INCLUSIONS OF SIMPLE C^{∗}-ALGEBRAS

MASAKI IZUMI

Abstract. We prove that if a conditional expectation from a simple C^{∗}-algebra onto its C^{∗}-subalgebra satisfies the Pimsner-Popa inequality, there exists a quasibasis. As an application, we establish the Galois correspondence for outer actions of finite groups (and more generally finite dimensional Kac algebras) on simple C^{∗}- algebras. We also introduce the notion of sectors for stable simple C^{∗}-algebras and purely infinite simple C^{∗}-algebras in the Cuntz standard form, and discuss several applications, including their relationship to K-theory.

1. Introduction

In [18], V. F. R. Jones introduced the notion of indices for inclusions of II_{1} factors, which opened a completely new aspect of operator algebras involving various other fields of mathematics and mathematical physics. Since then, several attempts have been made to generalize the notion of indices to broader classes of operator algebras. In [33], M. Pimsner and S. Popa introduced the notion of the probabilistic index for a conditional expectation, which is the best constant of so called the Pimsner-Popa inequality. In [23], H. Kosaki discussed another way to define indices for general inclusions of factors using the theory of operator valued weights (see also [1]).

Though the probabilistic index works perfectly for analytic purposes even in the case of C^{∗}-algebras (see, e.g. [35]), it is not always suitable for algebraic operations such as basic construction in the C^{∗}-case. In [42], Y. Watatani proposed to assume existence of a quasi-basis for a conditional expectation, a generalization of the Pimsner-Popa basis in the von Neumann algebra case, to analyze inclusions of C^{∗}-algebras. With a quasi-basis, Watatani successfully introduced a C^{∗}-version of basic construction [42], which is closely related to K-theory of C^{∗}-algebras [42, Capter III], [20]. However, he also pointed out that there exists a rather annoying example satisfying the Pimsner-Popa inequality that has no quasi-basis [42, Example 2.9.1].

One of the purposes of the present notes is to show that for simple C^{∗}-algebras, the Pimsner-Popa inequality always assures existence of a quasi-basis (Theorem 3.2). For this, we employ a second dual method, and apply M. Baillet, Y. Denizeau, and J.-F. Havet’s result [1] in the W^{∗}-case. As far as the Pimsner-Popa inequality holds, the index of a conditional expectation in the sense of [23] and [1] can be defined as a central element of the second dual. However, it does not even belong to the multiplier algebra in general. Indeed, our main results tell that this phenomenon occurs in the above mentioned example. Nevertheless, when C^{∗}-algebras forming an inclusion are simple, the index in the above sense happens to be a scalar, and the C^{∗}-basic construction works. Using a similar method, we also give a prescription for the C^{∗}-basic construction to the non-unital case, which was not treated in [42]. As an application, we show that a minimal conditional expectation factors through every intermediate C^{∗}-subalgebra for an inclusion of simple C^{∗}-algebra with a finite index (no matter what definition we adopt). In particular, we establish the Galois correspondence of outer actions of finite groups (more generally finite dimensional Kac algebras) on simple C^{∗}-algebras exactly in the same way as in the case of factors [30], [16].

Work supported by Mathematical Sciences Research Institute, JSPS, and Japanese Ministry of Education, Science, Sports and Culture.

As it is well-known in the case of factors, the bimodule approach (see, for example,

[9] and references in it) to the categorical aspect of subfactors is equivalent to the sector approach inspired by the algebraic quantum field theory [27]. In [20], T. Kajiwara and Y. Watatani developed the theory of C^{∗}-bimodules of finite indices and discuss their relationship to K-theory. We introduce here the notion of sectors for two classes of C^{∗}-algebras as an alternative approach: simple σ-unital stable C^{∗}- algebras and simple purely infinite unital C^{∗}-algebras in the Cuntz standard form. As an application, we give a K-theoretical obstruction for a particular combinatorial invariants to be realized in a simple C^{∗}-algebra with particular K-groups. In the final section, we show a Kishimoto’s type theorem (cf. [21]) for conditional expectations of finite indices as anther application.

The main part of these notes is based on a series of lectures the author gave in Tokyo Metropolitan University in December 1997, and the author would like to thank H. Takai for giving him the opportunity. He is grateful to all the audiences of the lectures, and especially H. Osaka who took notes carefully. Some statements of this paper have already been applied in [31]. Typing of this paper was finished while the author stayed in Mathematical Sciences Research Institute, and he would like to thank them for their hospitality.

2. Preliminaries

First we summarize notations we use in this paper. For a C^{∗}-algebra A, A_{sa} and A_{+} mean the self-adjoint part and the positive part of A respectively. We denote by S(A), P (A), and Q(A) the set of states on A, the set of pure states on A, and the set of positive linear functional of A with norm less than or equal to 1 respectively. For φ ∈ S(A), (π_{φ}, H_{φ}, Ω_{φ}) is the the GNS triple of φ. When A is not unital, M(A) denotes the multiplier algebra of A, which is often regarded as a C^{∗}-subalgebra of the second dual A^{∗∗}. The strict topology of M(A) is defined by the family of seminorms {p_{a}}_{a∈A} where

p_{a}(x) = ||ax|| + ||xa||.

The set of compact operators on ℓ^{2}(ℕ) is denoted by 𝕂.

Throughout this section, A ⊃ B denotes an inclusion of C^{∗}-algebras with a conditional expectation E from A onto B.

2.1. Indices for C^{∗}-algebras. First we introduce two numerical indices for E.

Definition 2.1. Let A, B, and E be as above. We define

Ind_{p} E = inf\{λ > 0; Id − \frac{1}{λ}E is positive\}.
Ind_{cp} E = inf\{λ > 0; Id − \frac{1}{λ}E is completely positive\}.

When there are no such number λ satisfying the first condition (respectively the second condition), we set Ind_{p} E = ∞ (respectively Ind_{cp} E = ∞).

Ind_{p} E was introduced by Pimsner and Popa in [33], and is called the probabilistic index of E. Importance of the condition Ind_{cp} E < ∞ was pointed out by M. Baillet, Y. Denizeau, and J.-F. Havet in [1, Théorèm 3.5]. Namely, they proved that when A and B are von Neumann algebras, Ind_{cp} E < ∞ occurs if and only if the dual operator valued weight is bounded.

By definition, Ind_{p} E ≤ Ind_{cp} E always holds. On the other hand, it was shown by M. Frank and E. Kirchberg in [10, Theorem 1] that Ind_{p} E is finite if and only if Ind_{cp} E is finite. Following their argument, we can show the following:

Lemma 2.2. Let A, B, and E be as above. If B has no non-zero finite dimensional representation, then Ind_{p} E = Ind_{cp} E.

Proof. It suffices to show the statement when Ind_{p} E < ∞, and we assume this. Recall that the second dual B^{∗∗} of B is naturally regarded as a von Neumann subalgebra of the second dual A^{∗∗} of A [32, Corollary 3.7.9]. Let E^{∗∗} be the normal extension of E to the second dual A^{∗∗}. Then, E^{∗∗} is a conditional expectation from A^{∗∗} onto B^{∗∗} with the same indices as E. Let z_{A^{∗∗}} and z_{B^{∗∗}} be the central projections of A^{∗∗} and B^{∗∗} corresponding to the discrete parts of A^{∗∗} and B^{∗∗} respectively. Then, thanks to [10, Proposition 1.3], [34, 1.1.2], we have z_{A^{∗∗}} = z_{B^{∗∗}}. Let M = A^{∗∗}z_{A^{∗∗}}, N = B^{∗∗}z_{B^{∗∗}}, and F be the restriction of E^{∗∗} to M. Since the direct sum of all irreducible representations of A is faithful, it suffices to show the statement for F. By assumption B has no non-zero finite dimensional representation, and N is a direct sum of infinite dimensional type I factors. Thus, there exists a system of matrix units {e_{ij}}_{i,j∈ℕ} in N such that {e_{ii}}_{i∈ℕ} is a partition of unity. Let F_{e} be the restriction of F to eMe where e = e_{11}. Then, we have

(M ⊃ N, F) ≅ (eMe ⊗ 𝔹(ℓ^{2}(ℕ)), eNe ⊗ 𝔹(ℓ^{2}(ℕ)), F_{e} ⊗ Id).

Therefore, Ind_{p} F and Ind_{cp} F coincide.

Now we introduce an index in terms of a quasi-basis following Watatani [42].

Definition 2.3 (Unital case). Let A and B be unital C^{∗}-algebras, and E be as above.

(1) A quasi-basis for E is a finite set {(u_{i}, v_{i})}_{i=1}^{n} ⊂ A×A such that for every a ∈ A, the following holds:

a = ∑_{i=1}^{n} u_{i}E(v_{i}a) = ∑_{i=1}^{n} E(au_{i})v_{i}.

(2) When {(u_{i}, v_{i})}_{i=1}^{n} is a quasi-basis for E, we define Ind_{w} E by

Ind_{w} E = ∑_{i=1}^{n} u_{i}v_{i}.

When there is no quasi-basis, we write Ind_{w} E = ∞. Ind_{w} E is called the Watatani index of E.

Remark 2.4. Several remarks about the above definitions are in order.

(1) Ind_{w} E does not depend on the choice of the quasi-basis in the above formula, and it is a central element of A [42, Proposition 1.2.8].

(2) Once we know that there exists a quasi-basis, we can choose one of the form {(w_{i}, w_{i}^{∗})}_{i=1}^{m}, which shows that Ind_{w} E is a positive element [42, Lemma 2.1.6].

(3) Ind_{cp} E ≤ ||Ind_{w} E|| always holds, which was first proved in [33] in the case of factors.

(4) When, E_{1} and E_{2} are faithful conditional expectations from A onto B, then E_{1} has a quasi-basis if and only if E_{2} does. In other words, the property that E has a quasi-basis is actually a property of the inclusion A ⊃ B.

(5) Though the above definition looks fine even in the non-unital case, Proposition 2.5 below shows that a quasi-basis exists only if both A and B are unital. It is also easy to show that a simple unitization trick does not resolve this problem at all.

Several examples of conditional expectations of Cuntz-Krieger algebras with quasibases are found in [13], [15].

2.2. C^{∗}-basic construction. In this subsection, we recall Watatani’s notion of the C^{∗}-basic construction. We assume that E is faithful in what follows. For general theory of Hilbert C^{∗}-modules, we refer to [25].

We introduce pre-Hilbert B-module structure into A by defining a B-valued inner product

⟨a_{1}, a_{2}⟩_{B} := E(a_{1}^{∗}a_{2}).

We denote by E_{E} and η_{E} the Hilbert B-module completion of A and the natural inclusion map from A into E_{E}. As usual we use notations 𝕂_{B}(E_{E}) and 𝕃_{B}(E_{E}) for the set of “compact operators” on E_{E} and the set of bounded B-module maps with adjoints respectively. Then, the Jones projection e_{E} ∈ 𝕃_{B}(E_{E}) is defined by

e_{E}η_{E}(a) = η_{E}(E(a)), a ∈ A.

A is embedded into 𝕃_{B}(E_{E}) as the left multiplication operators. Note that 𝕂_{B}(E_{E}) is the norm closure of the linear spans of Ae_{B}A in 𝕃_{B}(E_{E}), which is called the reduced C^{∗}-basic construction and is denoted by A_{1}.

A word of caution: in general we do not have A_{1} ⊃ A even when Ind_{p} E < ∞ holds. Indeed, Watatani proved the following in [42]:

Proposition 2.5. Let the notations be as above. Then, (1) Assume that D is a C^{∗}-algebra with a copy of A as a C^{∗}-subalgebra and that

there exists a projection p ∈ D satisfying

pap = E(a)p, a ∈ A,

such that B ∋ b ↦ bp is an injective map. Then there exists an isomorphism π from A_{1} into D such that π is an identity map on A and π(e_{B}) = p [42, Proposition 2.2.7].

(2) E has a quasi-basis if and only if Ind_{p} E < ∞ and A_{1} is unital (equivalently 𝕂_{B}(E_{E}) = 𝕃_{B}(E_{E})) [42, Proposition 2.1.5], [37].

When the condition of (2) is satisfied, there exists the dual conditional expectation E_{1} from A_{1} onto A satisfying E_{1}(e_{B}) = (Ind_{w} E)^{−1}. If {(u_{i}, v_{i})}_{i=1}^{n} is a quasi-basis for E, E_{A} has a quasi-basis {(s_{i}, t_{i})}_{i=1}^{n} with

s_{i} = u_{i}e_{E}(Ind_{w} E)^{1/2}, t_{i} = (Ind_{w} E)^{1/2}e_{E}v_{i}.

2.3. Non-unital case. In the case of non-unital C^{∗}-algebras, there is no way to define indices using a quasi-basis inside the algebra as mentioned in Remark 2.4, (5). However, our primary needs are the dual inclusion and the dual conditional expectation rather than the quasi-basis itself, and we still have a good chance to get the former even when we give up the latter. The aim of this subsection is to supplement Watatani’s C^{∗}-basic construction by using a second dual argument in the non-unital case. We first prepare some basic facts.

We regard B^{∗∗} as a von Neumann subalgebra of A^{∗∗} as before, and also regard M(A) and M(B) as C^{∗}-subalgebras of A^{∗∗} and B^{∗∗} respectively. In general, there is no inclusion relation between M(A) and M(B). However, when AB = A holds (which is equivalent to that some (and all) approximate unit of B is an approximate unit of A), we have M(A) ⊃ M(B) [32, 3.12.12].

Lemma 2.6. Let A, B, and E be as above satisfying Ind_{p} E < ∞. Then, the following hold:

(1) A = AB. In consequence, M(B) is a unital subalgebra of M(A).

(2) The restriction of E^{∗∗} to M(A) is a conditional expectation from M(A) onto M(B).

Proof. (1). Let x ∈ A. Ind_{p} E < ∞ implies x^{∗}x ≤ (Ind_{p} E)E(x^{∗}x), and so for 0 < θ < 1/2 there exists u ∈ A such that x = uE(x^{∗}x)^{θ} [32, Proposition 1.4.5]. Therefore, the statement holds.

(2). This follows from the characterization of the self-adjoint part M(A)_{sa} in A_{sa}^{∗∗} as

M(A)_{sa} = (Ã_{sa})^{m} ∩ (Ã_{sa})_{m},

(see [32, Theorem 3.12.9] for the notation).

