Classification of Simple C^{∗}-algebras of Tracial Topological Rank Zero ^{∗}

Huaxin Lin Department of Mathematics University of Oregon Eugene, Oregon 97403-1222

Abstract We give a classification theorem for unital separable simple nuclear C^{∗}-algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if A and B are two such C^{∗}-algebras and

(K_{0}(A), K_{0}(A)_{+}, [1_{A}], K_{1}(A))
≅ (K_{0}(B), K_{0}(B)_{+}, [1_{B}], K_{1}(B)),

then A ≅ B.

1 Introduction

There has been rapid progress in recent years in the program of classification of nuclear C^{∗}-algebras, in particular, in the case that C^{∗}-algebras are simple and have real rank zero. For example, Elliott and Gong ([EG2]) show that simple AH C^{∗}-algebras of real rank zero (with slow dimension growth) can be classified (up to isomorphisms) by their scaled ordered K_{0} and K_{1}. Kirchberg and Phillips ([K] and [P1]) show that separable nuclear purely infinite simple C^{∗}-algebras satisfying the Universal Coefficient Theorem (UCT) can also be classified by their K-theory. These two results are the highlights of the program of classification of nuclear C^{∗}-algebras initiated by G. A. Elliott (see [Ell2]). More recent results in the case of C^{∗}-algebras which have real rank one can be found in a paper of Elliott, Gong and Li ([EGL]). Many other mathematicians have contributed to this program. (See for example, [BEEK], [BBEK], [D1], [D2], [DE2], [DL1], [DL2], [DG], [Ell1], [Ell2], [Ell3], [Ell4], [EG1], [EE], [EG2], [EGLP], [EGS], [ER], [ES], [G1], [G2], [JS], [BK1], [BK2], [Li1], [Li2], [Ln1]-[Ln9], [Ln11]- [Ln12], [LP1], [LP2], [LP3], [LS], [LqP], [NT], [P1], [P2], [Ro1], [Ro2], [Ro3], [ER], [Su1], [Su2], [Th1],

[Th2] - an incomplete list). The application of the classification program began to spread to different fields, in particular, to the study of dynamical systems. In this paper we will only consider the case that C^{∗}-algebras are simple and have real rank zero and stable rank one.

∗Research partially supported by NSF grants DMS 9801482. AMS 1991 Subject Classification Numbers: Primary 46L05, 46L35. Key words: Classification of Simple C^{∗}-algebras, tracial topological rank zero.

During the development of classification of nuclear separable purely infinite simple C^{∗}-algebras, a special class of inductive limits of finite direct sums of even Cuntz-algebras was first classified by Rørdam ([Ro1]). Rørdam’s paper kicked off a series of work on classification of nuclear purely infinite simple C^{∗}-algebras. More general examples of purely infinite simple C^{∗}-algebras were classified, such as simple C^{∗}-algebras which are inductive limits of O_{n} tensored with C(S^{1}) ([LP1]) and those C^{∗}- algebras arising as crossed products of stable simple AF-algebra with a ℤ-action determined by an automorphisms ([Ro2]; see also [ER] and other papers listed above). These purely infinite simple C^{∗}-algebras exhaust all possible K-theory. Later Kirchberg and Phillips classified separable nuclear purely infinite simple C^{∗}-algebras (satisfying the UCT) without assuming an inductive limit structure or other special forms.

Naturally, it is important to classify a class of (unital) separable nuclear stably finite simple C^{∗}-algebras without assuming they are inductive limits of certain simple forms.

In [Ln7] and [Ln10], we introduced the notion of tracial topological rank. Recall that a simple unital C^{∗}-algebra A is said to have tracial topological rank zero (written T R(A) = 0), if for any ε > 0, any finite subset F ⊂ A and any a ∈ A_{+} \ {0}, there exists a nonzero projection p and a finite dimensional C^{∗}-subalgebra B ⊂ A with 1_{B} = p such that

(1) ∥px − xp∥ < ε,

(2) dist(pxp, F) < ε for all x ∈ F and

(3) 1 − p is equivalent to a projection in Her(a) = \overline{aAa}.

Roughly speaking, if an AF-algebra could be described as C^{∗}-algebra which can be approximated by finite dimensional C^{∗}-subalgebras in norm, then a C^{∗}-algebra A with T R(A) = 0 could be described as C^{∗}-algebras which can be approximated by finite dimensional C^{∗}-subalgebras in “measure,” or rather in trace. This also resembles a key structure of unital separable nuclear purely infinite simple C^{∗}-algebra : Every such C^{∗}-algebra A has the following property: for any ε > 0 any finite subset F ⊂ A, there exists a nonzero projection p and a C^{∗}-subalgebra B ≅ O_{2} such that

(1’) ∥px − xp∥ < ε and

(2’) dist(pxp, B) < ε for all x ∈ F.

((3’) Note that 1 − p is equivalent to a projection in Her(a) = \overline{aAa} since A is purely infinite simple.) This property plays a very important role in the classification theorem of Kirchberg and Phillips.

It is shown (in [Ln7]) that a unital separable simple C^{∗}-algebra A with T R(A) = 0 has real rank zero, stable rank one, weakly unperforated K_{0}(A) and is quasidiagonal. Simple AH-algebras with slow dimension growth have stable rank one (see [DNNP]) and are also quasidiagonal and have weakly unperforated K_{0}-groups. It is proved in [EG2] that all simple AH-algebras with slow dimension growth and with real rank zero have tracial topological rank zero. It seems that the class of simple C^{∗}-algebras with tracial topological rank zero is the right substitute for class of quasidiagonal simple C^{∗}-algebras with real rank zero, stable rank one and weakly unperforated K_{0}. It was shown in [Ln9] that unital separable simple C^{∗}-algebras which are inductive limits of type I C^{∗}-algebras with real rank zero, stable rank one, weakly unperforated K_{0} and unique tracial states have tracial topological rank zero. N. C. Phillips ([LqP]) shows certain simple crossed products arising from smooth minimal diffeomorphisms have tracial topological rank zero. In studying non-commutative shift, recently, A. Kishimoto ([Ki]) shows that the crossed product of the two sided infinite tensor product ⊗_{ℤ}A by the shifts has tracial topological rank zero.

Tracial topological rank zero (or TAF) for simple C^{∗}-algebras was introduced for the purpose of applying the so-called uniqueness theorem originally established in [Ln6]. An immediate consequence of the uniqueness theorem and the notion of TAF is that a unital separable simple nuclear C^{∗}-algebras of tracial topological rank zero with the same K-theory as that of UHF-algebra Q with K_{0}(Q) = ℚ is actually isomorphic to Q ([Ln7]). Further attempts were made to classify all separable unital simple nuclear C^{∗}-algebras with tracial topological rank zero. For example, it was shown that if A is a separable unital simple nuclear C^{∗}-algebras with RR(A) = 0 and K_{0}(A) = ℚ ⊕ kerτ, where τ is the unique trace (see [Ln8] and [Ln11]), then it is isomorphic to any other such C^{∗}-algebras. This was also obtained in [DE2] in the case that K_{0}(A) = ℚ. It is also shown in [Ln8] that A is a unital separable simple nuclear C^{∗}-algebra with T R(A) = 0 such that K_{0}(A) is a subring of ℚ (such as ℤ[1/2]) and K_{1}(A) is torsion free then it is isomorphic to a unital A𝕋-algebra with the same K-theory.

In this paper by combining several previous results, we give a classification theorem for unital separable simple nuclear C^{∗}-algebras with tracial topological rank zero which satisfy the UCT. C^{∗}- algebras in this class are classified (up to isomorphism) by their scaled ordered K_{0}-groups and K_{1}- groups.

The proof of classification theorem (5.2) as before consists of a uniqueness theorem, an existence theorem and an intertwining argument of Elliott. With the results in [Ln6] and the introduction of the notion of tracial topological rank (or TAF), the desired uniqueness theorem (5.1) was established in [Ln8]. However, the version of the (half) existence theorem (4.1) in [Ln8] (see also [DE2]) does not provide a true existence theorem for the purpose of classification of simple nuclear C^{∗}-algebras with tracial topological rank zero. Extra conditions on the order structure of K_{0}(A) were needed in [Ln8],

[DE2] and [Ln11]. The main technical result in this paper is to recover the order information lost in 4.1.

In Section 2 we show that a nuclear separable simple C^{∗}-algebras with tracial topological rank zero has an approximate structure which is similar to a standard inductive construction. Section 3 gives a special version of an interpolation property. In section 4 we establish the required existence theorem, by applying section 2 and 3 together with the half existence theorem (4.1 from [Ln8]) and a cutting lemma from our previous result in [Ln12]. After we establish the required existence theorem, the classification theorem can be established using the results already established in [Ln8].

Acknowledgements This papar was written while the author was visiting East China Normal University in Summer 2000 and visiting Mathematical Science Research Institute at Berkeley. Research at MSRI is supported in part by NSF grant DMS-9701755. It is also partially supported by a grant from the NSF, and the Zhi-Jiang Scholarship at the ECNU. The author benefited from conversation with Guihua Gong and N. C. Phillips. He would also like to thank G. A. Elliott for inspiration and encouragement during the work of this and related research.

2 An Approximation Construction

Recall that a unital simple C^{∗}-algebra A is said to have tracial topological rank zero (written T R(A) = 0), if for any finite subset F ⊂ A any ε > 0 and any nonzero element a ∈ A_{+}, there exists a finite dimensional C^{∗}-subalgebra B ⊂ A with p = 1_{B} such that

(1) ∥px − xp∥ < ε,

(2) dist(pxp, B) < ε for all x ∈ F and

(3) 1 − p is equivalent to a projection in \overline{aAa}.

2.1 We begin by reviewing a construction of C^{∗}-algebras of real rank zero. This construction appeared in several places ([BBEK], [Go] and [D1]). We present a construction of Dadarlat’s version. Recall that a C^{∗}-algebra B is said to be residually finite dimensional (RFD), if B has a separating family of finite dimensional irreducible representations. If B is separable RFD C^{∗}-algebra, then there is a sequence of finite dimensional irreducible representations {π_{n}} of B which separates A. Let B be such a RFD C^{∗}-algebra, {π_{n}} be a separating sequence of finite dimensional irreducible representations and {x_{n}} be a dense sequence of the unit ball in B. Suppose that the rank of π_{n} is r(n).

