av_{n}∥ = 0

for all a in A. Hence

(\begin{matrix}u_{n}^{∗} & 0 \\ 0 & v_{n}^{∗} \end{matrix}) (\begin{matrix}a & x^{∗} \\ x & b \end{matrix}) (\begin{matrix}u_{n} & 0 \\ 0 & v_{n} \end{matrix}) → (\begin{matrix}a & 0 \\ 0 & b \end{matrix})

for all a, b, x in A, and this shows that A is strongly purely infinite, cf. Lemma 5.2.

Suppose next that A is strongly purely infinite, nuclear, and that A either has an approximate unit consisting of projections or is stable. If A ⊗ K absorbs O_{∞}, then so does A by Proposition 8.5 (in either case), and A ⊗ K is strongly purely infinite by Proposition 5.11 (iii). It therefore suffices to consider the case, where A is stable.

Let B and V : B → M_{2}(A) be as in the proof of (ii) ⇒ (iii) of Proposition 8.4. Being isomorphic to A, B is nuclear, and V is approximately 2-step inner. Thus V is approximately 1-step inner by Theorem 7.21, and so A is isomorphic to A⊗O_{∞} by Proposition 8.4.

□

9 Summary and open problems

The main results of this paper are contained in Theorem 7.21 and in the theorem below: Theorem 9.1 Consider the following six properties of a separable C^{∗}-algebra A:

(i) A ≅ A ⊗ O_{∞}. (ii) A is strongly purely infinite ^{4}. (iii) A is purely infinite ^{5}. (iv) A is weakly purely infinite ^{6}.

(v) A_{ω} is traceless ^{7} for some free filter ω on ℕ. (vi) A is traceless. Then

(i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇔ (v) ⇒ (vi),

and (ii) ⇒ (i) if A is nuclear, and either stable or with an approximate unit consisting of projections, (iii) ⇒ (ii) if A is either simple, of real rank zero, or approximately divisible ^{8}, (iv) ⇒ (iii) if A has the global Glimm property ^{9}, and in particular if A is either simple, of real rank zero, or approximately divisible, (vi) ⇒ (v) if A is approximately divisible, Proof: The implications (i) ⇒ (ii) and (ii) ⇒ (i) are treated in Theorem 8.6.

(ii) ⇒ (iii) is proved in Proposition 5.4. It is shown in Corollary 6.9 that (iii) ⇒ (ii) if A is of real rank zero, and hence in particular if A is simple, because all simple purely infinite C^{∗}-algebras are of real rank zero. Proposition 5.14 shows that (iii) ⇒ (ii) for approximately divisible C^{∗}-algebras.

4A definition of being strongly purely infinite is given in Definition 5.1.

5A definition of being purely infinite is given in Definition 3.4.

6A definition of being weakly purely infinite is given in Definition 4.3.

7Traceless C^{∗}-algebras are defined in Definition 4.2.

8A definition of being approximately divisible can be found in Definition 4.1.

9A definition of the global Glimm property can be found in Definition 4.12

(iii) ⇒ (iv) follows from [16, Theorem 4.16], cf. the remark below Definition 4.3. The implication (iv) ⇒ (iii) is treated in Proposition 4.15, Corollary 4.16 and Proposition 4.18.

The equivalence (iv) ⇔ (v) is proved in Theorem 4.8 (i), and by Theorem 4.8 (ii) we have (iv) ⇒ (vi). Finally, the implication (vi) ⇒ (iv) is proved to hold for approximately divisible C^{∗}-algebras in [16, Theorem 5.9]. □

The implication (ii) ⇒ (i) does not hold in general. There is in [11, Theorem 1.4] an example of a simple, unital, purely infinite, separable C^{∗}-algebra A which is not approximately divisible. Hence A is not isomorphic to A ⊗ O_{∞} (as noted below Corollary 8.3). We do not have counter examples to any other implication of Theorem 9.1.

We summarize the situation when A is either simple, approximately divisible or of real rank zero. The conclusions of Corollary 9.2 below was obtained in 1994 by the first named author (and is published in [15]).

Corollary 9.2 Let A be a simple, separable, nuclear C^{∗}-algebra. Then A ≅ A ⊗ O_{∞} if and only if A is purely infinite.

Proof: If A is simple and purely infinite, then A is either stable or unital by [22]. Hence conditions (i) – (v) in Theorem 9.1 are equivalent for A. □

Corollary 9.3 Let A be a separable, nuclear C^{∗}-algebra that is either stable or admits an approximate unit consisting of projections. Then the following conditions are equivalent:

(i) A ≅ A ⊗ O_{∞},

(ii) A is approximately divisible and purely infinite,

(iii) A is approximately divisible and traceless.

Proof: The implications (iii) ⇒ (ii) ⇒ (i) follows from Theorem 9.1. If (i) holds then A is approximately divisible (this is easy to see from Corollary 8.3), and A is traceless (by Theorem 9.1). □

It is a consequence of the next corollary that all non-degenerate (simple and non-simple) Cuntz–Krieger algebras absorb O_{∞}. A Cuntz–Krieger algebra O_{A} is non-degenerate if its corresponding matrix has no irreducible component which is a permutation matrix. Nondegenerate Cuntz–Krieger algebras have real rank zero and all their non-zero projections are properly infinite.