Let π be a representation of B on H. The induced representation ind_{B}^{A}π is the representation of A, (which is actually extended to 𝕃_{B}(E_{E})), constructed as follows [36]: We introduce an inner product into the algebraic tensor product E_{E} ⊙_{B} H_{π} by

⟨ξ_{1} ⊗_{B} η_{1}, ξ_{2} ⊗_{B} η_{2}⟩ = ⟨⟨ξ_{2}, ξ_{1}⟩_{B}η_{1}, η_{2}⟩.

The induced representation ind_{B}^{A}π is the representation of A (and also 𝕃_{B}(E_{E})) on the Hilbert space completion E_{E}\overline{⊗}_{B}H_{π} of E_{E} ⊙_{B} H_{π} coming from the left multiplication on the first tensor component. For ψ ∈ S(A), we naturally have ind_{B}^{A}π_{ψ} = π_{ψ·E}.

Lemma 2.7. Let A, B, and E be as above, and (π_{S}, H_{S}) be the universal representation of B, i.e.

(π_{S}, H_{S}) = ( ⨁_{φ∈S(B)} π_{φ}, ⨁_{φ∈S(B)} H_{φ}).

If Ind_{p} E is finite, then ind_{B}^{A}π_{S} is quasi-equivalent to the universal representation of A.

Proof. It suffices to show that for every state φ ∈ S(A), π_{φ} is contained in ind_{B}^{A}π_{S}. Let ψ be the restriction of φ to B. Since Ind_{p} E is finite, we have

φ ≤ (Ind_{p} E)φ · E = (Ind_{p} E)ψ · E.

Therefore, π_{φ} is contained in ind_{B}^{A}π_{ψ}, and we get the result.

As mentioned at the beginning of this section, the condition Ind_{p} E < ∞ implies Ind_{cp} E < ∞, and so Ind_{cp} E^{∗∗} < ∞. Then, Baillet-Denizeau-Havet’s result [1, Théorèm 3.5] shows the following: E_{E^{∗∗}} is automatically a self-dual Hilbert W^{∗}- module, and the W^{∗}-algebra 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) is the weak closure of the linear span of A^{∗∗}e_{E^{∗∗}}A^{∗∗}. Moreover, there exists a bounded normal operator valued weight Ê^{∗∗} from 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) to A^{∗∗} satisfying Ê^{∗∗}(e_{E^{∗∗}}) = 1. Simple computation shows that Ê^{∗∗}−Id is a completely positive map. Following H. Kosaki’s idea in [23], they defined the index of E^{∗∗} by Ê^{∗∗}(1) ∈ Z(A^{∗∗}). Thanks to Lemma 2.7, we may consider A^{∗∗} the weak closure of the image of ind_{B}^{A}π_{S}. Therefore, there exists a natural faithful normal representation of 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) on E_{E^{∗∗}} \overline{⊗}_{B^{∗∗}}H_{S} = E_{E}\overline{⊗}_{B}H_{S}, and we regard 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) as a concrete von Neumann algebra acting on E_{E}\overline{⊗}_{B}H_{S}. On the other hand, A_{1} also naturally acts on E_{E}\overline{⊗}_{B}H_{S}. Therefore, we may identify e_{E^{∗∗}} with e_{E}. Note that A_{1} is weakly dense in 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) under this identification.

Now, we introduce Ind_{w} E for some class of inclusions of non-unital C^{∗}-algebras.

Theorem 2.8. Let A ⊃ B be an inclusion of C^{∗}-algebras, and E be a conditional expectation from A onto B with Ind_{p} E < ∞. Then, A_{1} ⊃ A holds if and only if Ê^{∗∗}(1) ∈ Z(M(A)). Moreover, if this is the case, the following hold:

(1) There exists a positive bounded A-A bimodule map Ê from A_{1} onto A satisfying

Ê(a_{1}e_{E}a_{2}) = a_{1}a_{2}, a_{1}, a_{2} ∈ A,
Ê(x) ≥ x, x ∈ A_{1+}.

For every approximate unit {u_{λ}} of A_{1}, {Ê(u_{λ})} converges to Ê^{∗∗}(1) in the strict topology.

(2) Let E_{1} be the conditional expectation from A_{1} onto A defined by

E_{1}(x) = Ê^{∗∗}(1)^{−1}Ê(x), x ∈ A_{1}.

Then, 𝕂_{A_{1}}(E_{E_{1}}) ⊃ A_{1} holds, i.e. the basic construction can be repeated.

Proof. Assume that A_{1} ⊃ A holds. Let {u_{λ}} be an approximate unit of A_{1}. Since A_{1} is weakly dense in 𝕃_{A^{∗∗}}(E_{E^{∗∗}}), {u_{λ}} strongly converges to 1 in 𝕃_{A^{∗∗}}(E_{E^{∗∗}}), and so {Ê(u_{λ})} strongly converges to Ê^{∗∗}(1) in A^{∗∗}. However, A_{1} ⊃ A implies that {Ê(u_{λ}a)} and {Ê(au_{λ})} converge to Ê(a) ∈ A in norm, which shows that {Ê(u_{λ})} converges to Ê^{∗∗}(1) ∈ Z(M(A)) in the strict topology.

Assume Ê^{∗∗}(1) ∈ Z(M(A)) now. Let a ∈ A and {u_{λ}} be an approximate unit of A_{1}. Then, we have

||(1 − u_{λ})a||^{2} = ||a^{∗}(1 − u_{λ})^{2}a|| ≤ ||a^{∗}(1 − u_{λ})a||
≤ ||a^{∗}Ê^{∗∗}(1 − u_{λ})a||.

Since Ê^{∗∗}(1) ∈ M(A) and Ê^{∗∗}(u_{λ}) ∈ A, {a^{∗}Ê^{∗∗}(1 − u_{λ})a} is a decreasing net in A strongly converging to 0 in A^{∗∗}. Thus, for any φ ∈ Q(A),

lim_{λ} φ(a^{∗}Ê^{∗∗}(1 − u_{λ})a) = 0.

As Q(A) is compact in weak* topology, the Dini theorem implies that the above convergence is uniform on Q(A), which means

lim_{λ} ||a^{∗}Ê^{∗∗}(1 − u_{λ})a|| = 0.

Thus, a = lim_{λ} u_{λ}a ∈ A_{1}.

(1). We set Ê to be the restriction of Ê^{∗∗} to A_{1}. Then, Ê has the desired properties. (2). Since A_{1} is an ideal of 𝕃_{B}(E_{E}) and A_{1} ⊃ A, we have Ae_{E} ⊂ A_{1}. Thanks to

Lemma 2.6, (1), we have Ê^{∗∗}(1) ∈ M(A) ⊂ M(A_{1}), and so Ae_{E}Ê^{∗∗}(1) ⊂ A_{1}. Thus, we get

𝕂_{A}(E_{E_{1}}) ⊃ Ae_{E}Ê^{∗∗}(1)e_{E_{1}}e_{E}A = Ae_{E}A,

which shows 𝕂_{A_{1}}(E_{E_{1}}) ⊃ A_{1}.

Remark 2.9. In the above, 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) ⊃ 𝕃_{B}(E_{E}) ⊃ A_{1} is identified with A_{1}^{∗∗} ⊃ M(A_{1}) ⊃ A_{1}. Indeed, (𝕃_{B}(E_{E}) ⊃ A_{1}) ≅ (M(A_{1}) ⊃ A_{1}) is standard [25]. A_{1}^{∗∗} ≅ 𝕃_{A^{∗∗}}(E_{E^{∗∗}}) follows from the fact that B is a full corner of A_{1} and the restriction of B to ind_{B}^{A}π_{S} is quasi-equivalent to the universal representation π_{S}.

Definition 2.10. Let the notations be as above. When A_{1} ⊃ A holds, we define the Watatani index Ind_{w} E of E by Ê^{∗∗}(1) ∈ Z(M(A)). We define the dual conditional expectation E_{1} from A_{1} onto A by

E_{1}(x) = (Ind_{w} E)^{−1}Ê(x), x ∈ A_{1}.

E_{1} satisfies Ind_{p} E_{1} ≤ ||Ind_{w} E||. When A_{1} ⊃ A does not hold, we write Ind_{w} E = ∞.

¿From Watatani’s original definition, it is not so clear whether the inclusions with finite Watatani indices are closed under natural operations, such as taking tensor product with 𝕂 or passing to full corners. However, thanks to Theorem 2.9, we immediately get the following because these two operations behave very well with the second dual:

Lemma 2.11. Let A ⊃ B be an inclusion of C^{∗}-algebras, and E be a conditional expectation from A onto B with finite Ind_{w} E. Then,

(1) Let E ⊗ Id be the conditional expectation from A ⊗ 𝕂 onto B ⊗ 𝕂. Then, we have Ind_{w} (E ⊗ Id) = (Ind_{w} E) ⊗ 1.

(2) Let p ∈ M(B) be a non-zero projection, such that pBp is a full corner of B, and E_{p} be the restriction of E to pAp, which is a conditional expectation from A onto B. Then, we have Ind_{w} (E_{p}) = (Ind_{w} E)p.

Proof. (1). This follows from (A ⊗ 𝕂)^{∗∗} = A^{∗∗}\overline{⊗}𝔹(ℓ^{2}(ℕ)). (2). This follows from the fact that when pBp is a full corner of B, pAp is a full corner of A as well, and the central supports of p in B^{∗∗} and A^{∗∗} are 1.

Remark 2.12. In the above situation, it is easy to show

M(A ⊗ 𝕂) ∩ (B ⊗ 𝕂)′ = (M(A) ∩ B′) ⊗ ℂ.

When A and B are σ-unital C^{∗}-algebras, we further have

M(pAp) ∩ (pBp)′ = (M(A) ∩ B′)p.

Indeed, since (pBp ⊗ K) is stable, L. G. Brown’s theorem [3, Theorem 4.23] implies that there exists an isometry V ∈ M(B ⊗ 𝕂) ⊂ M(A ⊗ 𝕂) such that V V^{∗} = p ⊗ 1. Thus, Ad(V) is an isomorphism from A ⊗ 𝕂 ⊃ B ⊗ 𝕂 onto pAp ⊗ 𝕂 ⊃ pBp ⊗ 𝕂. Let x ∈ M(A) ∩ B′. Then, have

V (x ⊗ 1)V^{∗} = V V^{∗}(x ⊗ 1) = p(x ⊗ 1),

which shows M(pAp) ∩ (pBp)′ = (M(A) ∩ B′)p.

We give a sufficient condition for E to have finite Ind_{w} E in terms of a quasi-basis in M(A).

Lemma 2.13. Let A ⊃ B be an inclusion of non-unital C^{∗}-algebra and E be a conditional expectation from A onto B with Ind_{p} E < ∞. We assume that there exists a sequence {(u_{i}, v_{i})}_{i=1}^{∞} ⊂ M(A) × M(A) such that if we set

T_{n}(a) = ∑_{i=1}^{n} u_{i}E(v_{i}a), a ∈ A,

{||T_{n} − Id||}_{n} converges to zero. Then, we have A_{1} ⊃ A. In particular, if the restriction of E^{∗∗} to M(A) has a quasi-basis, then A_{1} ⊃ A holds.

Proof. Note that M(A) acts on E_{E} by left multiplication. Since the norm of A and E_{E} are equivalent, the above condition is equivalent to

lim_{n→∞} ||1 − ∑_{i=1}^{n} u_{i}e_{E}v_{i}|| = 0.

Therefore we have

a_{1}a_{2} = ∑_{i=1}^{∞} a_{1}u_{i}e_{E}v_{i}a_{2} ∈ A_{1}, a_{1}, a_{2} ∈ A,

where convergence is in norm topology.

The most interesting inclusions of non-unital C^{∗}-algebras satisfying the assumption of Theorem 2.9 are those of simple C^{∗}-algebras with finite Ind_{p} E, which will be discussed in the next section. We present two rather easy but instructive examples here.

Example 2.14. Let X be a locally compact Hausdorff space with a free action of a finite group G. We set A = C_{0}(X), B = C_{0}(X/G), and E to be the average over G. Then, it is a routine work to show that E has Ind_{w} E = #G, and that the basic construction A_{1} is the crossed product C_{0}(X) ⋊ G (cf. [42, Proposition 2.8.1]).

Example 2.15. Let M(n, ℂ) be the n by n matrix algebra and {e(n)_{ij}}_{i,j=1}^{n} be the canonical matrix units. We set A_{n} = M(n, ℂ), B_{n} = p_{n}A_{n}p_{n} + (1 − p_{n})A_{n}(1 − p_{n}), and

E_{n}(x) = p_{n}xp_{n} + (1 − p_{n})x(1 − p_{n}),

where p_{n} = e(n)_{11}. Then, E_{n} is a conditional expectation from A_{n} onto B_{n}. Simple computation shows that {(u_{i}, u_{i}^{∗})}_{i=1}^{m} is a quasi basis if and only if the following holds:

1 = ∑_{i=1}^{m} u_{i}p_{n}u_{i}^{∗} = ∑_{i=1}^{m} u_{i}(1 − p_{n})u_{i}^{∗}.

It is easy to construct such elements and we get Ind_{w} E_{n} = 2. Let

V = ∑_{i=1}^{m} u_{i}p_{n} ⊗ e(m)_{1i}.

Then, we have V V^{∗} = 1 ⊗ e(m)_{11} and V^{∗}V ≤ p_{n} ⊗ 1_{m}. Thus, taking the trace of the both sides, we get n ≤ m.

Now, we set A = ⊕_{n=1}^{∞}A_{n}, B = ⊕_{n=1}^{∞}B_{n}, E = ⊕_{n=1}^{∞}E_{n}, where the direct sums are taken in the C^{∗}-sense. Then, it is easy to show that M(A) = A^{∗∗} is the von Neumann algebra direct sum, and the same thing is true for B as well. Therefore we have Ind_{w} E = 2. However, the above estimate implies that the extension of E to M(A) does not have a quasi-basis consisting of finitely many elements.

3. The simple case

We begin this section with recalling an example discussed in [42, Example 2.9.1]. Let λ be a positive number larger than or equal to 4, and {e_{i}}_{i=1}^{∞} be a sequence of projections satisfying the Jones relation, i.e.

e_{i}e_{j} = e_{j}e_{i}, |i − j| > 1,

e_{i}e_{i+±1}e_{i} = \frac{1}{λ}e_{i}.