Let A_{1} = B. Set R(2) = 1 + k(1)r(1) and A_{2} = M_{R(2)}, Define a homomorphism h_{1} : A_{1} → A_{2} by

h_{1}(a) = diag(a, π_{1}(a), ..., π_{1}(a)) for a ∈ A_{1},

where π_{1} repeats k(1) times. Let A_{3} = M_{R(3)}(A_{2}), where R(3) = 1 + k(2, 1)r(1) + k(2, 2)r(2), and define h_{2} : A_{2} → A_{3} by

h_{2}(a) = diag(a, π̃_{1}(a), ...., π̃_{1}(a), π̃_{2}(a), ..., π̃_{2}(a))

for a ∈ A_{2}, where π̃_{1} = π_{1} ⊗ id_{k(1)r(1)}, π̃_{2} = π_{2} ⊗ id_{k(1)r(1)} and π̃_{1}(a) repeats k(2, 1) times and π̃_{2}(a) repeats k(2, 2) times. Continuing this fashion, we obtain a sequence of homomorphisms h_{n} : A_{n} → A_{n+1} and set A = lim_{n}(A_{n}, h_{n}). If k(n, i) → ∞ as n → ∞ for each i, as in [Ln7], A is simple and T R(A) = 0. Let ϕ_{n,n+m} = h_{m+n} ∘ ⋯ h_{n}. Then

ϕ_{1,n}(a) = ( \begin{matrix}a \\ & Φ_{1}^{(n)}(a) \\ & & Φ_{2}^{(n)}(a) \\ & & & . & . & . \\ & & & & & & Φ_{n}^{(n)}(a) \end{matrix} )

for a ∈ A_{1}, where Φ_{1}^{(n)}(a) is s(1, n) many copies of π_{1}(a), Φ_{2}(a) is s(2, n) many copies of π_{2}(a),..., Φ_{n} is s(n, n) copies of π_{n}(a). From the construction, inf_{n}{s(k, n)r(k)/ ∑_{i=1}^{n} s(i, n)r(i)} is positive for each k and ∑_{i=1}^{n} s(i, n)r(i)/(1 + ∑_{i=1}^{n} s(i, n)r(i)) → 1.

It is clear that this construction can be made even more general (for example, A_{k} is a finite direct sum of matrix algebra over B with different size) but still have the similar properties. It is clear while this construction is special, it is typical. But much more is true.

In what follows we will show that every nuclear separable simple C^{∗}-algebra A with T R(A) = 0 has a similar structure, or at least an approximated one. We need to replace homomorphisms by approximate multiplicative morphisms and π_{k} will be replaced by those whose ranges are contained in finite dimensional C^{∗}-algebras.

Lemma 2.2 (cf. 5.3 in [Ln7]) Let A be a separable unital nuclear simple C^{∗}-algebra with RR(A) = 0. Let {A_{n}} be a sequence of nuclear RFD C^{∗}-algebras such that A_{n} ⊂ A_{n+1} and A is the closure of ∪_{n}A_{n}. Fix ε_{n} > 0 which decreases to zero, a sequence of nonzero homomorphisms h_{n} from A_{n} to a finite dimensional C^{∗}-algebra F_{n} and a finite subset F_{n} ⊂ A_{n}. There exists a sequence of non-zero projections {e_{n}} and a sequence of monomorphisms h′_{n} : F_{n} → A with h′_{n}(1_{F_{n}}) = e_{n} satisfying the following:

(1) ∥e_{n}x − xe_{n}∥ < ε_{n} and

(2) ∥h′_{n} ∘ h_{n}(x) − e_{n}ae_{n}∥ < ε_{n} for all x ∈ F_{n}.

Proof: Fix an integer n. Let ε > 0 and a finite subset F_{n} = {x_{1}, x_{2}, ..., x_{k}} ⊂ A_{n} be given. Let I = kerh_{n} ⊂ A_{n} and B be the hereditary C^{∗}-subalgebra of A generated by I. Let C be the closure of A_{n} + B. We will show that for any c ∈ C, bc, cb ∈ B for all b ∈ B. It suffices to consider those elements c ∈ A_{n}. Note that B is the closure of IAI. Fix b ∈ B. For any ε > 0, there is a positive element e ∈ I such that

∥eb − b∥ < ε.

We also have ce ∈ I ⊂ B and ceb ∈ B for all c ∈ A_{n}. Therefore

∥cb − ceb∥ < ε.

This implies that cb ∈ B. Similarly, bc ∈ B. Thus C is a C^{∗}-subalgebra containing B as a (closed) ideal. Since C/B = A_{n}/B ∩ I = A_{n}/I ≅ F_{n}, we see that C/B ≅ F_{n}. To save notation, we may write F_{n} = C/B. Let π : C → C/B be the quotient map. Hence we may identify h_{n} with π|_{A_{n}}.

By 5.2 in [Ln8], every projection in F_{n} lifts to a projection in C. Note that B has real rank zero, so it admits an approximate identity consisting of projections. Now a standard argument (see Lemma 9.8 in [Ef]) shows that C contains a finite dimensional C^{∗}-subalgebra F′_{n} such that F′_{n} ≅ F_{n} and π(F′_{n}) = F_{n}. Thus there is a monomorphism h̃_{n} : F_{n} → F′_{n} such that π ∘ h̃_{n} = id_{F_{n}}. In particular, h̃_{n} ∘ π(x) − x ∈ B for all x ∈ C.

Let q = 1_{F′_{n}}, then (1 − q)C(1 − q) = (1 − q)B(1 − q). Write F′_{n} = M_{n_{1}} ⊕ M_{n_{2}} ⊕ ⋯ ⊕ M_{n_{k}} and assume that d_{i} ∈ M_{n_{i}} are minimal projections in M_{n_{i}}. Let {e_{0}^{(m)}} be an approximate identity for (1−q)B(1−q) consisting of projections. Let {c_{i}^{(m)}} be an approximate identity for d_{i}Bd_{i} consisting of projections, i = 1, 2, ..., k. Write the hereditary C^{∗}-subalgebra of C generated by M_{n_{i}} as M_{n_{i}}(d_{i}Cd_{i}). Put e_{i}^{(m)} = diag(c_{i}^{(m)}, c_{i}^{(m)}, ..., c_{i}^{(m)}) (c_{i}^{(m)} repeats n_{i} times) and E_{m} = e_{0}^{(m)} + ∑_{i=1}^{k} e_{i}^{(m)}. Then it is clear that {E_{m}} forms an approximate identity for B consisting of projections. Furthermore, by construction,

E_{m}x = xE_{m}

for each x ∈ F′_{n}. Since h̃_{n} ∘ h_{n}(x_{i}) − x_{i} ∈ B, for sufficiently large m,

∥(1 − E_{m})(h̃_{n} ∘ h_{n}(x_{i}) − x_{i})(1 − E_{m})∥ < ε/2, i = 1, 2, ..., k.

Moreover, since E_{m}(h̃_{n} ∘ h_{n}(x_{i})) = (h̃_{n} ∘ h_{n}(x_{i}))E_{m},

∥(1 − E_{m})x_{i} − x_{i}(1 − E_{m})∥ < ε/2, i = 1, 2, ..., k,

and h′_{n}(a) = (1 − E_{m})h̃_{n}(a)(1 − E_{m}) (a ∈ F_{n}) defines a homomorphism. Let e_{n} = (1 − E_{m}). Then h′_{n}(1_{F_{n}}) = e_{n},

∥e_{n}x_{i} − x_{i}e_{n}∥ < ε/2 and ∥h′_{n} ∘ h_{n}(x_{i}) − e_{n}x_{i}e_{n}∥ < ε

i = 1, 2, ..., k.

Definition 2.3 Let A and B be C^{∗}-algebras, let L : A → B be a contractive completely positive linear map, let ε > 0 and let F ⊂ A be a subset. L is said to be F-ε-multiplicative, if

∥L(xy) − L(x)L(y)∥ < ε

for all x, y ∈ F.

2.4 Let A be a separable unital nuclear simple C^{∗}-algebra with T R(A) = 0. It follows from a result of B. Blackadar and E. Kirchberg ([BK1] and [BK2]) that A is an inductive limit of RFD C^{∗}-algebras. Let {A_{n}} be a sequence of nuclear RFD C^{∗}-algebras such that A_{n} ⊂ A_{n+1} and A is the closure of ∪_{n}A_{n}. Fix a finite subset F_{1} ⊂ A_{1}, δ_{0} > 0 and a homomorphism h from A_{1} to a finite dimensional C^{∗}-subalgebra F_{0}. Suppose that {x_{1}, ..., x_{n}, ...} is a dense sequence of elements in the unit ball of A. For the convenience, we assume that x_{1} ∈ F_{1}. Applying Lemma 2.1, we obtain a nonzero projection e_{0} > 0 and a homomorphism h′ : F_{0} → A with e_{0} = h′(1_{F_{0}}) such that

(i_{0}) ∥e_{0}a − ae_{0}∥ < δ_{0}/8 and

(ii_{0}) ∥h′ ∘ h(a) − e_{0}ae_{0}∥ < δ_{0}/32 for all a ∈ F_{1}. Set H = h′ ∘ h.

Since T R((1 − e_{0})A(1 − e_{0})) = 0, there is a projection q′_{1} ≤ (1 − e_{0}) and a finite dimensional C^{∗}-subalgebra F′_{1} with 1_{F′_{1}} = q′_{1} such that

(i′_{0}) ∥q′_{1}x − xq′_{1}∥ < δ_{0}/8,

(ii′_{0}) dist(q′_{1}xq′_{1}, F′_{1}) < δ_{0}/32 for all x ∈ F_{1} and

(iii′_{0}) τ(1 − q′_{1}) < 1/4 for all tracial states on A.