Corollary 9.4 Let A be a separable, nuclear C^{∗}-algebra of real rank zero. Then the following conditions are equivalent:

(i) A ≅ A ⊗ O_{∞},

(ii) A is strongly purely infinite,

(iii) A is purely infinite,

(iv) A is weakly purely infinite,

(v) A_{ω} is traceless for all free filters ω on ℕ,

(vi) all non-zero projections in A are properly infinite.

Proof: Each C^{∗}-algebra of real rank zero admits an approximate unit consisting of projections, and so conditions (i) – (v) of Theorem 9.1 (and hence of the present corollary) are equivalent for separable, nuclear C^{∗}-algebras of real rank zero. The implication (iii) ⇒ (vi) follows from [16, Theorem 4.16] (saying that all non-zero positive elements in a purely infinite C^{∗}-algebra are properly infinite), and (vi) ⇒ (iii) follows from [16, Proposition 4.7] (since every hereditary sub-C^{∗}-algebra of a quotient of a C^{∗}-algebra of real rank zero again is of real rank zero). □

There are still several unanswered questions regarding the structure of infinite C^{∗}-algebras. We list some of the more intriguing of these open problems below:

Question 9.5 (Three kinds of pure infiniteness) Do we have

A strongly purely infinite ⟺ A purely infinite ⟺ A weakly purely infinite

for all C^{∗}-algebras A?

The two right-implications “⇒” in Question 9.5 are true and easy to prove (see Theorem 9.1). All weakly purely infinite C^{∗}-algebras are purely infinite if and only if all weakly purely infinite C^{∗}-algebras have the global Glimm property (see Proposition 4.15), or, equivalently if and only if all non-zero projections in a weakly purely infinite C^{∗}-algebra are infinite (see Proposition 4.19).

We do not know if the multiplier algebra of a purely infinite C^{∗}-algebra is purely infinite. We do not even know if its unit is infinite. Following the proof of Proposition 4.11, this would follow if we have an affirmative answer to the following:

Question 9.6 (Sums of properly infinite elements) Let a and b be positive elements in a C^{∗}-algebra A such that, for some δ > 0, the elements (a−ε)_{+} and (b−ε)_{+} are properly infinite for all ε ∈ [0, δ]. Does it follow that their sum a + b is properly infinite?

A partial answer to this question can be found in Lemma 4.9. Note that our assumption on a and b is slightly stronger than just asking these two elements to be properly infinite. For example, any strictly positive element a in the C^{∗}-algebra K of compact operators on an infinite dimensional Hilbert space is properly infinite (see [16, Proposition 3.7]), but (a − ε)_{+} is not properly infinite for ε > 0.

There are strongly purely infinite C^{∗}-algebras that are neither stable nor have an approximate unit consisting of projections. Take for example C_{0}(ℝ) ⊗ O_{∞} (which by the way clearly is O_{∞}-absorbing). The implication (ii) ⇒ (i) of Theorem 9.1 therefore does not apply to all separable, nuclear C^{∗}-algebras. Nonetheless, we have no (nuclear, separable) counter example to this implication. We therefore ask:

Question 9.7 (Morita equivalence of O_{∞}-absorbsion) Suppose that A and B are stably isomorphic C^{∗}-algebras and that A ≅ A ⊗ O_{∞}. Does it follow that B ≅ B ⊗ O_{∞}?

One can answer Question 9.7 in the affirmative if one can prove that the inductive limit of any sequence A_{1} → A_{2} → ⋯ of O_{∞}-absorbing, separable C^{∗}-algebras (not necessarily with non-degenerate connecting mappings) is O_{∞}-absorbing.

We know that extensions of weakly purely infinite and of purely infinite C^{∗}-algebras again are weakly purely infinite, respectively, purely infinite. What is the situation for strongly purely infinite C^{∗}-algebras?

Question 9.8 (Extensions of O_{∞}-absorbing C^{∗}-algebras) Given an extension

\xymatrix{0 \ar[r] & I \ar[r] & A \ar[r] & B \ar[r] & 0}

of C^{∗}-algebras. Suppose that I and B are strongly purely infinite. Does it follow that A is strongly purely infinite? Can one conclude that A ≅ A ⊗ O_{∞} if we know that I ≅ I ⊗ O_{∞} and B ≅ B ⊗ O_{∞}?

It is shown in Proposition 8.5 that I and B are O_{∞}-absorbing if A is O_{∞}-absorbing.

Question 9.9 (Traceless C^{∗}-algebras) Let ω be (any) free filter on ℕ, and let A be a C^{∗}-algebra. Can one conclude that A_{ω} is traceless if A is traceless?

With a positive answer to Question 9.9, a C^{∗}-algebra is weakly purely infinite if and only if it is traceless. This would have particularly interesting consequences for simple C^{∗}-alge-bras. Indeed, all simple, weakly purely infinite C^{∗}-algebras are purely infinite, and a simple C^{∗}-algebra is traceless if it admits no (densely defined) quasitrace. Hence, an affirmative answer to Question 9.9 would also answer in the affirmative the old problem whether all simple C^{∗}-algebras must either have a quasitrace or be purely infinite. It is the guess of the authors that there could be a (non-nuclear) counter example to Question 9.9.

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Humboldt Universität zu Berlin, Institut für Mathematik, Unter den Linden 6, D-10099 Berlin, Germany, and

Universität Münster, Mathematisches Institut, Einsteinstraße, D-48149 Münster, Germany

E-mail address: kirchbrg@mathematik.hu-berlin.de

Department of Mathematics, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

E-mail address: rordam@math.ku.dk
Internet home page: www.math.ku.dk/ rordam/