Let A ⊃ B be the C^{∗}-algebra generated by {1}∪{e_{i}}_{i=1}^{∞} and {1}∪{e_{i}}_{i=2}^{∞} respectively with the Markov trace τ [18]. Then, there exists a trace preserving conditional expectation E from A onto B. It is shown in [42, Example 2.9.1] that although Ind_{p} E = λ, there exists no quasi-basis for E. One big difference from the case of λ < 4, where there exists a quasi-basis, is that A and B are not simple for λ ≥ 4.

The purpose of this section is to show that this type of an annoying phenomenon never occurs when A or B is simple.

3.1. Existence of Watatani indices. The following rather easy lemma turns out to be extremely useful for our purpose:

Lemma 3.1. Let A be a simple C^{∗}-algebra and {a_{λ}} ⊂ A_{+} be a bounded increasing net converging to a ∈ Z(A^{∗∗}) in strong topology. Then, a is a scalar and

(1) if A is unital, {a_{λ}} converges in norm.

(2) if A is non-unital, {a_{λ}} converges in strict topology of M(A).

Proof. Suppose that a is not a scalar. Then, there would exist a positive number ϵ and a non-zero central projection z ∈ Z(A^{∗∗}) such that za ≠ 0 and ||az|| ≤ ||a|| − ϵ. This implies ||a_{λ}z|| ≤ ||a|| − ϵ. However, since A is simple, every non-zero representation of A is faithful. Thus, we would get

sup_{λ}{||a_{λ}z||} = sup_{λ}{||a_{λ}||} = ||a||,

which is contradiction. Therefore, a is a scalar. The rest of the statement follows from the same argument as in the proof of Theorem 2.8 using the Dini theorem.

Theorem 3.2. Let A be a simple C^{∗}-algebra and B be a C^{∗}-subalgebra of A. If E is a conditional expectation from A onto B with Ind_{p} E < ∞, then Ind_{w} E is finite and it is a scalar. In particular, when A is unital with Ind_{p} E < ∞, E has a quasi-basis.

Proof. We use the same notation as in the proof of Theorem 2.8. Let {u_{λ}} be an approximate unit of A_{1}. Then, {Ê^{∗∗}(u_{λ})} is an increasing net in A_{+} converging to Ê^{∗∗}(1) ∈ Z(A^{∗∗}) in A^{∗∗}. Thus, Lemma 3.1 implies that Ê^{∗∗}(1) is a scalar and Theorem 2.8 implies that A_{1} ⊃ A with Ind_{w} E = Ê^{∗∗}(1).

The following is a generalization of M. Rieffel’s result [38, Therem 3.1] to the case of inclusions with finite indices.

Theorem 3.3. Let A ⊃ B be an inclusion of C^{∗}-algebras with a conditional expectation E from A onto B with Ind_{p} E < ∞. Then,

(1) If B is simple, A is a finite direct sum of simple closed two-sided ideals.

(2) If A is simple, B is a finite direct sum of simple closed two-sided ideals.

Proof. (1). First we assume that A is unital. Let J be a closed two-sided ideal of A, and {u_{λ}} be an approximate unit of J. Since {u_{λ}} strongly converges to a central projection z in Z(A^{∗∗}) [32, 3.11.10], {E(u_{λ})} converges to a central element E^{∗∗}(z) ∈ Z(B^{∗∗}) in B^{∗∗}. Thus, Lemma 3.1 implies that E^{∗∗}(z) is a scalar and {E(u_{λ})} converges in norm. However,

0 ≤ z − u_{λ} ≤ (Ind_{p} E)E^{∗∗}(z − u_{λ})

implies that {u_{λ}} converges to z in norm, and so z ∈ J and J = Az. Since the restriction of E to Z(A) is a state satisfying the Pimsner-Popa inequality, Z(A) is finite dimensional [42, Proposition 2.7.3]. Thus, A is a finite direct sum of simple C^{∗}-algebras.

When A is non-unital, the same argument as above implies that for every b ∈ B, {b^{∗}(z − u_{λ})^{2}b} converges in norm. Thanks to Lemma 2.6, (1), we have AB = A. Thus, {u_{λ}} converges to z in strict topology. The same argument as above shows that Z(M(A)) is finite dimensional, and A is a finite direct sum of simple C^{∗}-algebras.

(2). Thanks to Theorem 3.2, we can apply (1) to A_{1} ⊃ A, and so A_{1} is a finite direct sum of simple C^{∗}-algebras. Since A_{1} is strongly Morita equivalent to B, we get the result.

Corollary 3.4. Let A ⊃ B be an inclusion C^{∗}-algebras, and E be a conditional expectation from A onto B with Ind_{p} E < ∞. Assume that B is simple. Then Ind_{w} E is finite, and if moreover A is unital, E has a quasi-basis.

Proof. Thanks to Theorem 3.3, A is a finite direct sum of simple C^{∗}-algebras. First we assume that A is unital. Let {z_{i}}_{i=1}^{n} be the set of minimal central projections of A. Then, E(z_{i}) is a positive scalar, say c_{i}. We define a conditional expectation E_{i} from Az_{i} onto Bz_{i} by

E_{i}(a) = \frac{1}{c_{i}} E(a)z_{i}, a ∈ Az_{i}.

Then, Ind_{p} E_{i} ≤ c_{i}(Ind_{p} E). Thus, we can apply Theorem 3.2 to E_{i} and there exist quasi-bases {(u_{k}^{i}, u_{k}^{i}^{∗})}_{k=1}^{m_{k}} for E_{i}.

\{(\frac{1}{\sqrt{c_{i}}} u_{k}^{i}, \frac{1}{\sqrt{c_{i}}} u_{k}^{i}^{∗})\}_{i,k}

is a quasi-basis for E with

Ind_{w} E = ∑_{i=1}^{n} \frac{1}{c_{i}} (Ind_{w} E_{i})z_{i}.

When A is not unital, passing to the second dual, we get the same formula

Ê^{∗∗}(1) = ∑_{i=1}^{n} \frac{1}{c_{i}} (Ind_{w} E_{i})z_{i} ∈ Z(M(A)),

where E_{i} is defined in the same way using z_{i} ∈ Z(M(A)). Thus, Theorem 2.8 shows that Ind_{w} E exists.

3.2. The stable case. For an inclusion of non-unital C^{∗}-algebras A ⊃ B with a conditional expectation E, we denote by E^{M} the restriction of E^{∗∗} to M(A). Then thanks to Lemma 2.6, when Ind_{p} E is finite, E^{M} is a conditional expectation from M(A) onto M(B). Even when Ind_{w} E is finite, Example 2.15 shows that E^{M} does not have a quasi-basis (consisting of finitely many elements) in general. In this subsection, we show that for a stable and σ-unital B, Ind_{p} E < ∞ implies existence of a quasi-basis for E^{M}.

Lemma 3.5. Let A ⊃ B be an inclusion of C^{∗}-algebras with a conditional expectation E from A onto B. Assume that Ind_{p} E is finite. Then,

(1) If B is stable, so is A.

(2) If B is σ-unital, so is A.

Proof. Note that a C^{∗}-algebra D is stable if and only if there exists a system of matrix units {e_{ij}}_{i,j∈ℕ} in M(D) such that

∑_{i=1}^{∞} e_{ii} = 1

holds in strict topology. Thus, (1) and (2) follow from Lemma 2.6, (1).

Proposition 3.6. Let A ⊃ B be an inclusion of stable σ-unital C^{∗}-algebras with a conditional expectation E from A onto B with Ind_{p} E < ∞. Assume that A is simple. Then, there exist isometries V ∈ M(A_{1}), W ∈ M(A) such that V V^{∗} = e_{E} and

e_{E}W = (Ind_{w} E)^{1/2}V.

{((Ind_{w} E)^{1/2}W^{∗}, (Ind_{w} E)^{1/2}W)} is a quasi-basis of E^{M} satisfying

E^{M}(W W^{∗}) = \frac{1}{Ind_{w} E} .

Proof. The following is a modification of Pimsner-Popa’s argument in [33]. We use the same notation as in the proof of Theorem 2.8. Thanks to Theorem 3.2 and Lemma 3.5, A_{1} is stable and σ-unital. Since e_{E} is a projection in 𝕃_{B}(E_{E}) = M(A_{1}) such that e_{E}A_{1}e_{E} = Be_{E} is a stable σ-unital full corner, L. G. Brown’s theorem [3, Theorem 4.23] implies that there exists an isometry V ∈ M(A_{1}) satisfying V V^{∗} = e_{E}. We set

W = (Ind_{w} E)^{−1/2}Ê^{∗∗}(V).

Note that W belongs to M(A) thanks to Lemma 2.6, (2) and Remark 2.9. We claim that for every x ∈ 𝕃_{A^{∗∗}}(E_{E^{∗∗}}), e_{E}x = e_{E}Ê^{∗∗}(e_{E}x) holds [33]. Indeed, one can easily check it for x ∈ A^{∗∗}e_{E}A^{∗∗}. Since A^{∗∗}e_{E}A^{∗∗} is weakly dense in 𝕃_{A^{∗∗}}(E_{E^{∗∗}}), the claim holds. Applying this to V, we get

V = (Ind_{w} E)^{1/2}e_{E}W.

Since V is isometry, we have

1 = (Ind_{w} E)W^{∗}e_{E}W. (3.1)

Applying Ê^{∗∗} to the both side, we see that W is an isometry. In a similar way, using V V^{∗} = e_{E}, we get

E^{M}(W W^{∗}) = \frac{1}{Ind_{w} E} .

Now we show the quasi-basis property. Let x ∈ M(A). Then (3.1) implies

xe_{E} = (Ind_{w} E)W^{∗}E^{M}(W x)e_{E}.

Applying Ê^{∗∗} to the both sides, we get

x = (Ind_{w} E)W^{∗}E^{M}(W x),

which shows that {((Ind_{w} E)^{1/2}W^{∗}, (Ind_{w} E)^{1/2}W)} is a quasi-basis for E^{M}.

As a corollary we get the following:

Corollary 3.7. Let A ⊃ B be an inclusion of σ-unital infinite dimensional C^{∗}- algebras with a conditional expectation E from A onto B. If A is simple, we have

Ind_{p} E = Ind_{cp} E = Ind_{w} E.

Proof. We already know that if one of these quantities is finite, so are all. Thus, we assume that they are finite. Thanks to Theorem 3.3, B is a finite direct sum of infinite dimensional simple C^{∗}-algebras, and so B does not have a non-zero finite dimensional irreducible representation. Therefore, Lemma 2.2 implies Ind_{p} E = Ind_{cp} E. On the other hand, in general we have Ind_{p} E ≤ Ind_{w} E.

Note that Ind_{cp} E and Ind_{w} E do not change if we replace (A, B, E) with

(A ⊗ 𝕂, B ⊗ 𝕂, E ⊗ Id).

Thus, we may assume that B is stable. Since A is dense in M(A) in the strict topology, Ind_{p} E = Ind_{p} E^{M} holds. Therefore, Proposition 3.6 implies

Ind_{w} E ≤ Ind_{p} E^{M} = Ind_{p} E,

which finishes the proof.

3.3. The purely infinite case. There are several properties inherited through an inclusion of a finite index (cf. [31]). Nuclearity and exactness are typical examples of such. We show that pure infiniteness in the sense of J. Cuntz [6] is also inherited as an application of Theorem 3.3. A slightly different proof using the SP-property is found in [31]. In the case of crossed product inclusions, this type of results were proven by A. Kishimoto-A. Kumjian [22], and J. A. Jeong-K. Kodaka-H. Osaka [17].

The next lemma was announced by E. Kirchberg in 1994, which can be shown in a similar way as in [39, Theorem 3.2] using [39, Lemma 3.1]. Let ω ∈ βℕ \ ℕ be a free ultra-filter. For a C^{∗}-algebra A, we set

c_{ω}(A) = {(x_{n}) ∈ ℓ^{∞}(ℕ, A); lim_{n→ω} ||x_{n}|| = 0},

A^{ω} = ℓ^{∞}(ℕ, A)/c_{ω}(A).

Lemma 3.8 (Kirchberg). Let A be a unital infinite dimensional C^{∗}-algebra. Then, A is purely infinite simple if and only if A^{ω} is simple.

Theorem 3.9. Let A ⊃ B be an inclusion of C^{∗}-algebras with a conditional expectation E from A onto B with Ind_{p} E < ∞. If either A or B is simple purely infinite, the other is a finite direct sum of simple purely infinite C^{∗}-algebras.

Proof. Thanks to Theorem 3.2, we may assume that B is simple purely infinite by passing to the dual inclusion if necessary. Furthermore, Theorem 3.3 shows that we may assume that A and B are unital and simple by passing to corners. Let E^{ω} be the conditional expectation from A^{ω} onto B^{ω} naturally induced by E. Then, we also have Ind_{p} E^{ω} < ∞. Since B^{ω} is simple due to Lemma 3.8, Theorem 3.3 implies that A^{ω} is a finite direct sum of a simple unital C^{∗}-algebras. Suppose that A^{ω} is not simple. Then, there would exist a central projection z ∈ A^{ω} with z ≠ 0, 1. By the usual perturbation argument, we can take a representing sequence (z_{n}) of z such that for all n, z_{n} is a projection with z_{n} ≠ 0, 1. However, since A is simple, there exists a sequence (a_{n}) ∈ ℓ^{∞}(ℕ, A) such that ||a_{n}|| = 1, z_{n}a_{n} = a_{n}, and a_{n}(1 − z_{n}) = a_{n} hold for all n. Let a = (a_{n}) + c_{ω}(A) ∈ A^{ω}. Then, a ≠ 0 though a does not commute with z which is contradiction. Thus, A^{ω} is simple and A is purely infinite.

A unital simple purely infinite C^{∗}-algebra A is said to be in the Cuntz standard from if [1] = 0 holds in K_{0}(A). In the same way as in the proof of Proposition 3.6, (or more easily using nice K-theoretical properties of unital purely infinite C^{∗}-algebras shown in [6]), we can show the following:

Proposition 3.10. Let A ⊃ B be inclusion of unital simple purely infinite C^{∗}- algebras in the Cuntz standard form with a conditional expectation E from A onto B with Ind_{p} E < ∞. Then, there exist isometries V ∈ A_{1}, W ∈ A such that V V^{∗} = e_{E} and

e_{E}W = (Ind_{w} E)^{1/2}V.