Set q_{1} = q′_{1} + e_{0} and F_{1} = F′_{1} ⊕ h(F_{0}). Thus

(i) ∥q_{1}x − xq_{1}∥ < δ_{0}/4,

(ii) dist(q_{1}xq_{1}, F_{1}) < δ_{0}/16 for all x ∈ F_{1} and

(iii) τ(1 − q_{1}) < 1/4 for all tracial states τ on A.

Let F_{2} be the union of {x_{2}} ∪ F_{1} and a set S_{1} of standard generators of F_{1}. Since A is nuclear, there is a contractive completely positive linear map L′_{1} : q_{1}Aq_{1} → F_{1} such that L′_{1}|_{F_{1}} = id_{F_{1}}. Set L_{1}(a) = L′_{1}(q_{1}aq_{1}) for all a ∈ A. Then L_{1} is a contractive completely positive linear map from A to F_{1} and L_{1}|_{F_{1}} = id_{F_{1}}. Note that L_{1} is {x_{1}}-δ_{0}/2-multiplicative. There is δ′_{2} > 0 such that for any S_{1}-δ′_{2}-multiplicative contractive completely positive linear map L from F_{1} there is a homomorphism h from F_{1} such that

∥L(a) − h(a)∥ < δ_{0}/16

for all a ∈ {L_{1}(x_{1})} ∪ S_{1}. Let δ_{2} = min{δ′_{2}, δ_{0}/4}. Since T R(A) = 0, there is a projection q_{2} and a finite dimensional C^{∗}-subalgebra F_{2} with 1_{F_{2}} = q_{2} such that

(i’) ∥q_{2}x − xq_{2}∥ < δ_{2}/4 for all x ∈ F_{2},

(ii’) dist(q_{2}xq_{2}, F_{2}) < δ_{2}/16 for all x ∈ F_{2} and

(iii’) τ(1 − q_{2}) < 1/8 for tracial states on A.

Since A is nuclear, there is a contractive completely positive linear map L′_{2} : q_{2}Aq_{2} → F_{2} such that L′_{2}|_{F_{2}} = id_{F_{2}}. Set L_{2}(a) = L′_{2}(q_{2}aq_{2}) for all a ∈ A. Note that L_{2} is a contractive completely positive linear map from A to F_{2} such that L_{2}|_{F_{2}} = id_{F_{2}} and it is F_{2}-δ_{2}-multiplicative. Thus there is a homomorphism h_{2} : F_{1} → F_{2} such that

∥L_{2}(a) − h_{2}(a)∥ < δ_{0}/16

for all a ∈ {L_{1}(x_{1})} ∪ S_{1}. Since L_{2}(z) ≠ 0 (by (i’),(ii’))and (iii’)) if z ∈ S_{1}, h_{2} has to be injective.

Continuing this way, we obtain a sequence of finite subsets F_{0}, F_{1}, ..., F_{n}, ... in the unit ball of A which is dense, a sequence of decreasing positive numbers δ_{n} (with δ_{n} < δ_{n−1}/4^{n}), a sequence of projections {q_{n}} ⊂ A, a sequence of finite dimensional C^{∗}-subalgebras F_{n} with 1_{F_{n}} = q_{n} and a sequence of homomorphisms h_{n+1} → F_{n} → F_{n+1} such that

(1) ∥q_{n}x − xq_{n}∥ < δ_{n}/4 for F_{n}

(2) dist(q_{n}x_{i}q_{n}, F_{n}) < δ_{n}/16, i = 1, ..., n.

(3) τ(1 − q_{n}) < 1/2^{n} for all tracial states τ on A,

(4) F_{n+1} ⊃ S_{n}, where S_{n} is a set of standard generators of F_{n},

(5) ∥L_{n+1}(a)−h_{n+1}(a)∥ < 1/16^{n} for all a ∈ {L_{n}(F_{n})}∪{S_{n}}, where L_{n} : A → F_{n} is a contractive completely positive linear map such that L_{n}|_{F_{n}} = id_{F_{n}}, n = 1, 2, ....

Let D_{n,1}, D_{n,2}, ..., D_{n,m(n)} be simple summands of F_{n} and π_{n,i} : F_{n} → D_{n,i} be the quotient map. Let Ψ_{n} : A → (1 − q_{n})A(1 − q_{n}) defined by Ψ_{n}(a) = (1 − q_{n})a(1 − q_{n}) for a ∈ A. Note that Ψ_{n} are F_{n}-δ_{n}/2-multiplicative. Define J_{n} : A → A by J_{n}(a) = L_{n}(a) ⊕ Ψ_{n}(a) (for a ∈ A). Note that J_{n} is F_{n}-δ_{n}/2-multiplicative. Set J_{m,n} = J_{n} ∘ J_{n−1} ∘ ⋯ ∘ J_{m} and h_{m,n} = h_{n} ∘ h_{n−1} ∘ ⋯ h_{m+1} : F_{m} → F_{n}. Note also that J_{m,n} is F_{m}-δ_{m}-multiplicative. To save notation, we will also use L_{n}, Ψ_{n}, J_{n}, J_{m,n}, h_{m}, and h_{m,n} for L_{n} ⊗ id_{M_{k}}, Ψ_{n} ⊗ id_{M_{k}}, J_{n} ⊗ id_{M_{k}} and J_{m,n} ⊗ id_{M_{k}}, h_{m} ⊗ id_{M_{k}} and h_{m,n} ⊗ id_{M_{k}}.

The following has been discussed in many recent papers (cf. [Ln8], [Ln12] and [DE2]). We present here for the sake of notation.

Definition 2.5 Let A be a nuclear C^{∗}-algebra and B be a σ-unital C^{∗}-algebra. Let T (A, B) be the set of those extensions of B by A whose six-term exact sequence in K-theory gives pure extensions

0 → K_{i}(B) → K_{i}(E) → K_{i}(A) → 0 i = 0, 1

(i.e., every finitely generated subgroup of K_{i}(B) splits). Following Rørdam, set KL(A, B) = Ext(A, SB)/T (A, SB). Let β ∈ KL(A, B), then β gives an homomorphism from K_{0}(A) → K_{0}(B). Suppose that both A and B are stably finite. Denote by KL(A, B)_{+} those β which induce positive homomorphisms.

Let C_{n} be a commutative C^{∗}-algebra with K_{0}(C_{n}) = ℤ/nℤ and K_{1}(C_{n}) = 0. Suppose that A is a C^{∗}-algebra. Then K_{i}(A, ℤ/kℤ) = K_{i}(A ⊗ C_{k}). Let P(A) be the set of all projections in M_{∞}(A), M_{∞}(C(S^{1}) ⊗ A), M_{∞}((A ⊗ C_{m}˜)) and M_{∞}((C(S^{1}) ⊗ A ⊗ C_{m}˜)). We have the following commutative diagram ([Sc2]):

K_{0}(A) → K_{0}(A, ℤ/kℤ) → K_{1}(A)
↑_{k}  ↓_{k}
K_{0}(A) ← K_{1}(A, ℤ/kℤ) ← K_{1}(A)

As in [DL1], we use the notation

\underline{K}(A) = ⊕_{i=0,1,n∈ℤ_{+}}K_{i}(A; ℤ/nℤ).

By Hom_{Λ}(\underline{K}(A), \underline{K}(B)) we mean all homomorphisms from \underline{K}(A) to \underline{K}(B) which respect the direct sum decomposition and the so-called Bockstein operations (see [DL1]). It follows from [DL1] that if A satisfies the Universal Coefficient Theorem, then Hom_{Λ}(\underline{K}(A), \underline{K}(B)) = KL(A, B).

Let A and B be two C^{∗}-algebras and L : A → B a completely positive linear map. Then L induces maps from A ⊗ C_{m} → B ⊗ C_{m}, from C(S^{1}) ⊗ A ⊗ C_{m} to C(S^{1}) ⊗ B ⊗ C_{m}, namely, L ⊗ id. For convenience, we will also denote the induced map by L. Given a projection p ∈ P(A), if L is G-ε-multiplicative with sufficiently large G and sufficiently small ε, L(p) is close to a projection. Let L(p)′ be that projection. Fix finite subsets of P_{1} ⊂ P(A). It is easy to see that L(p)′ and L(q)′ are in the same equivalence class of projections of P(A), if p and q are in P_{1} and are in the same equivalence class of projections of P(A), provided that F is sufficiently large and ε is sufficiently small. We use [L](p) for the class of projections containing [L](p)′. In what follows, whenever we write [L](p), we assume that F is sufficiently large and ε is sufficiently small so that [L](p) is well-defined on P_{1}. Furthermore, abusing the language, we write [L]([p]) as well as [L](p), where [p] is the equivalence class containing p. Suppose that q is in P(A) with [q] = k[p] for some integer k, by adding sufficiently many elements (partial isometries) in F, we can assume that [L](q) = k[L](p). Suppose that G is a finitely generated group generated by P and G = ℤ^{n} ⊕ ℤ/k_{1}ℤ ⊕ ⋯ ℤ/k_{m}ℤ. Let g_{1}, g_{2}, ..., g_{n} be free generators of ℤ^{n} and t_{i} ∈ ℤ/k_{i}ℤ be the generator with order k_{i}, i = 1, 2, ..., m. Since every element in K_{0}(C) (for any unital C^{∗}-algebra C) may be written as [p_{1}] − [p_{2}] for projections p_{1}, p_{2} ∈ A ⊗ M_{l}, for some l > 0, with sufficiently large F and sufficiently small ε, one can define [L](g_{j}) and [L](t_{i}). Moreover (with sufficiently large F and sufficiently small ε), the order of [L](t_{i}) divides k_{i}. Then we can define a map [L]|_{G} by defining [L](∑_{i}^{n} n_{i}g_{i} +∑_{j}^{m} m_{j}t_{j}) = ∑_{i}^{k} n_{i}[L](g_{i})+∑_{j}^{m} m_{j}[L](t_{j}). Thus [L] is a group homomorphism on G. Note, in general, [L]|_{P} may not coincide with [L]|_{G} on P. However, if F is large enough and ε is small enough, they coincide. In what follows, if P is given, we say [L]|_{G} is well-defined and write [L]|_{G} if [L]|_{P} is well-defined, [L]|_{G} is well-defined and is a homomorphism and [L]|_{P} = [L]|_{G} on P. We will also use [L]|_{P} for [L]|_{[P]} whenever it is convenient.

Definition 2.6 Let S be a compact convex set and Aff(S) be the space of all affine continuous functions on S. Aff(S) has the following order:

Aff(S)_{+} = {f ∈ Aff(S) : f(s) > 0 s ∈ S} ∪ {0}.

This is the order we use in this paper.

Let A be a simple C^{∗}-algebra with T R(A) = 0 and with the tracial space T (A). If τ ∈ T (A), we extend it to a trace on A ⊗ M_{n} on every n by using τ ⊗ T r, where T r is the standard trace on M_{n}. We will use τ for the extension.

Denote by ρ_{A} : K_{0}(A) → Aff(T (A)) the homomorphism defined by ρ_{B}([p])(τ) = τ(p) for all τ ∈ T (A) and for projections p ∈ A ⊗ M_{n} (n = 1, 2, ....). We will often write τ([p]) for ρ_{A}([p]).

We now return to the construction in 2.4.

Lemma 2.7 Let P ⊂ M_{k}(A) be a finite subset of projections. Assume that P ⊂ M_{k}(A_{1}) and F_{1} is sufficiently large and δ_{0} is sufficiently small. With notation in 2.4, we may assume that [L_{1+n}∘J_{1,n}]|_{P} and [L_{n+1} ∘ J_{1,n}]|_{G} are well defined, where G is the subgroup determined by P, and

lim_{n→∞} τ([L_{1+n} ∘ J_{1,n}]([p])) = τ([p])

and the convergent is uniform on T (A). Furthermore,

|τ(h_{1+n} ∘ ⋯ ∘ h_{1}([p])) − τ(h_{n} ∘ ⋯ ∘ h_{1}([p])| < 1/2^{n+1}

for all τ ∈ T (A) and

lim_{n→∞} τ(h_{1+n} ∘ ⋯ h_{1}([p]) ≥ (1 − ∑_{k=1}^{∞} 2^{k+1})τ(h_{1}([p])) > 0.

for p ∈ P and for all tracial states on A (and the convergent is uniform on T (A)).

Proof: Choose a finite subset F_{1} ⊂ A_{1} and δ_{0} > 0 so that [L]|_{P} and [L]|_{G} are well defined provided that L is a F_{1}-δ_{0}-multiplicative contractive completely positive linear map. Without loss of generality, we may assume that δ_{0} < 1/2. This implies that [p] = [J_{1,n}(p)] for p ∈ P. In particular τ([p]) = τ([J_{1,n}(p)]) for p ∈ P. The first limit formula together with the uniformness of the convergence follows from the fact that τ(1 − q_{n}) < 1/2^{n+1} for all tracial states τ and Theorem 6.8 in [Ln10] (see also 3.4 and 3.6 in [Ln7]) that A has the fundamental comparison property of Blackadar. Second formula also follows from the the same fact that τ(1 − q_{k}) < 1/2^{k+1}, k = 1, 2, ....

2.8 Fix a finite subset P of projections of A. Let N be a sufficiently large integer. Then

[L_{N+1} ∘ J_{N}] = [L_{N+1} ∘ L_{N}] ⊕ [L_{N+1} ∘ Ψ_{N}]
= [h_{N+1} ∘ (diag(π_{N,1}, π_{N,2}, ..., π_{N,m(N)})) ∘ [L_{N}] ⊕ [L_{N+1} ∘ Ψ_{N}]

on P, and

[L_{N+2} ∘ J_{N,N+1}] = [L_{N+2} ∘ L_{N+1} ∘ J_{N}] ⊕ [L_{N+2}] ∘ [Ψ_{N+1} ∘ J_{N}]
= [L_{N+2} ∘ L_{N+1} ∘ L_{N}] ⊕ [L_{N+2} ∘ L_{N+1} ∘ Ψ_{N}] ⊕ [L_{N+2}] ∘ [Ψ_{N+1} ∘ J_{N}]
= [h_{N,N+2} ∘ (diag(π_{N,1}, π_{N,2}, ..., π_{N,m(N)})) ∘ [L_{N}]
⊕[L_{N+2} ∘ L_{N+1} ∘ Ψ_{N}] ⊕ [L_{N+2}] ∘ [Ψ_{N+1} ∘ J_{N}].

Moreover,

[L_{N+n} ∘ J_{N,N+n−1}] = [h_{N,N+n} ∘ (diag(π_{N,1}, π_{N,2}, ..., π_{N,m(N)})) ∘ [L_{N}]
⊕[L_{N+n} ∘ Ψ_{N+n−1} ∘ J_{N,N+n−2}] ⊕ [L_{N+n} ∘ L_{N+n−1} ∘ Ψ_{N+n−2} ∘ J_{N,N+n−3}]
⊕ ⋯ ⊕ [L_{N+n} ∘ ⋯ L_{N+1} ∘ Ψ_{N}].

on P. Denote by ψ̃_{(N,i)} = π_{N,i}∘[L_{N}], ψ̃_{(N+1,i)} = π_{N+1,i}∘L_{N+1}∘Ψ_{N}, ..., ψ̃_{(N+n,i)} = π_{N+n−1,i}∘L_{N+n−1}∘ Ψ_{N+n−2}. Let c_{(N+n,i,m)}(τ) = τ(h_{N+n,N+n+m} ∘ ψ̃_{N,i}(1_{A})) > 0. Moreover, we have computed (in 2.7) that c_{(N+n,i,m)}(τ) → c_{(N+n,i)}(τ) > 0 uniformly (as m → ∞) on T (A).

Rearranging {ψ̃_{N+m,i}} (m = 1, 2, ....) as ψ_{j}, j = 1, 2, ... Set s(k) = ∑_{l=1}^{k} m(N + l − 1). Suppose that the image of ψ_{j} is isomorphic to M_{r(j)}. So the rank of the image of ψ_{j} is r(j).

2.9 Identify K_{0}(M_{r(j)}) with ℤ. Let g_{j} be the element in K_{0}(A) corresponding to the minimal projection in ψ_{j}(A). Fix p ∈ P. Set z = tr ∘[ψ_{j}]([p]), where tr is the normalized trace on matrix algebras. Then (r(j)z)g_{j} ∈ K_{0}(A). If ψ_{j} = ψ̃_{N+m,i}, define g_{j}^{(n)} = [h_{N+m,N+m+n}](g_{j}) and a_{j}^{(n)} = c_{(N+m,i,n)}. Then τ(g_{j}^{(n)}) = a_{j}^{(n)}(τ))/r(j).

We have the following lemma Lemma 2.10 For any p ∈ P,

τ([p]) = lim_{k→∞} ∑_{j=1}^{s(k)} a_{j}^{(k)}tr ∘ [ψ_{j}]([p]) uniformly on T (A),

where tr denotes the normalized standard trace on matrix algebras. Furthermore, a_{j}^{(k)} > 0 and a_{j}^{(k)} → a_{j} > 0 uniformly on T (A) (as k → ∞),j = 1, 2, ..., s(k).

Proof: The expression of τ and the uniformness of the convergence follow from 2.7 and the fact that τ([Ψ_{N} + n](1)) = τ(1 − q_{N+n}) → 0. The assertion that a_{j} > 0 follows from the fact that each c_{(m,i,n)} > 0.

Corollary 2.11 Let G_{0} = G ∩ K_{0}(A) and let ρ̃: G → l^{∞}(ℚ) be defined by

g ↦ (tr ∘ [ψ_{1}](g), ..., tr ∘ [ψ_{n}], ...).

Then, kerρ̃ ⊂ kerτ.

Proof: Clearly from 2.10, if tr ∘ [ψ_{j}](g) = 0 for every j, then τ(g) = 0.

2.12 Summery: The above construction (roughly) says, if ε > 0, δ > 0, finite subset of projections in A_{N} ⊗ M_{k} (for some k > 0) and finite subset F ⊂ A_{N} are given, we may write (a ∈ F)

a ≈_{ε} ( \begin{matrix}Ψ_{n}(a) \\ & Φ_{1}^{(n)}(a) \\ & & Φ_{2}^{(n)}(a) \\ & & & . & . & . \\ & & & & & & Φ_{s(n)}^{(n)}(a) \end{matrix} ) ,

where Φ_{1}^{(n)} = h_{N,N+n} ∘ ψ_{1}, Φ_{s(k)}^{(n)} = h_{N+n−1,N+n} ∘ ψ_{s(n)}. For any tracial state τ, we have τ(Φ_{j}^{(n)})(a) = a_{j}^{n}(τ)·tr(ψ_{1}(a)). It is important that a_{j}^{(n)} converges to some positive element a_{j} ∈ Aff(T (A)) uniformly on T (A) and

|τ([p]) − ∑_{j=1}^{s(n)} a_{j}^{(n)}tr([ψ_{j}](p)| < δ

for all τ ∈ T (A) and for p ∈ P.

This concludes this section.

3 A Technical Lemma

3.1 Let S be a compact convex set. Let r be a positive integer or r = ∞ and 𝔻 ⊂ Aff(S) be a subgroup. Set l_{r}^{∞}(𝔻) be set of r-tuple of elements in 𝔻 (or the set of bounded sequences in 𝔻).

If f = {f_{n}} ∈ l_{r}^{∞}(𝔻), we write

∥f∥_{∞} = sup{∥f_{n}∥ : n = 1, 2, ..., }.

The following lemma will not be used in this paper. But it makes the next lemma much more transparent.

Let {x_{ij}} be a r × ∞ matrix, a_{j}^{(n)} > 0 such that a_{j}^{(n)} → a_{j} > 0, j = 1, 2, .... Suppose that

(x_{ij})_{r×∞}(a_{j}^{(n)})_{∞×1} → z = (z_{j})_{1×∞}.