{((Ind_{w} E)^{1/2}W^{∗}, (Ind_{w} E)^{1/2}W)} is a quasi-basis of E satisfying

E(W W^{∗}) = \frac{1}{Ind_{w} E} .

Proof. Since A_{1} is also unital purely infinite simple being strongly Morita equivalent to B, the only point is to show that [1] = [e_{E}] = 0 in K_{0}(A_{1}). As A_{1} ⊃ A is a unital inclusion and [1] = 0 holds in K_{0}(A), we have [1] = 0 in K_{0}(A_{1}). Let π be an isomorphism from B into the corner e_{E}A_{1}e_{E} defined by π(b) = be_{B}. Then, [e_{E}] = π_{∗}([1_{B}]) = 0 holds as well. Thanks to Cuntz’ observation in [6], 1 and e_{E} are equivalent in A_{1}, and we get a desired isometry V. The rest of the argument is the same as the proof of the Lemma 3.6.

Remark 3.11. Even if we drop the condition that A and B are in the Cuntz standard form in the above, we still have a one-element quasi-basis consisting of a multiple of an isometry W as above. The condition we have to give up in this case is

E^{M}(W W^{∗}) = \frac{1}{Ind_{w} E} .

Indeed, using the fact that A_{1} is purely infinite, we can find an isometry V ∈ A_{1} satisfying V V^{∗} ≤ e_{E}, which is enough to get such W.

3.4. Finite group actions. When A is a unital C^{∗}-algebra and α is an action of G on A. We say that α is outer if for each g ≠ e, α_{g} is not implemented by a unitary of M(A). In general, it is not so clear when the fixed point inclusion A ⊃ A^{α} has a quasi-basis while the crossed product inclusion A ⋊_{α} G ⊃ A always does (see [42]). However, when A is simple, we have the following (cf. [42, Proposition 2.8.6, 2.8.8]):

Corollary 3.12. Let A be a simple C^{∗}-algebra and α be an action of a finite group G on A. Let E be the condition expectation from A onto the fixed point subalgebra A^{α} given by the average over G. Then,

(1) Ind_{w} E is finite. In particular, when A is unital, E has a quasi-basis.

(2) Moreover, if α is outer, Ind_{w} E = #G and the basic construction is identified with the crossed product A ⋊_{α} G.

Proof. (1). This immediately follows from Theorem 3.2.

(2). We define a projection e ∈ M(A ⋊_{α} G) by

e = \frac{1}{#G} ∑_{g∈G} u_{g},

where {u_{g}}_{g∈G} is the implementing unitary representation of α in M(A ⋊_{α} G). Since α is outer, the crossed product A ⋊_{α} G is simple [21], and so e(A ⋊_{α} G)e is a full corner of A ⋊_{α} G. Thus, A_{1} ⊃ A ⊃ A^{α} is isomorphic to

e((A ⊗ 𝔹(ℓ^{2}(G))) ⋊_{α⊗Ad(ρ)} G)e ⊃ e(A ⊗ 𝔹(ℓ^{2}(G)))e ⊃ e(A ⋊_{α} G)e,

where ρ is the right regular representation of G. Therefore, the statement follows from Lemma 2.11, (2).

Remark 3.13. When α is outer in the above situation, it is straightforward to show M(A_{1}) ∩ A′ = ℂ because we have A_{1} = A ⋊_{α} G. This also implies that the relative commutant M(A) ∩ A^{α}′ is trivial. Indeed, for x ∈ M(A) ∩ A^{α}′, we can define an element R_{x} ∈ 𝕃_{B}(E_{E}) ∩ A′ = M(A_{1}) ∩ A′ by

R_{x}η_{E}(a) = η_{E}(ax), a ∈ A.

However, R_{x} ∈ M(A_{1}) ∩ A′ = ℂ implies x ∈ ℂ.

The above corollary assures only existence of a quasi-basis, and says nothing about the required number of elements for a quasi-basis in terms of the index. Indeed, the following example shows that we have no control about it.

Example 3.14. We choose a sequence {n_{k}}_{k=1}^{∞} of integers larger than 1 satisfying

∑_{k=1}^{∞} \frac{1}{n_{k}} < ∞.

Let A be the UHF algebra defined by

A = ⨂_{k=1}^{∞} M(n_{k}, ℂ),

where M(n, ℂ) is the n by n matrix algebra. We choose a rank one projection p_{k} from each M(n_{k}, ℂ) and define self-adjoint unitaries u_{k} by u_{k} = 1 − 2p_{k}. We define a G = ℤ/2ℤ action α by

α = ⨂_{k=1}^{∞} Ad(u_{k}).

Then, α is an outer action of A. We set B to be the fixed point algebra A^{α} and define E by

E(x) = \frac{x + α(x)}{2}.

Then, we have Ind_{w} E = 2. Since α is outer, B is a simple C^{∗}-algebra, and the inclusion A ⊃ B is irreducible, i.e. A∩B′ = ℂ. We show that the required number of elements in the quasi-basis for E depends on the sequence {n_{k}}_{k=1}^{∞}, and has nothing to do with the index of E.

Let τ be the unique normalized trace, and M be the weak closure of A in the GNS representation of A. We use the same symbols τ and α for their normal extensions. We denote by ||a||_{2} the L^{2}-norm of M defined by τ. We set

v_{n} = ⨂_{k=1}^{n} u_{k}.

Then, simple calculation shows that for n < m we have

||v_{m} − v_{n}||_{2}^{2} = 2(1 − ∏_{k=n+1}^{m}(1 − \frac{2}{n_{k}})).

Therefore, {v_{k}}_{k=1}^{∞} converges to a self-adjoint unitary operator v ∈ M in ∗-strong topology. Thus, the extension of α to M is the inner automorphism implemented by v. Let e_{+} and e_{−} be the spectral projections of v corresponding to 1 and −1 respectively. Then, we have

τ(e_{−}) = \frac{1}{2}(1 − ∏_{k=1}^{∞}(1 − \frac{2}{n_{k}})).

Let {(u_{i}, u_{i}^{∗})}_{i=1}^{n} be a quasi-basis for E. Since E is given by

E(x) = \frac{x + vxv^{∗}}{2} = e_{+}xe_{+} + e_{−}xe_{−},

we get

1 = ∑_{i=1}^{n} u_{i}e_{+}u_{i}^{∗} = ∑_{i=1}^{n} u_{i}e_{−}u_{i}^{∗}.

Then, the same argument as in Example 2.15 implies

\frac{1}{τ(e_{−})} ≤ n.

As we can make τ(e_{−}) arbitrarily close to 0 by choosing the sequence {n_{k}}_{k=1}^{∞}, we see that there is no upper bound of the number n in terms of Ind_{w} E even for inclusions of simple C^{∗}-algebras with trivial relative commutant. In this example, A_{1} = A⋊_{α}ℤ/2ℤ has two extreme traces while A has a unique trace τ. Therefore, even though the Jones projection takes value 1/2 in the trace extension τ · E_{1}, e_{E} is not necessarily “big” in K_{0}(A_{1}), which also explains this phenomenon.

4. Sectors

Theorem 3.2 enables us to perform various algebraic constructions in the case of inclusions of simple C^{∗}-algebras. In the rest of the paper, we present such applications. Our results seem to be new even in the case of crossed product and fixed point inclusions for finite group actions when intermediate C^{∗}-subalgebras are considered (see Section 6).

The purpose of this section is to develop the sector theory of C^{∗}-algebras (cf. [27]). We treat only two classes of C^{∗}-algebras:

C_{1} : Simple stable σ-unital C^{∗}-algebras

C_{2} : Simple unital purely infinite C^{∗}-algebras in the Cuntz standard form.

This is not at all a constraint because for a given inclusion of simple σ-unital C^{∗}- algebras, we can find an inclusion consisting of C^{∗}-algebras in the above classes by taking a tensor product with 𝕂 (or by passing to corners if the algebras are purely infinite) without changing the combinatorial nature (e.g. higher relative commutants) of the inclusions. This is one of the non-trivial consequences of Theorem 3.2. We mainly discuss C_{1} because C_{2} is treated in the same way. Thanks to Corollary 3.7, we do not need to distinguish the three different definitions of indices in these classes, and we simply denote them by Ind E.

For A, B ∈ C_{1}, we denote by Mor(B, A)_{0} the set of ∗-homomorphism ρ from B to A such that the image ρ(B) has a finite index in A. ρ has a strictly continuous extension to M(B), which is still denoted by ρ for simplicity. We denote by E_{ρ} the minimal expectation from A onto ρ(B), which can be shown to exist in the same way as in [11], [42, Theorem 2.12.3], using M(A) ∩ ρ(B)′ instead of A ∩ ρ(B)′ (for the properties of minimal expectations, see also [24], [28], [19]). We define the statistical dimension d(ρ) of ρ by

d(ρ) = \sqrt{Ind E}.

ρ is said to be irreducible if M(A) ∩ ρ(B)′ = ℂ holds. For two ρ_{1}, ρ_{2} ∈ Mor(B, A)_{0}, we define the space of intertwiners between ρ_{1} and ρ_{2} by

(ρ_{1}, ρ_{2}) = {V ∈ M(A); V ρ_{1}(x) = ρ_{2}(x)V, x ∈ B}.

When ρ_{1} is irreducible, (ρ_{1}, ρ_{2}) has Hilbert space structure with an inner product given by

V_{2}^{∗}V_{1} = ⟨V_{1}, V_{2}⟩1, V_{1}, V_{2} ∈ (ρ_{1}, ρ_{2}).

We say that ρ_{1}, ρ_{2} ∈ Mor(B, A)_{0} are equivalent if there exists a unitary u ∈ M(A) such that Ad(u)·ρ_{1} = ρ_{2}. We denote by Sect(A, B) the quotient space of Mor(B, A)_{0} by this equivalence relation (note that the order of A and B are reversed in theses two symbols). A member of Sect(A, B) is said to be an A − B sector. We introduce 3 categorical operations into Sect(A, B), which is actually similar to KK-operations.

The product of two sectors [ρ_{1}] ∈ Sect(A, B) and [ρ_{2}] ∈ Sect(B, C) is defined by composition:

[ρ_{1}][ρ_{2}] := [ρ_{1} · ρ_{2}].

For two [ρ_{1}], [ρ_{2}] ∈ Sect(A, B), the direct sum is defined as follows: Since A is stable, there exist two isometries S_{1}, S_{2} ∈ M(A) satisfying the O_{2} relation S_{1}S_{1}^{∗} + S_{2}S_{2}^{∗} = 1. We define ρ ∈ Mor(B, A)_{0} by

ρ(x) = S_{1}ρ_{1}(x)S_{1}^{∗} + S_{2}ρ_{2}(x)S_{2}^{∗},

and set [ρ_{1}] ⊕ [ρ_{2}] := [ρ].

Lemma 4.1. Let the notations be as above. Then, for every projection p ∈ (ρ, ρ), ρ ∈ Mor(B, A)_{0}, there exists an isometry V ∈ M(A) such that V V^{∗} = p. In consequence, [ρ] is uniquely decomposed into a direct sum of irreducible sectors.

Proof. Let E_{p} be the conditional expectation from pAp onto ρ(B)p defined by

E_{p}(x) = \frac{1}{E_{ρ}(p)}E_{ρ}(x)p, x ∈ pAp.

Then, we have Ind_{p} E_{p} < ∞. As ρ(B)p is stable, pAp is stable thanks to Lemma 3.5. Thus, L. G. Brown’s theorem [3, Theorem 4.23] implies that there exists such V as above. Note that the space (ρ, ρ) = M(A) ∩ ρ(B)′ is finite dimensional [42, Proposition 2.7.3]. Let

M(A) ∩ ρ(B)′ = ⨁_{i=1}^{n} A_{i}

be decomposition of M(A) ∩ ρ(B)′ into the simple components, and {e(i)_{kl}}_{k,l=1}^{m_{i}} be a system of matrix units of A_{i}. For each i, we choose an isometry V (i) ∈ M(A) satisfying V (i)V (i)^{∗} = e(i)_{11}, and set V (i)_{k} = e(i)_{k1}V (i). Then, {V (i)_{k}}_{i,k} satisfy

ρ(x) = ∑_{i=1}^{n} ∑_{k=1}^{m_{k}} V (i)_{k}ρ_{i}(x)V (i)_{k}^{∗},

where ρ_{i} is defined by

ρ_{i}(x) = V (i)^{∗}ρ(x)V (i).

Since Ad(V (i)) is an isomorphism from A ⊃ ρ_{i}(B) onto e(i)_{11}Ae(i)_{11} ⊃ ρ(B)e(i)_{11}, ρ_{i} belongs to Mor(B, A)_{0}. Therefore, we get

[ρ] = ⨁_{i=1}^{n} m_{i}[ρ_{i}].

It is easy to show that ρ_{i} is irreducible and the decomposition is unique.

Now, we introduce conjugate sectors. For this purpose, we construct canonical endomorphisms first [27].

Lemma 4.2. Let A ⊃ B be an inclusion of C^{∗}-algebras in C_{1} of a finite index, and E be the minimal conditional expectation from A onto B. Then, there exists an isomorphism γ_{1} from A_{1} onto B such that if γ is the restriction of γ_{1} to A, there exist two isometries V ∈ (Id_{A_{1}}, γ_{1}) and W ∈ (Id_{A}, γ) satisfying

V^{∗}W = γ_{1}(V)^{∗}W = \frac{1}{\sqrt{Ind E}} ,

E_{1}(x) = W^{∗}γ_{1}(x)W, x ∈ A_{1}.

E(a) = γ_{1}(V^{∗})γ(a)γ_{1}(V), a ∈ A.

Proof. Let V and W be the isometries defined in Proposition 3.6. Then, we have V^{∗}W = (Ind E)^{−1/2} and E_{1}^{M}(V) = (Ind E)^{−1/2}W. Since e_{E}A_{1}e_{E} = Be_{E} and V V^{∗} = e_{E} hold, there exists an surjective isomorphism from γ_{1} satisfying V xV^{∗} = γ_{1}(x)e_{E} for x ∈ A_{1}, which implies

V x = γ_{1}(x)V. (4.1)

Therefore, when γ_{1} is regarded as a map from A_{1} into itself, V ∈ (Id_{A}, γ_{1}). Applying E_{1}^{M} to the both sides of (4.1) for x ∈ A, we get W ∈ (Id, γ). Note that γ_{1} is given by

γ_{1}(x) = (Ind E)E_{1}^{M}(V xV^{∗}), x ∈ A_{1}.