We want to find finitely many non-negative b_{1}, ..., b_{n} for some integer n > 0 such that

(x_{ij})_{r×n}(b_{j})_{n×1} = (z_{j})_{n×1}.

The following lemma states that much more than this is true. Note that both constant δ and integer K are important in the lemma.

Lemma 3.2 Let S be a compact convex set and Aff(S) be the space of all affine continuous functions on S. Let 𝔻 be a dense ordered subgroup of Aff(S). Let {x_{ij}}_{0<i≤r, 0<j<∞} be a r×∞ matrix having rank r and with each x_{ij} ∈ ℚ, and let {a_{j}^{(n)}} be sequences of positive elements in 𝔻 such that a_{j}^{(n)} → a_{j}( > 0) uniformly on S as n → ∞. For each n,

(x_{ij})_{r×n}v_{n} = y_{n},

where v_{n} = (a_{j}^{(n)})_{n×1} is an n × 1 column vector and y_{n} = (b_{i}^{(n)}) ∈ 𝔻^{r} is an r × 1 column vector. Suppose that y_{n} → z for some z = (z_{j})_{r×1} ∈ 𝔻^{r} uniformly (in Aff(S)^{r} norm). Then there is δ > 0 and a positive integer K > 0 satisfying the following: For some sufficiently large n, there is u = (c_{j})_{n×1} ∈ (1/K)(𝔻_{+}^{n}) ( where c_{j}(τ) > 0 for all τ ∈ S

or c_{j} = 0 ) such that

(x_{ij})_{r×n}u = z′

if z′ ∈ 𝔻_{+}^{r} and ∥z − z′∥_{∞} < δ.

Proof: To save notation, without loss of generality, we may assume that (x_{ij})_{r×r} has rank r. Set A_{n} = (x_{ij})_{r×n} (n ≥ r). Then there exists an invertible matrix B ∈ M_{r}(ℚ) (which does not depend on n) such that BA_{n} = C_{n}, where C_{n} = (c_{ij})_{r×n}, c_{ii} = 1 for i = 1, 2, ..., r, and c_{ij} = 0 if i ≠ j, j = 1, 2, ..., r, and c_{ij} ∈ ℚ. Since B ∈ M_{r}(ℚ), there is a positive integer K > 0 such that K · B ∈ M_{r}(ℤ). Moreover, K · C_{n} ∈ M_{r×n}(ℤ).

Let I_{r} be the r × r identity matrix. We may write

C_{n} = (I_{r}, D′_{n}),

where D′_{n} is a r × (n − r) matrix. Note that K · D′_{n} ∈ M_{r×(n−r)}(ℤ). Thus we have

C_{n}v_{n} = By_{n} and I_{r}v′_{n} = By_{n} − D_{n}v_{n},

where v′_{n} = (a_{1}^{(n)}, a_{2}^{(n)}, ..., a_{r}^{(n)}) (as a column) and D_{n} = (0, D′_{n}) is a r × n matrix. Note that for any n × 1 column vector v with the form (t_{1}, t_{2}, ..., t_{r}, a_{r+1}^{(n)}, a_{r+2}^{(n)}, ..., a_{n}^{(n)}), D_{n}v = D_{n}v_{n}. Since a_{j}^{(n)} → a_{j} > 0 uniformly on S and S is compact, there is N_{1} > 0 such that

a_{j}^{(n)}(τ) ≥ inf{a_{j}(τ)/2 : τ ∈ S} > 0 )s ∈ S)

for all n ≥ N_{1} and j = 1, 2, ..., r. Let 0 < ε < min{inf{a_{j}(τ)}/8 : τ ∈ S, j = 1, 2, ..., r}. There is N_{2} > 0 such that

∥By_{n} − Bz∥_{∞} < ε/4

if n ≥ N_{2}. There is δ > 0 depending only on B such that, if ∥z − z′∥_{∞} < δ,

∥Bz − Bz′∥_{∞} < ε/4

(B is determined by {x_{ij}}). Therefore

∥By_{n} − Bz′∥_{∞} < ε/2

for all n ≥ N_{2}. Set N = max{N_{1}, N_{2}}. Let

u′ = Bz′ − D_{n}v_{n},

where u′ = (c_{1}, c_{2}, .., c_{r}) (column vector). Since I_{r}v′_{n} = v′_{n} and I_{r}u′ = u′, we have

∥u′ − v′_{n}∥_{∞} = ∥(Bz′ − D_{n}v_{n}) − (By_{n} − D_{n}v_{n})∥ < ε

if n ≥ N. Therefore c_{j} > 0 for j = 1, 2, ..., r. Set u = (c_{1}, ..., c_{r}, a_{r+1}^{(n)}, a_{r+2}^{(n)}, ..., a_{n}^{(n)}). Then

I_{r}u′ = Bz′ − D_{n}u

(n ≥ N). Since I_{r}u′ = Bz′ − D_{n}u, Bz′ ∈ (1/K)𝔻^{r} and K · D_{n} ∈ M_{r×(n−r)}(ℤ)

(c_{1}, c_{2}, ..., c_{r}) = u′ = I_{r}u′ ∈ (1/K)(𝔻^{r})_{+}.

Since D_{n} = C_{n} − I_{r}, we have

C_{n}u = Bz′.

Finally, since B is invertible, we have

A_{n}u = z′.

3.3 Let G be a group and A = (a_{ij})_{m×k} ∈ M_{m×k}(ℤ). Viewing y = (g_{1}, ..., g_{k}) ∈ G^{k} as a column, one defines Ay = (∑_{j=1}^{k} a_{1j}g_{j}, ∑_{j=1}^{k} a_{2j}g_{j}, ..., ∑_{j=1}^{k} a_{mj}g_{j}). Thus A maps G^{k} to G^{m}. If A ∈ M_{m×k}(ℤ) and B ∈ M_{r×m}(ℤ), then B(Ay) = (BA)(y). Note that if B ∈ M_{r×m}(ℚ) but BA ∈ M_{r×k}(ℤ) does not imply B(Ay) = (BA)(y) in general, since G might have torsion.

Lemma 3.4 Let S be a compact convex set and Aff(S) be the set of all affine continuous functions on S. Let 𝔻 be a dense ordered subgroup of Aff(S) and G be an ordered group with the order determined by a surjective homomorphism ρ : G → 𝔻, i.e.,

G_{+} = {g ∈ G : ρ(g)(τ) > 0} ∪ {0}.

Let {x_{ij}}_{0<i≤r, 0<j<∞} be a r × ∞ matrix having rank r and with each x_{ij} ∈ ℚ_{+}, r(j) ∈ ℕ such that r(j)x_{ij} ∈ ℤ_{+} for all i, j, g_{j}^{(n)} ∈ G such that ρ(g_{j}^{(n)}) = a_{j}^{(n)}(τ)/r(j) (τ ∈ S), where {a_{j}^{(n)}} is a sequence of positive elements in 𝔻 such that a_{j}^{(n)} → a_{j}( > 0) uniformly on S as n → ∞. For each n, (s(n) ≥ n)

(r(j)x_{ij})_{r×s(n)}ṽ_{n} = ỹ_{n},

where ṽ_{n} = (g_{j}^{(n)})_{s(n)×1} is a s(n) × 1 column and ỹ_{n} = (b̃_{j}^{(n)}) ∈ G^{r} is a r × 1 column vector. Set b_{i}^{(n)} = ρ(b̃_{j}^{(n)}) and y_{n} = (b_{i}^{(n)}). Suppose that y_{n} → z on S uniformly on S for some z = (z_{j})_{r×1} ∈ 𝔻^{r} (in Aff(S)^{r} norm).

Then there is δ > 0 and a positive integer K > 0 satisfying the following:

For some sufficiently large n, there is u = (c̃_{j})_{s(n)×1} ∈ G_{+}^{s(n)} (ρ(c̃_{j}) > 0 or ρ(c̃_{j}) = 0 ) such that

(r(j)x_{ij})_{r×s(n)}ũ = z̃′

if z̃′ ∈ G^{r} and there is z̃′′ ∈ G^{r} such that K^{3}z̃′′ = z̃′ and ∥z − Mz′∥_{∞} < δ, where z̃′ = (z̃′_{1}, ..., z̃′_{r}) such that z′_{j} = ρ(z̃′_{j}), j = 1, 2, ..., r and M is a positive integer.

Proof: The proof is a repetition of the proof of 3.2 with some modification. The important difference is that G may have torsion. We proceed as in the proof of 3.2. To save notation, without loss of generality, we may assume that (x_{ij})_{r×r} has rank r. Set A′_{n} = (r(j)x_{ij})_{r×s(n)} ∈ M_{r×s(n)}(ℤ) (s(n) ≥ r). Then there exists an invertible matrix B′ ∈ M_{r}(ℚ) (which does not depend on n) such that B′A′_{n} = C_{n}, where C = (c_{ij})_{r×s(n)}, c_{ii} = 1 for i = 1, 2, ..., r, and c_{ij} = 0 if i ≠ j, j = 1, 2, ..., r, and c_{ij} ∈ ℚ. Since B′ ∈ M_{r}(ℚ), there is a positive integer K > 0 such that K ·B′ ∈ M_{r}(ℤ). Moreover, K · C_{n} ∈ M_{r×s(n)}(ℤ). We may also assume that K · (B′)^{−1} ∈ M_{r}(ℤ).

Let I_{r} be the r × r identity matrix. We may write

C_{n} = (I_{r}, D′_{n}),

where D′_{n} is a r × (s(n) − r) matrix. Note that K · D′_{n} ∈ M_{r×(s(n)−r)}(ℤ). Since 𝔻 is dense in Aff(S), there are ξ_{n} ∈ G^{s(n)} such that ξ_{n} = (d̃_{j}^{(n)})_{s(n)×1} and

∥K^{3}ρ(d̃_{j}^{(n)}) − a_{j}^{(n)}/Mr(j)∥ < \frac{1}{s(n)^{2} · 2^{n}} , j = 1, 2, ..., s(n), n = 1, 2, ....