Thus,

γ_{1}(V^{∗})W = (Ind E)E_{1}^{M}(V V^{∗}V^{∗})W = \sqrt{Ind E}E_{1}^{M}(V V^{∗})
= \frac{1}{\sqrt{Ind E}} .

For a_{1}, a_{2} ∈ A, we have

W^{∗}γ_{1}(a_{1}e_{E}a_{2})W = a_{1}W^{∗}γ_{1}(V V^{∗})W a_{2} = \frac{1}{Ind E} a_{1}a_{2} = E_{1}(a_{1}e_{E}a_{2}),

which shows E_{1}(x) = W^{∗}γ_{1}(x)W. As Proposition 3.6 implies A = BW W^{∗}B, to prove E(a) = γ_{1}(V^{∗}aV), it suffices to show it for a = b_{1}W W^{∗}b_{2}, b_{1}, b_{2} ∈ B. Since γ_{1} is a map onto B, there exist x_{1}, x_{2} ∈ A_{1} such that γ_{1}(x_{i}) = b_{i}, i = 1, 2. Thus, we get

γ_{1}(V^{∗}b_{1}W W^{∗}b_{2}V) = γ_{1}(x_{1}V^{∗}W W^{∗}V x_{2}) = \frac{1}{Ind E} γ_{1}(x_{1}x_{2})
= E(b_{1}W W^{∗}b_{2}),

which finishes the proof.

Remark 4.3. It was pointed out in [20, Theorem 5.12] that in general, downward basic construction does not exist even for an inclusion of simple C^{∗}-algebras of a finite index. However, the above lemma shows that when B is stable, it always exists. Indeed, we define a conditional expectation E_{−1} from B onto γ(A) by γ · E_{1} · γ_{1}^{−1} and set e_{E_{−1}} = W W^{∗}. Then, we have Be_{E_{−1}}B = A and

e_{E_{−1}}be_{E_{−1}} = W W^{∗}γ_{1}(γ_{1}^{−1}(b))W W^{∗} = W E_{1}(γ_{1}^{−1}(b))W^{∗} = E_{−1}(b)e_{E_{−1}}, b ∈ B.

Therefore, we get the downward tower

A ⊃ B ⊃ γ(A) ⊃ γ(B) ⊃ · · · .

γ is called the canonical endomorphism for A ⊃ B. In the above argument, the only choice made was that of V, and V could be replaced with uV, where u is a unitary in M(B). Thus, γ is uniquely determined by the inclusion as a B − A sector. When γ is regarded as a map from B to itself, it induces a map on the K-groups of B, which is nothing but the transfer map discussed in [42] and [20].

Lemma 4.4. For A, B ∈ C_{1} and ρ ∈ Mor(B, A)_{0} there exists \overline{ρ} ∈ Mor(A, B)_{0}, unique up to equivalence, such that the following holds: there exist isometries R_{ρ} ∈ (Id_{B}, \overline{ρ}·ρ) and \overline{R}_{ρ} ∈ (Id_{A}, ρ · \overline{ρ}) such that

\overline{R}^{∗}ρ(R) = R_{ρ}^{∗}\overline{ρ}(\overline{R}_{ρ}) = \frac{1}{d(ρ)}.

Moreover, d(ρ) = d(\overline{ρ}) holds.

Proof. We apply the previous lemma to A ⊃ ρ(B) and set \overline{ρ} = ρ^{−1} · γ, R_{ρ} = ρ^{−1}(γ_{1}(V)), and \overline{R}_{ρ} = W. Then, these have the desired properties. Note that the conditions in the statement imply that

B ⊃ \overline{ρ}(A) ⊃ \overline{ρ} · ρ(B)

is the basic construction (regardless the particular construction we made). Therefore, we get d(\overline{ρ}) = d(ρ).

Now, we show uniqueness. Assume that (σ, S, \overline{S}) satisfies the same property for (\overline{ρ}, R_{ρ}, \overline{R}_{ρ}) in the statement. Then, σ(\overline{R}_{ρ}^{∗})S ∈ (\overline{ρ}, σ). We show that σ(\overline{R}_{ρ}^{∗})S is a nonzero multiple of a unitary, which will finish the proof. We set E(x) = ρ(S^{∗}σ(x)S), x ∈ A. Then, E is a conditional expectation from A onto ρ(A) with a quasi-basis {(d(ρ)S^{∗}, d(ρ)S)}. Since Ind E = Ind E_{ρ} and E_{ρ} is the minimal expectation, we get E = E_{ρ}. Thus,

ρ(S^{∗}σ(\overline{R}_{ρ}\overline{R}_{ρ}^{∗})S) = E_{ρ}(\overline{R}_{ρ}\overline{R}_{ρ}^{∗}) = ρ(R_{ρ}^{∗}\overline{ρ}(\overline{R}_{ρ}\overline{R}_{ρ}^{∗})R_{ρ}) = \frac{1}{d(ρ)^{2}} ,

which shows that d(ρ)σ(\overline{R}_{ρ}^{∗})S is an isometry. In a similar way, we get E_{σ}(x) = σ(\overline{S}^{∗}ρ(x)\overline{S}), x ∈ B. Therefore,

E_{σ}(d(ρ)^{2}σ(\overline{R}_{ρ}^{∗})SS^{∗}σ(\overline{R}_{ρ})) = d(ρ)^{2}σ(\overline{R}_{ρ}^{∗})E_{σ}(SS^{∗})σ(\overline{R}_{ρ}) = 1.

Since E_{σ} is faithful, d(ρ)σ(\overline{R}_{ρ}^{∗})S is a unitary.

The sector [\overline{ρ}] is called the conjugate sector of [ρ].

With isometries R_{ρ}, \overline{R}_{ρ}, it is possible to show that basically, every combinatorial result known for factors holds also for inclusions of C^{∗}-algebras in C_{1}, such as the Frobenius reciprocity discussed in [15] (c.f. [20]). Moreover, Remark 2.11 shows that taking tensor product with 𝕂 does not change higher relative commutants, which allows us to talk about principal graphs, standard invariants and so on.

Definition 4.5. Let A ⊃ B be an inclusion of simple σ-unital C^{∗}-algebras with a finite index. We denote by ι the inclusion map from B ⊗ 𝕂 into A ⊗ 𝕂. We call the set of sectors generated by [ι] and [\overline{ι}] the sectors associated with A ⊃ B. When it only contains finitely many irreducibles, A ⊃ B is said to be of finite depth.

As already pointed out in [42], [20], the most significant part of sectors in the C^{∗}-case is that they give rise to KK-elements, which obey particular fusion rules if one starts from an inclusion with a particular combinatorial nature.

Remark 4.6. All the arguments in this section also work for inclusions of C^{∗}-algebras in the class C_{2}. The only difference in the proof is to use the following observation, which was already used in Proposition 3.9: When A ⊃ B is an inclusion of C^{∗}-algebras in C_{2} of a finite index, [p] = 0 holds in K_{0}(A) for every projection p ∈ A ∩ B′. This follows from the fact that p is the image of the unit of B via the inclusion map of pAp ⊃ Bp.

As a corollary of Lemma 4.2 (for the C_{2} case), we get the following, which is announced in [34] too without a proof:

Corollary 4.7. Let M ⊃ N be an inclusion of type III factors with separable preduals. If a conditional expectation E from M onto N satisfies Ind_{p} E < ∞, E is automatically normal.

Proof. Note that every type III factor belongs to C_{2}. Lemma 4.2 implies that there exist an endomorphism γ of M and an isometry R ∈ M such that we have

E(x) = R^{∗}γ(x)R, x ∈ M.

Thanks to Takesaki’s theorem [41], every endomorphism of a von Neumann algebra with a separable predual is automatically normal, and we get the result.

Example 4.8. Let A ⊃ B be an inclusion of simple σ-unital C^{∗}-algebras with the principal graph A_{4} (or in other words, of an index (3 + \sqrt{5})/2). Then, [ι][\overline{ι}] is decomposed into two sectors [Id] and [σ], and [σ] satisfies the following fusion rule [12]:

[σ]^{2} = [Id] ⊕ [σ].

This implies that there exist two elements x_{i} ∈ End(K_{i}(A)), i = 0, 1 such that x_{i}^{2} = x_{i} + 1 holds. More specifically, we assume that A is stably isomorphic to the Cuntz algebra O_{n+1} [5]. Then, we have K_{0}(A) ≅ ℤ/nℤ, K_{1}(A) = {0}. Therefore the Diophantine equation

x^{2} ≡ x + 1, mod n

should have a solution. Here is the list of small integers having this property:

1, 5, 11, 19, 29, 31, 41, · · · .

Now we construct examples realizing some of the above groups. In [13, Example 3.1], an endomorphism ρ ∈ End(O_{2}) whose image has an index (3 + \sqrt{5})/2 is constructed by the following formula:

ρ(S_{1}) = \frac{1}{d}S_{1} + \frac{1}{\sqrt{d}}S_{2}S_{2},

ρ(S_{2}) = (\frac{1}{d}S_{1} − \frac{1}{d}S_{2}S_{2})S_{2}^{∗} + S_{2}S_{1}S_{1}^{∗}.

where d = (1 + \sqrt{5})/2. ρ and E_{ρ} commute with the following 𝕋 = ℝ/2πℤ-action:

λ_{t}(S_{1}) = e^{2it}S_{1}, λ_{t}(S_{2}) = e^{it}S_{2}.

Let G_{n} be a cyclic subgroup of 𝕋 of order n. Then, the restriction of E_{ρ} to the fixed point algebra O_{2}^{G_{n}} is a conditional expectation onto ρ(O_{2}^{G_{n}}). Since E_{ρ}(S_{1}S_{1}^{∗}) = 1/d^{2} holds and S_{1}S_{1}^{∗} ∈ O_{2}^{G_{n}}, the index of the restriction is the same as Ind E = (3+\sqrt{5})/2. It is a routine work to show that O_{2}^{G_{n}} is stably isomorphic to the Cuntz-Krieger algebra O_{D^{n}} [8], where

D = [ \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ] .

Thus, thanks to [7], the K-groups of O_{2}^{G_{n}} are given by

K_{0}(O_{2}^{G_{n}}) = ℤ^{2}/(1 − D^{n})ℤ^{2},
K_{1}(O_{2}^{G_{n}}) = {0}.

For example,

K_{0}(O_{2}^{G_{2}}) = {0}, K_{0}(O_{2}^{G_{3}}) = ℤ/2ℤ ⊕ ℤ/2ℤ, K_{0}(O_{2}^{G_{4}}) = ℤ/5ℤ,
K_{0}(O_{2}^{G_{5}}) = ℤ/11ℤ, K_{0}(O_{2}^{G_{6}}) = ℤ/4ℤ ⊕ ℤ/4ℤ, K_{0}(O_{2}^{G_{7}}) = ℤ/29ℤ.

It is possible to construct various inclusions of Cuntz-Krieger algebras for each paragroup in a similar way using [15]. For example, the principal graph E_{8} can be realized in the Cuntz-Krieger algebra (in fact, it is stably isomorphic to the Cuntz algebra O_{32} thanks to [40]) with K_{0} isomorphic to ℤ/31ℤ. Since descendant sectors of the E_{8} inclusion contains a sector with the principal graph A_{4} [12], we see that ℤ/31ℤ can be also realized in the above list.

5. Normal extensions of conditional expectations

There seems to be a common belief, at least among some specialists, saying that “when a conditional expectation leaves a state invariant, it should have a normal extension to the weak closure in the GNS representation of the state”. Probably, it comes from an analogy to the case of group actions, or depth 2 inclusions. In this section, we provide a counter example for this statement as an application of sector theory developed in the last section.

The following shows that there is no real distinction between induction and reduction when an inclusion has a finite index.

Lemma 5.1. Let A, B be C^{∗}-algebras in C_{1} (or C_{2}), and σ, π be representations of A and B respectively.

(1) Let ρ ∈ Mor(B, A)_{0}, then the map

Hom(σ · ρ, π) ∋ X ↦ Xσ(\overline{R}_{ρ}) ∈ Hom(σ, π · \overline{ρ})

is a vector space isomorphism.

(2) If A ⊃ B has a finite index, the following holds:

σ|_{B} ≅ ind_{A}^{A_{1}}σ · γ_{1}^{−1},
ind_{B}^{A}π ≅ π · γ,

where γ and γ_{1} are as in Lemma 4.2.

Proof. (1) It is straightforward to show that the inverse map is given by

Hom(σ, π · \overline{ρ}) ∋ Y ↦ d(ρ)π(R_{ρ}^{∗})Y ∈ Hom(σ · ρ, π),

which shows the statement.

(2). Since A_{1} = V^{∗}V A_{1} ⊂ V^{∗}e_{E}Ae_{E}A = V^{∗}A, there exists an isometry U from H_{σ} onto E_{E_{1}} \overline{⊗}_{A}H_{σ} such that

Uσ(a)ξ = \sqrt{Ind E}V^{∗}a ⊙_{A} ξ, a ∈ A, ξ ∈ H_{σ},

which shows the first statement. The second one can be shown in the same way using A = W^{∗}B.

Lemma 5.2. Let A ⊃ B be an inclusion of C^{∗}-algebras with a conditional expectation E from A onto B satisfying Ind_{p} E < ∞. For every pure state ψ ∈ P (B), there exist pure state extensions {φ_{i}}_{i=1}^{n} ⊂ P (A) with mutually orthogonal supports in A^{∗∗} and positive numbers {c_{i}}_{i=1}^{n} such that

ψ · E = ∑_{i=1}^{n} c_{i}φ_{i}.

Moreover, (Ind_{p} E)^{−1} ≤ c_{i} and n ≤ Ind_{p} E hold. In particular, we have

ind_{B}^{A}π_{ψ} ≅ ⨁_{i=1}^{n} π_{φ_{i}}.