Let w̃_{n} = K^{3}ξ_{n} and ρ(d̃_{j}^{(n)}) = d_{j}^{(n)}. Then K^{3}d_{j}^{(n)} → K^{3}d_{j} = a_{j}/Mr(j) > 0 uniformly on S. Set ỹ′_{n} = A′_{n}w̃_{n} = K^{3}A′_{n}ξ_{n}. Note that A′_{n}ξ_{n} ∈ G^{r} (A′_{n} ∈ M_{r×s(n)}(ℤ)). Let ρ^{(s(n))} : G^{s(n)} → 𝔻^{s(n)} be defined by ρ^{(s(n))}(g_{1}, ..., g_{s(n)}) = (ρ(g_{1}), ..., ρ(g_{s(n)})) for g ∈ G. Let y′_{n} = ρ^{(s(n))}(ỹ′_{n}). Then, from the construction above, y′_{n} → z/M uniformly on S.

We have

C_{n}w̃_{n} = K^{3}C_{n}ξ_{n} = K^{3}B′A′_{n}ξ_{n}(= B′ỹ′_{n}) and I_{r}ṽ′_{n} = B′ỹ′_{n} − D_{n}w̃_{n},

where ṽ′_{n} = (K^{3}d̃_{1}^{(n)}, ..., K^{3}d̃_{r}^{(n)}) (as a column) and D_{n} = (0, D′_{n}) is a r × s(n) matrix. Note that for any s(n) × 1 column vectors ṽ and v with the form (s_{1}, ..., s_{r}, K^{3}d̃_{r+1}^{(n)}, K^{3}d̃_{r+2}^{(n)}, ..., K^{3}d̃_{s(n)}^{(n)}) and (t_{1}, t_{2}, ..., t_{r}, K^{3}d_{r+1}^{(n)}, K^{3}d_{r+2}^{(n)}, ..., K^{3}d_{s(n)}^{(n)}), D_{n}ṽ = D_{n}w̃_{n} and D_{n}v = D_{n}w_{n}, where ρ^{(s(n))}(ṽ) = v and ρ^{(s(n))}(w̃_{n}) = w_{n}.

Since d_{j}^{(n)} → d_{j} > 0 uniformly on S, there is an N_{1} > 0 such that

d_{j}^{(n)} ≥ inf{d_{j}/2(τ) : τ ∈ S} > 0

for all n ≥ N_{1} and j = 1, 2, ..., r. Let 0 < ε < min{inf{d_{j}(τ) : τ ∈ S}/8K^{3} : j = 1, 2, ..., r}. There is N_{2} > 0 such that

∥B′y_{n} − B′z∥_{∞} < ε/2

if n ≥ N_{2}. Thus there is N_{3} > 0 such that

∥B′(y′_{n}) − B′(z/M)∥_{∞} < ε/2

if n ≥ N_{3} (y′_{n} → z uniformly).

There is δ > 0 depending only on B′ such that, if ∥z − Mz′∥_{∞} < δ,

∥B′y′_{n} − B′z′∥_{∞} < ε/2.

if n ≥ N_{3}. Now let z̃ and z̃′′ be as described in the lemma. Set N = max{N_{1}, N_{2}, N_{3}}. Let

B′z̃′ − D_{n}w̃_{n} = K^{3}Bz̃′′ − K^{3}D_{n}ξ_{n} = K^{2}u′,

where u′ = (c̃_{1}, c̃_{2}, .., c̃_{r}) ∈ G^{r} (column). Set u′′ = K^{2}u′. Let ρ^{(r)}(u′) = (c_{1}, c_{2}, ..., c_{r}) ∈ 𝔻^{r} We may write

I_{r}u′′ = B′z̃′ − D_{n}w̃_{n}.

Since I_{r}ṽ′_{n} = ṽ′_{n} and I_{r}u′ = u′, we have

∥ρ^{(r)}(u′′) − ρ^{(r)}(ṽ′_{n})∥_{∞} = ∥ρ^{(r)}(B′z̃′ − D_{n}w̃_{n}) − ρ^{(r)}(B′ỹ′_{n} − D_{n}w̃_{n})∥_{∞} < ε

if n ≥ N. Since ρ^{(r)}(ṽ′_{n}) = (K^{3}d_{1}^{(n)}, K^{3}d_{2}^{(n)}, ..., K^{3}d_{r}^{(n)}), therefore c_{j}(τ) > 0 for all τ ∈ S and j = 1, 2, ..., r. Set

ũ = (K^{2}c̃_{1}, ..., K^{2}c̃_{r}, K^{3}d̃_{r+1}^{(n)}, K^{3}d̃_{r+2}^{(n)}, ..., K^{3}d̃_{s(n)}^{(n)}) and ū = (c̃_{1}, ..., c̃_{r}, Kd̃_{r+1}^{(n)}, ..., Kd̃_{s(n)}^{(n)}).

Then ū ∈ G_{+}^{s(n)} and

I_{r}u′′ = K^{3}B′z̃′′ − K^{2}D_{n}ū = B′z̃′ − D_{n}ũ

(n ≥ N). Since C_{n} = I_{r} + D_{n}, we have

C_{n}ũ = K^{2}C_{n}ū = K^{2}ū + K^{2}D_{n}ū = I_{r}ũ + D_{n}ũ
= I_{r}u′′ + D_{n}w̃_{n} = K^{3}B′z̃′′ = B′z̃′.

We have K(B′)^{−1}, KB′, KB′ ∈ M_{r}(ℤ). Therefore

(B′)^{−1}C_{n}ũ = K(B′)^{−1}(KC_{n})ū = K^{2}A_{n}ū = A_{n}ũ

and

(B′)^{−1}B′z̃ = (K · (B′)^{−1})(K · (B′))(Kz̃′′) = (K^{2}I_{r})(Kz̃′′) = z̃′.

Hence

A_{n}ũ = z̃′.

4 An Existence Theorem

The following theorem was proved in [Ln7]. A more general version was obtained by M. Dadarlat and S. Eilers ([DE2]).

Theorem 4.1 (Theorem 5.9 in [Ln8]) Let A be a separable C^{∗}-algebra satisfying UCT such that A is the closure of an increasing sequence {A_{n}} of RFD C^{∗}-algebras and B be a unital nuclear separable C^{∗}-algebra. Then, for any α ∈ Hom_{Λ}(\underline{K}(A), \underline{K}(B)), there exist two sequences of completely positive contractions ϕ_{n}^{(i)} : A → B ⊗ K (i = 1, 2) satisfying the following:

(1) ∥ϕ_{n}^{(i)}(ab) − ϕ_{n}^{(i)}(a)ϕ_{n}^{(i)}(b)∥ → 0 as n → ∞,

(2) for each n, the images of ϕ_{n}^{(2)} are contained in a finite dimensional C^{∗}-subalgebra of B ⊗ K and for any finite subset P ⊂ P(A), [ϕ_{n}^{(2)}]|_{P} and [ϕ_{n}^{(2)}]|_{G} are well defined for all large n, where G is the subgroup generated by P,

(3) for each finite subset of P ⊂ P(A), there exists m > 0 such that

[ϕ_{n}^{(1)}]|_{G} = α + [ϕ_{n}^{(2)}]|_{G}

for all n ≥ m.

(4) For each n, we may assume that ϕ_{n}^{(2)} is a homomorphism on A_{n}.

(The condition that B is nuclear can be replaced by the condition that each A_{n} is nuclear).

It is shown in [Ln7] that if A is a simple nuclear separable C^{∗}-algebras with T R(A) = 0, then A is strong NF. It follows a result from B. Blackadar and E. Kirchberg ([BK1] and [BK2]) that every strong NF C^{∗}-algebra is an inductive limit of residually finite dimensional (RFD) C^{∗}-algebras. Thus the above theorem can be applied to simple nuclear separable C^{∗}-algebras with T R(A) = 0 which satisfies the UCT.

However the term ϕ_{n}^{(2)} prevents us to apply the theorem directly.

The following is proved in [Ln12]:

Lemma 4.2 (Lemma 1.8 in [Ln12]) Let A be a unital simple AH-algebra with slow dimension growth and with real rank zero. Let G_{0} be a finitely generated subgroup of K_{0}(A) with decomposition G_{0} = G_{00} ⊕ G_{01}, where G_{00} ⊂ kerρ_{A} and G_{01} is a finite generated free group such that (ρ_{A})|_{G_{01}} is injective. Suppose that P ⊂ \underline{K}(A) is a finite subset which generates a subgroup G such that G ∩ K_{0}(A) ⊃ G_{0}.

Then, for any ε > 0, any finite subset F ⊂ A, any 1 > r > 0, and any integer K, there is an F-ε-multiplicative map L : A → A satisfying the following:

(1) [L]|_{P} and [L]|_{G} are well-defined and [L]|_{G} is positive on G,

(2) [L]|_{G∩kerρ_{A}} = id|_{G∩kerρ_{A}}, [L]|_{G∩K_{0}(A,ℤ/kℤ)} = id|_{G∩K_{0}(A,ℤ/kℤ)}, [L]|_{G∩K_{1}(A)} = id|_{G∩K_{1}(A)} and [L]|_{G∩K_{1}(A,ℤ/kℤ)} = id|_{G∩K_{1}(A,ℤ/kℤ)} for those k with G ∩ K_{i}(A, ℤ/kℤ) ≠ ∅ (i = 0, 1),

(3) ρ_{A} ∘ [L](g) ≤ rρ_{A}(g) for all g ∈ G ∩ K_{0}(A),

(4) Let g_{1}, g_{2}, .., g_{l} be positive generators of G_{01}. Then, there are f_{1}, ..., f_{l} ∈ K_{0}(A)_{+} such that

g_{i} − [L](g_{i}) = Kf_{i}, i = 1, 2, ..., l.

Combining the above two results and 3.4, we prove the existence theorem 4.3.