Proof. We denote by S_{ψ}(A) and P_{ψ}(A) the set of state extensions of ψ to A, and that of the pure state extensions of ψ to A respectively. P_{ψ}(A) is the set of extreme points of S_{ψ}(A), and S_{ψ}(A) is the weak∗ closure of the convex hull of P_{ψ}(A). For a state φ ∈ S(A) and a representation π of A, we denote by s(φ) and c(π) the support of φ in A^{∗∗} and the central support of π in Z(A^{∗∗}). We use the same symbols φ and π for their normal extensions to A^{∗∗} for simplicity.

Let φ ∈ P_{ψ}(A). Then, we have

ψ · E = φ · E ≥ \frac{1}{Ind_{p} E} φ.

In particular,

ψ · E(c(π_{φ})) ≥ ψ · E(s(φ)) ≥ \frac{1}{Ind_{p} E} , (5.1)

which proves the second part of the statement.

(5.1) implies that there exist finitely many mutually disjoint irreducible representations π_{1}, π_{2}, ⋯, π_{m} of A such that whenever ω ∈ P_{ψ}(A), π_{ω} is equivalent to π_{j} for some j. We may assume ψ · E(c(π_{j})) ≠ 0 for all j. Let h_{j} be the density matrix for ψ · E(c(π_{j})·), that is, h_{j} is a trace class operator in 𝔹(H_{π_{j}}) satisfying

ψ · E(c(π_{j})x) = Tr(h_{j}π_{j}(x)), x ∈ A.

Let s(h_{j}) be the support of h_{j}. Since ψ is pure, any state ω ∈ S(A) satisfying cs(h_{j}) ≥ s(ω) for some number c > 0 is in S_{ψ}(A). Thus, (5.1) implies that h_{j} is a finite rank operator. We define a finite dimensional C^{∗}-algebra D by

D = ⨁_{j=1}^{m} 𝔹(s(h_{j})H_{π_{j}}).

Then, (5.1) also implies that there exists a continuous convex map θ : S(D) ⟶ S_{ψ}(A) given by

θ(ϕ)(x) = ϕ(⨁_{j=1}^{m} s(h_{j})π_{j}(x)s(h_{j})), ϕ ∈ S(D), x ∈ A.

Since S(D) is compact, the image of θ is compact in weak∗ topology. Moreover, as the image of θ contains P_{ψ}(A), θ is a surjection, which implies

ψ · E(x) = ∑_{j=1}^{m} ψ · E(c(π_{j})x) = ∑_{j=1}^{m} Tr(h_{j}π_{j}(x)), x ∈ A.

Further decomposing h_{j}, we get the decomposition of the desired form.

The following theorem tells that the depth 2 case is a rather fortunate exception.

Theorem 5.3. Let A ⊃ B be an inclusion of C^{∗}-algebras in either C_{1} or C_{2}, and E be a conditional expectation from A onto B with a finite index.

(1) For a state ψ ∈ S(B), there exists a normal extension of E to π_{ψ·E}(A)′′ if and only if π_{ψ}·γ is quasi-equivalent to π_{ψ}·γ^{2}, where γ is the canonical endomorphism for A ⊃ B.

(2) Let ρ ∈ End(O_{2}) be as in Example 4.8. Then, there exists an E_{ρ}-invariant pure state φ such that E_{ρ} does not have a normal extension to π_{φ}(O_{2})′′.

Proof. (1). Since we have

γ(x) = (Ind E)E(W xW^{∗}), x ∈ A,
E(x) = γ_{1}(V)^{∗}γ(x)γ_{1}(V), x ∈ A,

E has a normal extension if and only if γ does, which is further equivalent to that π_{ψ·E} is quasi-equivalent to π_{ψ·E} · γ. On the other hand, Lemma 5.1, (2) implies

π_{φ·E} ≅ ind_{B}^{A}π_{ψ} ≅ π_{ψ} · γ,

which shows the statement.

(2). Note that [ρ] is a self-conjugate sector obeying the fusion rule

[ρ]^{2} = [Id] ⊕ [ρ]. (5.2)

We claim that there exist two disjoint irreducible representations π_{1}, π_{2} of A such that π_{1} · ρ ≅ π_{2} and π_{2} ≅ π_{1} ⊕ π_{2}. We choose an arbitrary irreducible representation π. Since Ind E_{ρ} = (3+\sqrt{5})/2 < 3, Lemma 5.2 implies that π·ρ is decomposed into at most two irreducible representations, and there are two cases: (i) π · ρ is irreducible. (ii) There exist two irreducible representations σ_{1}, σ_{2} such that π · ρ ≅ σ_{1} ⊕ σ_{2}. We first consider the case (i). We set π_{1} = π, π_{2} = π · ρ. Then, (5.2) implies

π_{2} · ρ = π · ρ^{2} ≅ π ⊕ π · ρ ≅ π_{1} ⊕ π_{2},

which also shows π_{1} and π_{2} are disjoint. We consider the case (ii) now. Lemma 5.1,

(1) implies that σ_{1} · ρ and σ_{2} · ρ contain π. On the other hand, as before (5.2) implies π · ρ^{2} ≅ π ⊕ σ_{1} ⊕ σ_{2}. This means that either σ_{1} or σ_{2} is equivalent to π and we may assume σ_{2} ≅ π. Therefore, we get π · ρ ≅ σ_{1} ⊕ π and σ_{1} · ρ ≅ π. π_{1} := σ_{1} and π_{2} := π satisfy the claim.

Let A = O_{2} and B := ρ(A). We take a pure state ψ ∈ P (B) satisfying π_{ψ} ≅ π_{1}·ρ^{−1}, and set φ = ψ · E. Then, since γ = ρ^{2}, we have

π_{φ} ≅ π_{ψ} · ρ^{2} ≅ π_{1} · ρ ≅ π_{2},

which shows that φ is pure. We also have

π_{ψ} · γ^{2} ≅ π_{2} · ρ^{2} ≅ π_{1} ⊕ π_{2} ⊕ π_{2}.

This shows that π_{φ} and π_{φ}·γ are not quasi-equivalent, and E_{ρ} does not have a normal extension to π_{φ}(A)′′ = 𝔹(H_{φ}).

6. Intermediate C^{∗}-subalgebras

In this section, following the strategy developed in [16], we investigate the structure of intermediate C^{∗}-subalgebras of an inclusion of simple C^{∗}-algebras of a finite index.

Proposition 6.1. Let A ⊃ B be an inclusion of simple σ-unital C^{∗}-algebras, and E be a conditional expectation from A onto B of a finite index. Then, for every intermediate C^{∗}-algebra A ⊃ C ⊃ B, there exists a conditional expectation F from A onto C of a finite index.

Proof. Let e be a minimal projection of 𝕂. If there exists a conditional expectation F from A ⊗ 𝕂 onto C ⊗ 𝕂 of a finite index, the restriction of F to (1 ⊗ e)(A ⊗ 𝕂)(1 ⊗ e) gives a desired expectation. Therefore, we may assume that A, B, and C are stable from the beginning. Thanks to Proposition 3.6 and the argument in the proof of Corollary 3.4, there exists a quasi-basis {(u_{i}, u_{i}^{∗})}_{i=1}^{n} for the restriction of E^{M} to M(C). We define a completely positive map F from A to C by

F (x) = (Ind_{w} E|_{C})^{−1} ∑_{i,j=1}^{n} u_{i}E(u_{i}^{∗}xu_{j})u_{j}^{∗},

which satisfies the following for a ∈ A_{+}:

F (a) ≥ (Ind_{w} E|_{C})^{−1} ∑_{i,j=1}^{n} u_{i}E(u_{i}^{∗}xu_{j})u_{j}^{∗}(Ind_{w} E|_{C})^{−1}

≥ \frac{1}{Ind E} (Ind_{w} E|_{C})^{−1} ∑_{i,j=1}^{n} u_{i}u_{i}^{∗}au_{j}u_{j}^{∗}(Ind_{w} E|_{C})^{−1}

= \frac{1}{Ind E} a.

We show that F is a conditional expectation from A onto C. Let {(v_{k}, v_{k}^{∗})}_{k=1}^{m} be another quasi-basis for E^{M}|_{M(C)}. Then, we have

F (x) = ∑_{k=1}^{m} ∑_{i,j=1}^{n} v_{k}E^{M}(v_{k}^{∗}u_{i})E(u_{i}^{∗}xu_{j})u_{j}^{∗}

= ∑_{k=1}^{m} ∑_{i,j=1}^{n} v_{k}E(E^{M}(v_{k}^{∗}u_{i})u_{i}^{∗}xu_{j})u_{j}^{∗}

= ∑_{k=1}^{m} ∑_{j=1}^{n} v_{k}E(v_{k}^{∗}xu_{j})u_{j}^{∗}.

Let u be a unitary in M(C). Since {(u^{∗}u_{i}, u_{i}^{∗}u)}_{i=1}^{n} is also a quasi basis for E^{M}|_{M(C)}, we get F (ux) = uF (x), which shows that F is a C-C bimodule map. F^{∗∗}(1) = 1 implies that F has norm 1. Since F is identity on C, F is a conditional expectation.

To give the structure theorem of intermediate C^{∗}-subalgebras, we briefly explain the “crossed product decomposition” of A introduced in [16]. As mentioned before, the Frobenius reciprocity for sectors can be established in the same way as in [15, Section 2], and we use it freely in what follows. Let A ⊃ B be an inclusion of C^{∗}- algebras in C_{1}, and E be a conditional expectation from A onto B of a finite index. We assume that A ⊃ B is irreducible, that is, M(A) ∩ B′ is trivial. We denote by ι the inclusion map from B into A. Let

[\overline{ι}][ι] = ⨁_{r∈S} n_{r}[ρ_{r}],

be the irreducible decomposition. We arrange the index set S so that we have ρ_{e} = Id_{B} and \overline{[ρ_{r}]} = [ρ\overline{r}]. For simplicity, we write d(r) = d(ρ_{r}). For r ∈ S, we set

H_{r} = (ι, ι · ρ_{r}).

Then, the Frobenius reciprocity implies

dim H_{r} = dim(\overline{ι} · ι, ρ_{r}) = n_{r}.

Let {V (r)_{i}}_{i=1}^{n_{r}} be an orthonormal basis of H_{r}. Then, as in [16, Section 3], we can show that \{(\sqrt{d(r)}V (r)_{i}^{∗}, \sqrt{d(r)}V (r)_{i})\}_{r,i} is a quasi-basis for E. For x ∈ A, we set x(r)_{i} = d(r)E(V (r)_{i}x). Then, we have

x = ∑_{r∈S} ∑_{i=1}^{n_{r}} V (r)_{i}^{∗}x(r)_{i}.

Let C be an intermediate C^{∗}-subalgebra between A ⊃ B. Thanks to Theorem 3.3 and the assumption M(A) ∩ B′ = ℂ, C is simple. We set

K_{r} = H_{r} ∩ M(C).

We arrange the orthonormal basis {V (r)_{i}}_{i=1}^{n_{r}} such that {V (r)}_{i=1}^{m_{r}} is an orthonormal basis of K_{r}.

Lemma 6.2. Let the notations be as above. Then, we have

C = {x ∈ A; x(r)_{i} = 0, ∀i > m_{r}, r ∈ S}.

Proof. Let D be the right-hand side of the above equality. Then, C ⊃ D is obvious. Let F be a conditional expectation from A onto C whose existence is assured by Proposition 6.1. We claim that the restriction of F^{M} to H_{r} is the orthogonal projection onto K_{r}. Indeed, since F is a C-C bimodule map, F^{M} globally preserves H_{r}. For V_{1}, V_{2} ∈ H_{r}, we have ⟨F^{M}(V_{1}), V_{2}⟩ = V_{2}^{∗}F^{M}(V_{1}). Since it is already a scalar, it is equal to

F^{M}(V_{2}^{∗}F^{M}(V_{1})) = F^{M}(V_{2}^{∗})F^{M}(V_{1}) = F^{M}(F^{M}(V_{2}^{∗})V_{1}) = ⟨V_{1}, F^{M}(V_{2})⟩,

which shows the claim. This implies that for every x, we get

F (x) = ∑_{r∈S} ∑_{i=1}^{n_{r}} F (V (r)_{i}^{∗})x(r)_{i} = ∑_{r∈S} ∑_{i=1}^{m_{r}} V (r)_{i}^{∗}x(r)_{i} ∈ D.

Therefore, we get the statement.

The above lemma immediately implies the following:

Corollary 6.3. Let A ⊃ B be an irreducible inclusion of C^{∗}-algebras in C_{1} (or C_{2}), and E be a conditional expectation from A onto B of a finite index. Then, there exists a one-to-one correspondence between the following two objects: (1) The set of intermediate C^{∗}-subalgebras. (2) The set of systems of subspaces K_{r} ⊂ H_{r}, r ∈ S satisfying

(i) K_{r}K_{s} ⊂ ∑_{t∈S} BK_{t} (ii) K_{r}^{∗} ⊂ BK\overline{r}. The correspondence is given as follows: For an intermediate C^{∗}-subalgebra C, K_{r} is given by K_{r} = M(C) ∩ H_{r}. On the other hand, for a given system of subspaces K_{r} ⊂ H_{r}, r ∈ S, the corresponding subalgebra is ∑_{r∈S} BK_{r}.

As in [16], we apply this to depth 2 inclusions. Let A ⊃ B be an irreducible inclusion of simple σ-unital C^{∗}-algebras of a finite index, and

B ⊂ A ⊂ A_{1} ⊂ A_{2} ⊂ ⋯,

be the tower. We say that A ⊃ B is of depth 2 if M(A_{2}) ∩ B′ is a factor.

Corollary 6.4. Let A ⊃ B be a depth 2 inclusion of simple σ-unital C^{∗}-algebras. Then, there exists an action δ of a finite dimensional Kac algebra Q such that B is the fixed point algebra A^{δ}. There exists a one-to-one correspondence between the set of intermediate C^{∗}-subalgebras between A ⊃ B and the set of left coideal ∗ subalgebras of Q.

Proof. Let e be a minimal projection of 𝕂. If the statement holds for (A ⊗ 𝕂) ⊃ (B ⊗ 𝕂), the restriction of δ to (1 ⊗ e)(A ⊗ 𝕂)(1 ⊗ e) satisfies the conditions of the statement. Therefore, we may assume that A and B are stable. The rest of the statement follows from exactly the same arguments as in [15, Section 4] and [16, Section 4].

Remark 6.5. (1) As is pointed out in [20, Theorem 5.12], in general, depth two inclusions of C^{∗}-algebras cannot be characterized by crossed products instead of fixed point algebras as above. (2) One can get a similar statement for a group-subgroup inclusion using arguments in [14].

The Galois correspondence of group actions on C^{∗}-algebras has been discussed by several authors [4], [26] (see also [30] and references in it). For a finite group action on a simple C^{∗}-algebra, we get the following as a special case of the previous one:

Corollary 6.6. Let α be an outer action of a finite group G on a simple σ-unital C^{∗}-algebra A. Then, (1) There exists a one-to-one correspondence between the set of subgroups of G and

the set of intermediate C^{∗}-subalgebras between A ⋊_{α} G ⊃ A. The correspondence is given as follows: For a subgroup H, the corresponding intermediate C^{∗}-subalgebra is A ⋊_{α} H.

(2) There exists a one-to-one correspondence between the set of subgroups of G and the set of intermediate C^{∗}-subalgebras between A ⊃ A^{α}. The correspondence is given as follows: For a subgroup H, the corresponding intermediate C^{∗}-subalgebra is the fixed point subalgebra of A under the restriction of α to H.

7. A Kishimoto’s type theorem

Let B be a simple C^{∗}-algebra and α be an outer automorphism of B. In [21], A. Kishimoto proved the following statement (as a special case): For every element x ∈ M(B) and every non-zero hereditary C^{∗}-subalgebra C of B,

inf{||cxα(c)||; c ∈ C_{+}, ||c|| = 1} = 0.

There are a number of applications of this result: for example, it implies that the reduced crossed product of a simple (and purely infinite) C^{∗}-algebra by an outer action of a discrete group is again simple (and purely infinite) [21], [22].

As observed in [31], a counterpart of Kishimoto’s theorem for a conditional expectation E from A onto B would be the following statement: under some condition corresponding to outerness, for every element x ∈ M(A) and every non-zero hereditary C^{∗}-subalgebra C of B,

inf{||c(x − E(x))c||; c ∈ C_{+}, ||c|| = 1} = 0.

In fact, Osaka adopted this property as a definition of outerness of a conditional expectation in [31]. The goal of this section is to prove it for an irreducible inclusions of simple σ-unital C^{∗}-algebras of a finite index and finite depth. In the course of preparation, we prove Proposition 7.3, which is probably of enough interest in its own right.

For A ∈ C_{1}, we use the notations End(A)_{0} = Mor(A, A)_{0}, Sect(A) = Sect(A, A). ρ ∈ End(A)_{0} is said to be irreducible if M(A) ∩ ρ(A)′ = ℂ holds. We say that T = {ρ_{r}}_{r∈T_{0}} ⊂ End(A)_{0} is a closed system of irreducible endomorphisms if the following properties are satisfied:

(1) [ρ_{r}] = [ρ_{s}] implies r = s.

(2) There exists e ∈ T_{0} such that ρ_{e} = Id.

(3) There exist non-negative integers N_{r,s}^{t} for r, s, t ∈ T_{0} such that

[ρ_{r}][ρ_{s}] = ⨁_{t∈T_{0}} N_{r,s}^{t}[ρ_{t}].

(4) For each r ∈ T_{0}, there exists \overline{r} ∈ T_{0} such that \overline{[ρ_{r}]} = [ρ\overline{r}].

For each r ∈ T_{0}, we choose R_{r} ∈ (Id, ρ\overline{r} · ρ_{r}) and \overline{R}_{r} ∈ (Id, ρ_{r} · ρ\overline{r}) as in Lemma 4.4, and for r, s, t ∈ T_{0}, we choose an orthonormal basis {T (_{r,s}^{t})_{a}}_{a=1}^{N_{r,s}^{t}} of (ρ_{t}, ρ_{r} · ρ_{s}).

Now, we assume that A is acting on a Hilbert space H irreducibly, and set

L_{r} = {V ∈ 𝔹(H); V x = ρ_{r}(x)V }.

Thanks to Lemma 5.2, n_{r} := dim L_{r} is finite. We introduce an inner product into L_{r} by

⟨V_{1}, V_{2}⟩1 = V_{2}^{∗}V_{1}.

Lemma 5.1, (1) shows that we can define antilinear maps F_{r} : L_{r} ⟶ L\overline{r} and its inverse \overline{F}_{r} : L\overline{r} ⟶ L_{r} by

F_{r}(V) = \sqrt{d(r)}V^{∗}\overline{R}_{r},
\overline{F}_{r}(W) = \sqrt{d(r)}W^{∗}R_{r}.

Using these, we can introduce another inner product into L_{r} by

(V_{1}, V_{2}) = ⟨F (V_{2}), F (V_{1})⟩ = d(r)\overline{R}_{r}^{∗}V_{1}V_{2}^{∗}\overline{R}_{r}.

We choose an orthogonal basis {V (r)_{i}}_{i=1}^{n_{r}} of L_{r} with respect to the inner product ⟨·, ·⟩ that satisfies

(V (r)_{i}, V (r)_{j}) = c(r)_{i}δ_{i,j}.

Lemma 7.1. Let the notations be as above. We set

C_{(r,i),(s,j)}^{(t,k)} = ∑_{a=1}^{N_{r,s}^{t}}⟨T (_{r,s}^{t})_{a}^{∗}V (r)_{i}V (s)_{j}, V (t)_{k}⟩T (_{r,s}^{t})_{a} ∈ (ρ_{t}, ρ_{r} · ρ_{s}).

Then, the following hold: (1)

V (r)_{i}V (s)_{j} = ∑_{t,k} C_{(r,i),(s,j)}^{(t,k)}V (t)_{k}.

(2)

V (s)_{j}V (t)_{k}^{∗} = ∑_{r,i} \frac{d(r)c(s)_{j}}{d(t)} V (r)_{i}^{∗}C_{(r,i),(s,j)}^{(t,k)}.

Proof. (1). ∑_{t,a} T (_{r,s}^{t})_{a}T (_{r,s}^{t})_{a}^{∗} = 1 implies

V (r)_{i}V (s)_{j} = ∑_{t,a} T (_{r,s}^{t})_{a}T (_{r,s}^{t})_{a}^{∗}V (r)_{i}V (s)_{j}.

Since T (_{r,s}^{t})_{a}^{∗}V (r)_{i}V (s)_{j} ∈ L_{t}, we get

T (_{r,s}^{t})_{a}^{∗}V (r)_{i}V (s)_{j} = ∑_{t,k}⟨T (_{r,s}^{t})_{a}^{∗}V (r)_{i}V (s)_{j}, V (t)_{k}⟩V (t)_{k},

which shows the statement.

(2). Using Lemma 5.1, (1) and a similar argument as above, we get

V (s)_{j}V (t)_{k}^{∗} = \sqrt{d(s)}F_{s}(V (s)_{j})^{∗}R_{s}V (t)_{k}^{∗} = \sqrt{d(s)}F_{s}(V (s)_{j})^{∗}V (t)_{k}^{∗}ρ_{t}(R_{s})
= \sqrt{d(s)} ∑_{r,a} F_{s}(V (s)_{j})^{∗}V (t)_{k}^{∗}T (_{t,\overline{s}}^{r})_{a}T (_{t,\overline{s}}^{r})_{a}^{∗}ρ_{t}(R_{s})
= \sqrt{d(s)} ∑_{r,i,a} F_{s}(V (s)_{j})^{∗}V (t)_{k}^{∗}T (_{t,\overline{s}}^{r})_{a}V (r)_{i}V (r)_{i}^{∗}T (_{t,\overline{s}}^{r})_{a}^{∗}ρ_{t}(R_{s})

30 MASAKI IZUMI

Thanks to the Frobenius reciprocity, we may replace the orthonormal basis \{T (_{t\overline{s}}^{r})\}_{a=1}^{N_{t\overline{s}}^{r}} with

\{\sqrt{\frac{d(r)d(s)}{d(t)}}T (_{r,s}^{t})_{b}^{∗}ρ_{r}(\overline{R}_{s})\}_{b=1}^{N_{r,r}^{t}}.

Thus,

V (s)_{j}V (t)_{k}^{∗} = \frac{d(r)\sqrt{d(s)}^{3}}{d(t)} ∑_{r,i,b} F_{s}(V (s)_{j})^{∗}V (t)_{k}^{∗}T (_{r,s}^{t})_{b}^{∗}ρ_{r}(\overline{R}_{s})V (r)_{i}
× V (r)_{i}^{∗}ρ_{r}(\overline{R}_{s})^{∗}T (_{r,s}^{t})_{b}ρ_{t}(R_{s})
= \frac{d(r)\sqrt{d(s)}^{3}}{d(t)} ∑_{r,i,b} F_{s}(V (s)_{j})^{∗}V (t)_{k}^{∗}T (_{r,s}^{t})_{b}^{∗}V (r)_{i}\overline{R}_{s}
× V (r)_{i}^{∗}ρ_{r}(\overline{R}_{s}^{∗}ρ_{s}(R_{s}))T (_{r,s}^{t})_{b}
= \frac{d(r)\sqrt{d(s)}}{d(t)} ∑_{r,i,b} F_{s}(V (s)_{j})^{∗}V (t)_{k}^{∗}T (_{r,s}^{t})_{b}^{∗}V (r)_{i}\overline{R}_{s}V (r)_{i}^{∗}T (_{r,s}^{t})_{b}.

Since V (r)_{i}^{∗}T (_{r,s}^{t})_{b}V (t)_{k} ∈ L_{s}, we have

V (r)_{i}^{∗}T (_{r,s}^{t})_{b}V (t)_{k} = ∑_{l}⟨V (r)_{i}^{∗}T (_{r,s}^{t})_{b}V (t)_{k}, V (s)_{l}⟩V (s)_{l}
= ∑_{l} \overline{⟨T (_{r,s}^{t})_{b}^{∗}V (r)_{i}V (s)_{l}, V (t)_{k}⟩}V (s)_{l}.

Thus,

V (s)_{j}V (t)_{k}^{∗} = \frac{d(r)\sqrt{d(s)}}{d(t)} ∑_{r,i,l,} F_{s}(V (s)_{j})^{∗}V (s)_{l}^{∗}\overline{R}_{s}V (r)_{i}^{∗}C_{(r,i),(s,l)}^{(t,k)}
= \frac{d(r)}{d(t)} ∑_{r,i,l,} F_{s}(V (s)_{j})^{∗}F_{s}(V (s)_{l})V (r)_{i}^{∗}C_{(r,i),(s,l)}^{(t,k)}
= \frac{d(r)c(s)_{j}}{d(t)} ∑_{r,i} V (r)_{i}^{∗}C_{(r,i),(s,j)}^{(t,k)},

which shows the statement. Lemma 7.2. Let the notations be as above. Then, the following hold:

(1) Let r_{i}, s_{j}, t ∈ T_{0}, i = 1, 2, ⋯ m, j = 1, 2, ⋯, n. If x(t)_{k} belongs to

(ρ_{t} · ρ_{r_{1}} ⋯ ρ_{r_{m}}, ρ_{s_{1}} ⋯ ρ_{s_{n}}),

for every t ∈ T_{0}, 1 ≤ k ≤ n_{k}, and

∑_{t,k} x(t)_{k}V (t)_{k} = 0

holds, then, x(t)_{k} = 0 for all t, k.

(2) When ⟨T (_{r,s}^{t})_{a}^{∗}V (r)_{i}V (s)_{j}, V (t)_{k}⟩ ≠ 0, we have c(r)_{i}c(s)_{j} = c(t)_{k}.

(3)

∑_{r,i} \frac{d(r)}{c(r)_{i}} ⟨C_{(r,i),(s,j)}^{(t,k)}, C_{(r,i),(s,j)}^{(t,k′)}⟩ = \frac{d(t)}{c(t)_{k}} δ_{k,k′}.

Before starting the proof, we remark an outcome of (1) above. Assume that we are given V (s_{1})_{j_{1}}V (s_{2})_{j_{2}} ⋯ V (s_{n})_{j_{n}}V (r_{m})_{i_{m}}^{∗} ⋯ V (r_{2})_{i_{2}}^{∗}V (r_{1})_{i_{1}}^{∗}. Using Lemma 7.1, (1), (2), and Lemma 5.1, (1) repeatedly, we can reduce this element to the form ∑_{r,i} y(r)_{i}V (r)_{i}, y(r)_{i} ∈ A. Lemma 7.2, (1) asserts that no matter what way we take in the reduction process, the resulting “coefficients” {y(r)_{i}} are uniquely determined.

Proof. (1). We show x(e) = 0. The general statement can be obtained by applying this and Lemma 7.1, (2) to

∑_{t,k} x(t)_{k}V (t)_{k}V (r)_{i}^{∗} = 0.

We choose orthonormal bases

{X(r)_{a}}_{a} ⊂ (ρ_{r}, ρ_{r_{1}} ⋯ ρ_{r_{m}}),
{Y (r)_{b}}_{b} ⊂ (ρ_{r}, ρ_{s_{1}} ⋯ ρ_{s_{n}}).

Then,

0 = \overline{R}_{r}^{∗}Y (r)_{b}^{∗}( ∑_{t,k} x(t)_{k}V (t)_{k})X(r)_{a}\overline{R}_{r} = ∑_{t,k} \overline{R}_{r}^{∗}Y (r)_{b}^{∗}x(t)_{k}ρ_{t}(X(r)_{a}\overline{R}_{r})V (t)_{k}
= \overline{R}_{r}^{∗}Y (r)_{b}^{∗}x(e)X(r)_{a}\overline{R}_{r},

where we use

\overline{R}_{r}^{∗}Y (r)_{b}^{∗}x(t)_{k}ρ_{t}(X(r)_{a}\overline{R}_{r}) ∈ (ρ_{t}, Id) = δ_{t,e}ℂ.

Since Y (r)_{b}^{∗}x(e)X(r)_{a} ∈ (ρ_{r}, ρ_{r}) is already a scalar, we get Y (r)_{b}^{∗}x(e)X(r)_{a} = 0. If r ≠ s, we have Y (s)_{b}^{∗}x(e)X(r)_{a} ∈ (ρ_{r}, ρ_{s}) = {0}. Therefore,

x(e) = ∑_{r,s,a,b} Y (s)_{b}Y (s)_{b}^{∗}x(e)X(r)_{a}X(r)_{a}^{∗} = 0.