To establish the following theorem, we first apply 4.1 to obtain Φ_{1}. Then applying 4.2 to obtain L with a small r. We then try to construct h with its range contained in a finite dimensional C^{∗}- subalgebra, and hope that L ∘ Φ_{1} ⊕ h will be the right map. Given 4.1, the strategy is to construct h which is a homomorphism on a finite dimensional C^{∗}-subalgebra. So it is at least possible to have finite dimensional range and does not effect most other KK-theoretical information. While this strategy sounds, the technical difficulties remain to be overcome. Immediately, one would need to extend some certain homomorphism from a subgroup of K_{0}(C) (C is a finite dimensional C^{∗}-algebra) to a homomorphism on K_{0}(C). This surely requires some divisibility of the target group. The integer K in Lemma 4.2 comes to take care of that problem. However, the homomorphism on the group also has to be positive and the extension is also need to be positive. To make things worse, Φ_{1} does not even preserve the order. Furthermore, one should also be careful when maps with finite dimensional ranges are used. They may not give trivial maps on (the part of) K_{0}(A, ℤ/kℤ). Lemma 3.4 together with a careful and extensive use of the integer K in 4.2 will deal with these difficulties.

Here is the existence theorem:

Theorem 4.3 Let A be a unital separable simple nuclear C^{∗}-algebra with T R(A) = 0 which satisfies the UCT. Suppose that B is a unital simple AH-algebra with slow dimension growth, real rank zero and

(K_{0}(B), K_{0}(B)_{+}, [1_{B}], K_{1}(B)) ≅ (K_{1}(A), K_{1}(A)_{+}, [1_{A}], K_{1}(A)).

Let α ∈ KK(A, B)_{+} which carries the above isomorphism. Then, for any finite subset P ⊂ P(A), there is a sequence of contractive completely positive linear map H_{n} : A → B such that

(i) ∥H_{n}(ab) − H_{n}(a)H_{n}(b)∥ → 0 for all a, b ∈ A as n → ∞ and

(ii) [H_{n}]|_{P} = α|_{P} for all sufficiently large n.

Proof: To save the notation, we may assume

(K_{0}(B), K_{0}(B)_{+}, [1_{B}], K_{1}(B)) = (K_{1}(A), K_{1}(A)_{+}, [1_{A}], K_{1}(A)).

So we will identify these two 4-tuples. Let 𝔻 = ρ(K_{0}(A)). Note that 𝔻 is a dense ordered subgroup of Aff(T (A)).

Fix P and let P_{0} ⊂ P be so that P_{0} generates a (finitely generated) subgroup G_{0} so that G_{0} = G(P) ∩ K_{0}(A), where G(P) is the subgroup generated by P. We may assume that P ⊂ P(A_{1}). (Here we identify P with j(P), where j : A_{1} → A is the embedding.) We may assume that {p_{1}, ..., p_{l}} = P_{0}, where p_{i} ∈ M_{k}(A_{1}) are projections (for some k > 0). Let F_{0} be a finite subset of A_{1} and δ_{0} > 0 so that any contractive completely positive linear map L from A_{1} which is F_{0}-δ_{0}-multiplicative well defines [L]|_{P} and [L]|_{G(P)}. Let k_{0} be a positive integer such that

G(P) ∩ K_{i}(A, ℤ/kℤ) = ∅

for all k ≥ k_{0}, i = 0, 1.

Step (I): (Construct Φ_{n} and fix H)

It follows from 4.1 that there is a sequence of contractive completely positive linear map Φ_{n} : A → B ⊗ K such that

∥Φ_{n}(ab) − Φ_{n}(a)Φ_{n}(b)∥ → 0

for all a, b ∈ A as n → ∞, there is a sequence of contractive completely positive linear maps {Φ_{n}^{(0)}} from A to B so that their images contained in a finite dimensional C^{∗}-subalgebras and Φ_{n}^{(0)}|_{A_{n}} is a homomorphism, and

[Φ_{n}]|_{G} = α|_{G} + [Φ_{n}^{(0)}]|_{G} and [Φ_{n}]|_{P} = α|_{P} + [Φ_{n}^{(0)}]|_{P}

for all n. Fix Φ_{1}^{(0)}. Let H : A_{1} → A be a homomorphism such that H = h′ ∘ Φ_{1}^{(0)}, where h′ is a monomorphism from the image of Φ_{1}^{(0)} to A. The existence of such h′ is given by 2.1.

For any finite subset F ⊂ A and η > 0 (F_{0} ⊂ F and η < δ_{0}/2), to save notation, we may assume that Φ_{1} is F-η-multiplicative. Fix this H, F and η/4 (with N = 1), let q_{n}, Ψ_{n}, L_{n}, J_{n}, J_{n}, J_{m,n}, ψ_{n}, δ_{n} (with ∑_{n=1}^{∞} δ_{n} < δ_{0}/2) be as constructed in 2.4. (The importance of H will become clear later.)

Step (II): (Fix M, K and δ > 0 ) Let ρ̃: G(P) ∩ K_{0}(A) → l^{∞}(ℚ) be defined by

ρ̃([p_{i}]) = (tr(ψ_{1}(p_{i})), ..., tr(ψ_{m}(p_{i})), ...) for i = 1, ..., l.

Note that ρ̃(G_{0}) is a finitely generated torsion free group. Consider {ρ̃([p_{1}]), ρ̃([p_{2}]), ..., ρ̃([p_{l}])}. Suppose that the ℚ-linear span of the above l elements (in l^{∞}(ℚ)) has rank r. So a subset of {ρ̃([p_{1}]), ρ̃([p_{2}]), ..., ρ̃([p_{l}])} with r elements has ℚ- rank r. Without loss of generality, we may assume that {ρ̃([p_{1}]), ρ̃([p_{2}]), ..., ρ̃([p_{r}])} has rank r and its ℚ-span includes all ρ̃([p_{i}]), i = 1, ..., l. There is an integer M > 0 such that for any g ∈ ρ̃(G_{0}), Mg is in the subgroup of G̃_{0} generated by ρ̃([p_{1}]), ρ̃([p_{2}]), ..., ρ̃([p_{r}]). Let x_{ij} = tr(ψ̃_{j}([p_{i}])), i = 1, 2, ..., r and j = 1, 2, .... So we may assume that (x_{ij})_{r×r} has ℚ-rank r. Let g_{j}, g_{j}^{(n)}, a_{j}^{(n)} and a_{j} be as in 2.9. Let z_{i} = ρ_{A}([p_{i}]) ∈ 𝔻 and z = (z_{1}, ..., z_{r}). We keep the notation in 2.4, 2.8 and 2.9. We note that a_{j}^{(n)} ∈ 𝔻_{+} \ {0}, lim_{n→∞} a_{j}^{(n)} = a_{j} > 0 uniformly on S, ∑_{j=1}^{n} a_{j}^{(n)})x_{ij} → z_{i} (or (r(j)x_{ij})v_{n} → z) uniformly on S for i = 1, ..., r as n → ∞ (v_{n} = (ρ_{A}(g_{1}), ρ_{A}(g_{2}), ..., ρ_{A}(g_{s(n)}))), by 2.10. So Lemma 3.4 can be applied. Fix δ > 0 and integer K > 0 given by Lemma 3.4. We also note that since H was given before we construct q_{n} so summands of H appears in {ψ_{j}}. Therefore, if g ∈ kerρ̃, then g ∈ kerτ and g ∈ ker([Φ_{1}^{(0)}] ⊕ [Φ_{1}^{(0)}] ⊕ ⋯ ⊕ [Φ_{1}^{(0)}]) (for any finitely many copies of Φ_{1}^{(0)}).

Step (III). (Define Ψ_{1} )

Let Φ̃_{1}^{(0)} be a direct sum of MK^{3}(k_{0} +1)!−1 copies of Φ_{1}^{(0)}. Note that Φ̃_{1}^{(0)} is a homomorphism on A_{1}. Set Ψ_{1} = Φ_{1}⊕Φ̃_{1}^{(0)}. If F is a finite dimensional C^{∗}-algebra then one has the following commutative diagram:

[figure: K_{0}(F); K_{0}(F, ℤ/kℤ); K_{1}(F); K_{0}(F); K_{1}(F, ℤ/kℤ); K_{1}(F)]

where K_{0}(F, ℤ/kℤ) = K_{0}(F)/kK_{0}(F), K_{1}(F) = 0, K_{1}(F, ℤ/kℤ) = 0. Since Φ_{1}^{(0)} factors through a finite dimensional C^{∗}-subalgebra, it is easy to check that

[Φ_{1}^{(0)}]|_{K_{1}(A)∩G} = 0, and [Φ_{1}^{(0)}]|_{K_{1}(A,ℤ/kℤ)∩G} = 0.

Moreover,

(k_{0})![Φ_{1}^{(0)}]|_{K_{0}(A,ℤ/kℤ)∩G} = 0 (k ≤ k_{0}).

Therefore

[Ψ_{1}]|_{K_{1}(A)∩G} = α|_{K_{1}(A)∩G}, [Ψ_{1}]|_{K_{1}(A,ℤ/kℤ)∩G} = α|_{K_{1}(A,ℤ/kℤ)∩G},

and [Ψ_{1}]|_{K_{0}(A,ℤ/kℤ)∩G} = α|_{K_{0}(A,ℤ/kℤ)∩G}. Without loss of generality, we may assume that Ψ_{1}(1_{A}) is a projection in M_{R}(B).

Step (IV) (Cut Ψ_{1} according to M, K, k_{0} and δ)

We may assume that there are projections p̄_{1}, ..., p̄_{l} in M_{k}(B) (for some k > 0) such that

∥p̄_{i} − Ψ_{1}(p_{i})∥ < η < 1/4, i = 1, ..., l.

So [p̄_{i}] = [Ψ_{1}(p_{i})], i = 1, ..., l. There are projections q̄_{i} ≤ p̄_{i} such that [q̄_{i}] = K(k_{0} + 1)![Φ_{1}^{(0)}(p_{i})]. Set ē_{i} = p̄_{i} − q̄_{i}. Note that [ē_{i}] = α([p_{i}]), i = 1, 2, ..., l. Set P_{1} = {P ∪ [Ψ̃_{1}(P)]} ∪ P′_{0} ∪ α(P), where P′_{0} = {[q̄_{i}], ē_{i}, [p̄_{i}], [Φ_{1}^{(0)}([p_{i}]) : i = 1, ..., l}. Set G_{1} = G(P_{1}). We write G_{0} = G_{00} ⊕ G_{01}, where G_{00} = G_{0} ∩ kerτ and G_{01} ≅ τ(G_{0}). Let d_{1}, ..., d_{t} be positive and generate G_{01}.

By applying 4.2, for any ε > 0, any finite subset G ⊂ M_{R}(B), any 0 < r < δ < 1, there is a G-ε-multiplicative map L : M_{R}(B) → M_{R}(B) satisfying the following:

(1) [L]|_{P_{1}} and [L]|_{G_{1}} are well-defined and [L]|_{G_{1}} is positive on G_{1},

(2) [L]|_{G_{1}∩kerρ_{B}} = id|_{G_{1}∩kerρ_{B}}, [L]|_{G_{1}∩K_{0}(B,ℤ/kℤ)} = id|_{G_{1}∩K_{0}(B,ℤ/kℤ)}, [L]|_{G_{1}∩K_{1}(B)} = id|_{G∩K_{1}(B)} and [L]|_{G_{1}∩K_{1}(B,ℤ/kℤ)} = id|_{G_{1}∩K_{1}(B,ℤ/kℤ)} for those k with G_{1} ∩ K_{i}(B, ℤ/kℤ) ≠ ∅ (i = 0, 1),

(3) ρ_{B} ∘ [L](g) ≤ rρ_{B}(g) for all g ∈ G ∩ K_{0}(B),

(4) There are f_{1}, ..., f_{l} ∈ K_{0}(A)_{+} such that

α(d_{i}) − [L](α(d_{i})) = MK^{3}(k_{0} + 1)!f_{i}, i = 1, 2, ..., t.

(Note that we may identify α with identity map.) We choose r so that

R · rρ_{B}([Φ_{1}]([p_{i}])) ≤ (δ/2MK^{3}(k_{0} + 1)!)ρ(α([p_{i}])).

This is possible since τ(α([p_{i}]) = τ([p_{i}]) > 0 for all i (A and B are simple) and T (A) is compact. Therefore

τ ∘ [L] ∘ [Ψ_{1}]([p_{i}]) ≤ (δ/2)τ(α([p_{i}])) and α([p_{i}]) − [L ∘ Ψ_{1}]([p_{i}]) > 0

for all τ ∈ T (A), i = 1, ..., l.

Let [p_{i}] = ∑_{j}^{t} m_{j}^{(i)}d_{j} + s, where m_{j}^{(i)} ∈ ℤ and s ∈ G_{00}. Then by (2) above, α(s) − [L] ∘ α(s) = 0. Therefore

α([p_{i}]) − [L ∘ Ψ_{1}])([p_{i}]) = α(∑_{j} m_{j}^{(i)}d_{j}) − [L ∘ α](∑_{j} m_{j}^{(i)}d_{j}) − MK^{3}(k_{0} + 1)![L ∘ Φ_{1}^{(0)}]([p_{i}])
= MK^{3}(k_{0} + 1)!(∑_{j} m_{j}^{(i)}f_{j} − [L] ∘ [Φ_{1}^{(0)}]([p_{i}])) = MK^{3}(k_{0} + 1)!f′_{j}

for some f′_{j} ∈ K_{0}(B), j = 1, 2, ..., l.

Since K_{0}(B) is weakly unperforated, f′_{i} > 0 for i = 1, 2, ..., l. Set β : G(P) ∩ K_{0}(A) → K_{0}(B) by β([p_{i}]) = K^{3}(k_{0} + 1)!f′_{i}, i = 1, 2, ..., l.

Step (V): (Construct h) Let z̃′_{i} = β([p_{i}]) and z̃′ = (z̃′_{1}, z̃′_{2}, ..., z̃′_{r}) (as a column). Let z̃′′ = ((k_{0} + 1)!f′_{1}, ..., (k_{0} + 1)!f′_{r}). Then K^{3}z̃′′ = z̃′. Set z′_{i} = ρ(z̃′_{i}), z′ = (z′_{1}, ..., z′_{r}), Then

∥Mz′ − z∥_{∞} < δ (z = (z_{1}, ..., z_{r}))

It follows from 3.4 that there is ũ = (u_{1}, ..., u_{s(n)}) ∈ (K_{0}(B)^{s(n)})_{+} (for some s(n) ≥ r) such that

(r(j)x_{ij})ũ = z̃′.

Let D = ψ_{1}(A) ⊕ ψ_{2}(A) ⊕ ⋯ ⊕ ψ_{s(n)}(A). Define a homomorphism h_{0} : D → M_{k}(B) (for some k > 0) such that [h_{0}](e_{j}) = u_{j}, j = 1, ..., s(n), where e_{j} is a mininal projection in ψ_{j}(A). Such h_{0} exists since B has stable rank one and real rank zero. Define π : l^{∞}(ℚ) → l_{s(n)}^{∞}(ℚ) by

{x_{k}} ↦ (x_{1}, ..., x_{s(n)}).

So

[h_{0}](π ∘ ρ̃([p_{i}])) = β([p_{i}]), i = 1, ..., r.

Since kerρ̃ ⊂ kerτ ∘ α ∩ kerH, if x ∈ kerρ̃, then [L ∘ Ψ_{1}](x) = [L] ∘ α(x). Recall that we identify K_{0}(A) with K_{0}(B). By (2) above, since α(x) ∈ kerρ_{B}, [L] ∘ α(x) = α(x). Hence

α(x) − [L ∘ Ψ_{1}](x) = 0.

Therefore we may view that α−[L∘Ψ_{1}] gives a homomorphism on ρ̃(G_{0}). Also Mg is in the subgroup generated by ρ̃([p_{1}]), ..., ρ̃([p_{r}]) for any g ∈ ρ̃(G_{0}). Combining these two facts, we obtain

M[h_{0}](π ∘ ρ̃([p_{i}])) = Mβ([p_{i}]), i = 1, ..., r, ..., l.

Set h′_{0} = h_{0} ⊕ ⋯ h_{0}, M copies of h_{0}. Then

[h′_{0}](π ∘ ρ̃([p_{i}])) = α([p_{i}]) − [L ∘ Ψ_{1}]([p_{i}]), i = 1, ..., r, ..., l.

Set h = h′_{0} ∘ (ψ_{1} ⊕ ψ_{2} ⊕ ⋯ ⊕ ψ_{s(n)}). Then h is also F-η-multiplicative.

Moreover,

[h′_{0}] ∘ [ψ_{1} ⊕ ψ_{2} ⊕ ⋯ ⊕ ψ_{s(n)}]([p_{i}]) = [h′_{0}] ∘ (π ∘ ρ̃([p_{i}])) i = 1, 2, ..., l.

We also have

[h]|_{P∩K_{1}(A)} = 0 and [h]|_{P_{0}} = α|_{P_{0}} − [L] ∘ [Ψ_{1}]|)|_{P_{0}}.

Note that, since [h]([p_{i}]) = MK^{3}(k_{0} + 1)!f′_{i}, i = 1, ..., r. [h](g) = MK^{3}(k_{0} + 1)!g′ for any g ∈ G_{0} and some g′ ∈ K_{0}(B). Note since D is finite dimensional, we have the following (exact) commutative diagram.

[figure: K_{0}(D); K_{0}(D, ℤ/kℤ); K_{1}(D); K_{0}(D); K_{1}(D, ℤ/kℤ); K_{1}(D)]

Again, K_{1}(D) = 0, K_{1}(D, ℤ/kℤ) = {0} and K_{0}(D, ℤ/kℤ) = K_{0}(D)/kK_{0}(D) for all k. Since h factors through D, and [h](g) = (k_{0} + 1)!MK^{3}g′′, we conclude that

[h]|_{G∩K_{1}(A,ℤ/kℤ)} = 0 and [h]|_{G∩K_{0}(A,ℤ/kℤ)} = 0 for k ≤ k_{0}.

This implies that

[h]|_{P} = α|_{P} − [L] ∘ [Ψ_{1}]|_{P}.

Now define H_{1} = L ∘ Ψ_{1} ⊕ h. Then H_{1} is F-η-multiplicative and

[H_{1}]|_{P} = [h]|_{P} + [L] ∘ [Ψ_{1}]|_{P} = α|_{P}.

Finally, since [H_{1}(1_{A})] = [1_{B}], by conjegating a unitray in (B⊗K˜), we may assume that H_{1}(1_{A}) = 1_{B}.

5 The Main Theorem

We will use the following uniqueness theorem:

Theorem 5.1 (Theorem 2.3 in [Ln8]) Let A be a separable unital nuclear simple C^{∗}-algebra with T R(A) = 0 satisfying the UCT. Then, for any ε > 0, and any finite subset F ⊂ A, there exist δ > 0, a finite subset P ⊂ P (A) and a finite subset G ⊂ A satisfying the following: for any unital C^{∗}-algebra B of real rank zero and stable rank one with weakly unperforated K_{0}(B), and any two G-δ-multiplicative morphisms L_{1}, L_{2} : A → B with

[L_{1}]|_{P} = [L_{2}]|_{P},

there exists a unitary U ∈ B such that

ad(U) ∘ L_{1} ≈_{ε} L_{2} on F.

Theorem 5.2 Let A and B be two unital separable simple nuclear C^{∗}-algebras with T R(A) = T R(B) = 0 which satisfy the UCT. Then A ≅ B if and only if

(K_{0}(A), K_{0}(A)_{+}, [1_{A}], K_{1}(A)) ≅ (K_{0}(B), K_{0}(B)_{+}, [1_{B}], K_{1}(B)).

Proof: The “only if” part is obvious and known. We need to prove the “if ” part only. Using the terminology in [Ln8], Theorem 4.3 implies that both A and B are pre-classifiable. Then, by 5.1, Theorem 3.7 in [Ln8] says that A ≅ B if

(K_{0}(A), K_{0}(A)_{+}, [1_{A}], K_{1}(A)) ≅ (K_{0}(B), K_{0}(B)_{+}, [1_{B}], K_{1}(B)).

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