(2). Using Lemma 7.1, (1) and (2), we get

V (r)_{i}V (s)_{j}V (t)_{k}^{∗} = ∑_{t′,k′} C_{(r,i),(s,j)}^{(t′,k′)}V (t′)_{k′}V (t)_{k}^{∗}

= ∑_{t′,r′,k′,i′} \frac{d(r′)c(t′)_{k′}}{d(t)} V (r′)_{i′}^{∗}ρ_{r′}(C_{(r,i),(s,j)}^{(t′,k′)})C_{(r′,i′),(t′,k′)}^{(t,k)}.

On the other hand, using Lemma 7.1, (2) twice, we get

V (r)_{i}V (s)_{j}V (t)_{k}^{∗} = ∑_{u,l} \frac{d(u)c(s)_{j}}{d(t)} V (r)_{i}V (u)_{l}^{∗}C_{(u,l),(s,j)}^{(t,k)}

= ∑_{u,r′,l,i′} \frac{d(r′)c(r)_{i}c(s)_{j}}{d(t)} V (r′)_{i′}^{∗}C_{(r′,i′),(r,i)}^{(u,l)}C_{(u,l),(s,j)}^{(t,k)}.

Now, we compare the “V (e)^{∗}-coefficient” by applying (1), and get

c(t)_{k}C_{(r,i),(s,j)}^{(t,k)} = c(r)_{i}c(s)_{j}C_{(r,i),(s,j)}^{(t,k)},

which shows the statement.

(3). Lemma 7.1, (2) implies

V (t)_{l}V (t)_{k}^{∗} = ∑_{r,r′,i,i′} \frac{d(r)d(r′)c(s)_{j}^{2}}{d(t)^{2}} C_{(r′,i′),(s,j)}^{(t,l)}^{∗}V (r′)_{i′}V (r)_{i}^{∗}C_{(r,i),(s,j)}^{(t,k)}.

Using Lemma 7.1, (2) again and applying (1) in the same way as above, we get

\frac{c(t)_{k}}{d(t)} δ_{k,l} = ∑_{r,i} \frac{d(r)c(r)_{i}c(s)_{j}^{2}}{d(t)^{2}} C_{(r,i),(s,j)}^{(t,l)}^{∗}C_{(r,i),(s,j)}^{(t,k)}

= ∑_{r,i} \frac{d(r)c(t)_{k}^{2}}{c(r)_{i}d(t)^{2}} C_{(r,i),(s,j)}^{(t,l)}^{∗}C_{(r,i),(s,j)}^{(t,k)},

which finishes the proof.

Proposition 7.3. Let A be a C^{∗}-algebra in C_{1} (or C_{2}) acting irreducibly on a Hilbert space H, and T = {ρ_{r}}_{r∈T_{0}} ⊂ End(A)_{0} be a finite closed system of irreducible endomorphisms. We set

L_{r} = {V ∈ 𝔹(H); V x = ρ_{r}(x)V }.

Let D be the C^{∗}-algebra generated by ∪_{r∈T_{0}}AL_{r}. Then, D is simple and there exists a conditional expectation E of a finite index from D onto A satisfying E(aV) = 0 for every a ∈ A, V ∈ L_{r}, r ≠ e.

Proof. Our proof is based on R. Longo’s Q-system [29] and the characterization of the C^{∗}-basic construction. We set e(r)_{i} = d(r)/c(r)_{i} and

d = \sqrt{∑_{r,i} e(r)_{i}}.

Since A is stable, we can choose isometries {W (r)_{i}}_{r,i} ⊂ M(A) such that

∑_{r,i} W (r)_{i}W (r)_{i}^{∗} = 1.

We define an endomorphism γ of A by

γ(x) = ∑_{r,i} W (r)_{i}ρ_{r}(x)W (r)_{i}^{∗}.

Let S = W (e) and

T = \frac{1}{d} ∑_{r,s,t,i,j,k} \sqrt{\frac{e(r)_{i}e(s)_{j}}{e(t)_{k}}} W (r)_{i}ρ_{r}(W (s)_{j})C_{(r,i),(s,j)}^{(t,k)}W (t)_{k}^{∗}.

Then, thanks to Lemma 7.2, (3), S ∈ (Id, γ) and T ∈ (γ, γ^{2}) are isometries satisfying

S^{∗}T = γ(S^{∗})T = \frac{1}{d}.

We claim that T T^{∗} = γ(T^{∗})T holds. Indeed, straightforward calculation yields

T T^{∗} = \frac{1}{d^{2}} ∑_{r,s,t,u,i,j,k,l} \frac{\sqrt{e(r)_{i}e(s)_{j}e(u)_{l}e(v)_{m}}}{e(t)_{k}} W (r)_{i}ρ_{r}(W (s)_{j})C_{(r,i),(s,j)}^{(t,k)}C_{(u,l),(v,m)}^{(t,k)}^{∗}
× ρ_{u}(W (v)_{m}^{∗})W (u)_{l}^{∗}.

γ(T^{∗})T = \frac{1}{d} ∑_{r,t,u,i,k,l} \sqrt{\frac{e(r)_{i}e(t)_{k}}{e(u)_{l}}} W (r)_{i}ρ_{r}(T^{∗}W (t)_{k})C_{(r,i)(t,k)}^{(u,l)}W (u)_{l}^{∗}

= \frac{1}{d^{2}} ∑_{r,s,t,u,v,i,j,k,l,m} \sqrt{\frac{e(r)_{i}e(t)_{k}^{2}e(v)_{m}}{e(s)_{j}e(u)_{l}}} W (r)_{i}ρ_{r}(W (s)_{j}C_{(t,k)(v,m)}^{(s,j)}^{∗}ρ_{t}(W (v)_{m}^{∗}))
× C_{(r,i)(t,k)}^{(u,l)}W (u)_{l}^{∗}
= \frac{1}{d^{2}} ∑_{r,s,t,u,v,i,j,k,l,m} \sqrt{\frac{e(r)_{i}e(t)_{k}^{2}e(v)_{m}}{e(s)_{j}e(u)_{l}}} W (r)_{i}ρ_{r}(W (s)_{j}C_{(t,k)(v,m)}^{(s,j)}^{∗})
× C_{(r,i)(t,k)}^{(u,l)}ρ_{u}(W (v)_{m}^{∗})W (u)_{l}^{∗}.

Therefore, the claim is equivalent to

∑_{t,k} \frac{1}{e(t)_{k}} C_{(r,i),(s,j)}^{(t,k)}C_{(u,l),(v,m)}^{(t,k)}^{∗} = ∑_{t,k} \frac{e(t)_{k}}{e(s)_{j}e(u)_{l}} ρ_{r}(C_{(t,k)(v,m)}^{(s,j)}^{∗})C_{(r,i)(t,k)}^{(u,l)}.

This follows from comparison of the V (e)-coefficients in two different ways of expansion of V (r)_{i}V (s)_{j}V (v)_{m}^{∗}V (u)_{l}^{∗}. Therefore, the claim is proven. T T^{∗} = γ(T^{∗})T also implies T^{2} = γ(T)T because

(T^{2} − γ(T)T)^{∗}(T^{2} − γ(T)T) = 2 − T^{2∗}γ(T)T − T^{∗}γ(T^{∗})T^{2} = 0.

We set

B = {x ∈ A; T x = γ(x)T, T x^{∗} = γ(x^{∗})T }.

Then, B is an intermediate C^{∗}-subalgebra between A ⊃ γ(A) with T ∈ M(B). For x ∈ A, we define E(x) = T^{∗}γ(x)T, which is a conditional expectation from A onto B with a quasi-basis {(dS^{∗}, dS)}. We claim that A ⊃ B is irreducible, and consequently, B is simple. Indeed, let ι be the inclusion map from B into A. Then, Lemma 4.4 implies that [\overline{ι}][ι] = [γ]. Thus, the Frobenius reciprocity implies

dim(ι, ι) = dim(Id, γ) = 1,

which shows M(A) ∩ B′ = ℂ. Theorem 3.3 implies that B is simple.

Now we show D = A_{1}. Let

V = \frac{1}{d} ∑_{r,i} \sqrt{e(r)_{i}}W (r)_{i}V (r)_{i},

which is an isometry satisfying V x = γ(x)V for x ∈ A. We show that p = V V^{∗} satisfies the condition of Proposition 2.5, (1). Direct computation using Lemma 7.1,

(2) shows

V^{∗}T = \frac{1}{d^{2}} ∑_{r,s,t,i,j,k} \sqrt{\frac{e(r)_{i}^{2}e(s)_{j}}{e(t)_{k}}} V (r)_{i}^{∗}ρ_{r}(W (s)_{j})C_{(r,i),(s,j)}^{(t,k)}W (t)_{k}^{∗}

= \frac{1}{d^{2}} ∑_{r,s,t,i,j,k} \sqrt{\frac{e(r)_{i}^{2}e(s)_{j}}{e(t)_{k}}} W (s)_{j}V (r)_{i}^{∗}C_{(r,i),(s,j)}^{(t,k)}W (t)_{k}^{∗}

= \frac{1}{d^{2}} ∑_{s,t,j,k} \sqrt{e(t)_{i}e(s)_{j}}W (s)_{j}V (s)_{j}V (t)_{k}^{∗}W (t)_{k}^{∗}
= V V^{∗}.

On the other hand, Lemma 7.1, (1) implies

T V = \frac{1}{d^{2}} ∑_{r,s,t,i,j,k} \sqrt{e(r)_{i}e(s)_{j}}W (r)_{i}ρ_{r}(W (s)_{j})C_{(r,i),(s,j)}^{(t,k)}V (t)_{k}

= \frac{1}{d^{2}} ∑_{r,s,i,j} \sqrt{e(r)_{i}e(s)_{j}}W (r)_{i}ρ_{r}(W (s)_{j})V (r)_{i}V (s)_{j}

= \frac{1}{d^{2}} ∑_{r,s,i,j} \sqrt{e(r)_{i}e(s)_{j}}W (r)_{i}V (r)_{i}V (s)_{j}V (s)_{j}
= V^{2},

which shows pT = T p = V^{2}V^{∗}. Therefore, we get p ∈ B′ because p commutes with γ(A) and B = T^{∗}γ(A)T. As we have A = BS, to prove pxp = E(x)p, it suffices to show that pSp is equal to

E^{M}(S)p = T^{∗}γ(S)T p = \frac{1}{d}T p = \frac{1}{d}V^{2}V^{∗},

which can be shown by direct computation. Since B is simple, the map B ∋ b ↦ bp is faithful. Therefore, Proposition 2.5, (1) implies that we can identify the basic construction A_{1} with the C^{∗}-algebra generated by ApA. Obviously, we have an inclusion A_{1} ⊃ D. On the other hand,

AeA ⊃ AW (r)_{i}^{∗}eSA = AV (r)_{i}A = Aρ_{r}(A)V (r)_{i} = AV (r)_{i},

which shows D = A_{1}.

To finish the proof, it suffices to show that V (r)_{i} ∈ M(D) and to put E = E_{1}. However, this follows immediately from Lemma 7.1.

Remark 7.4. In the above, we actually have c(r)_{i} = 1. Indeed, thanks to the local index formula of the minimal conditional expectation E_{ρ_{r}} · E [11], [42], we have

E(V (r)_{i}V (r)_{i}^{∗}) = \frac{1}{d(r)},

which implies c(r)_{i} = 1.

Theorem 7.5. Let A be a C^{∗}-algebra in C_{1} (or C_{2}), and ρ ∈ End(A)_{0} be irreducible with [ρ] ≠ [Id]. If [ρ] and [\overline{ρ}] generate only finitely many irreducibles, then for every x ∈ M(A) and for every non-zero hereditary C^{∗}-subalgebra C of A,

inf{||cxρ(c)||; c ∈ C_{+}, ||c|| = 1} = 0

holds (i.e. ρ is properly outer in the sense of [2]).

Proof. Thanks to Kishimoto’s result [21, Lemma 1.1], when ρ is an automorphism, the statement holds. Therefore, we may assume that ρ is not an automorphism. Suppose that the statement does not hold. Then, in exactly the same way as in [21, Lemma 1.1], we can show that there would exist a ∈ M(A), a non-zero hereditary C^{∗}- subalgebra C ⊂ A, and a positive number δ such that for every unitary u ∈ C + ℂ1 and every pure state φ ∈ P (A) whose support s(φ) belongs to C^{∗∗}, the following holds:

||s(φ)u^{∗}aρ^{∗∗}(us(φ))|| ≥ δ (7.1)

This implies π_{φ}(ρ^{∗∗}(s(φ))) ≠ 0. Let

L_{ρ} = {V; V π_{φ}(x) = π_{φ}(ρ(x))V, x ∈ A}.

We claim that the map L_{ρ} ∋ V ↦ V Ω_{φ} ∈ H gives an isometry from L_{ρ} onto π_{φ}(ρ^{∗∗}(s(φ)))H_{φ}. Indeed, let ψ be the pure state on ρ(A) given by φ · ρ^{−1}. Then, π_{φ}(ρ^{∗∗}(s(φ)))H_{φ} is characterized as the set of vectors in H_{φ} inducing a scalar multiple of ψ on ρ(A), which shows that the above map is an isometry L_{ρ} into π_{φ}(ρ^{∗∗}(s(φ)))H_{φ}. On the other hand, for a given ξ ∈ π_{φ}(ρ^{∗∗}(s(φ)))H_{φ}, we can construct V ∈ L_{ρ} by setting

V π_{φ}(x)Ω_{φ} = π_{φ}(ρ(x))ξ, x ∈ A.

Thus, the claim holds.

Lemma 5.1 and Lemma 5.2 imply that L_{ρ} (and π_{φ}(ρ^{∗∗}(s(φ)))H_{φ} as well) is finite dimensional. Let {V_{i}}_{i=1}^{m} be an orthonormal basis of L_{ρ}. Then, thanks to the claim, we get

ρ^{∗∗}(s(φ)) = ∑_{i=1}^{m} V_{i}s(φ)V_{i}^{∗},

which implies

s(φ)u^{∗}aρ^{∗∗}(us(φ)u^{∗})a^{∗}us(φ) = ∑_{i=1}^{m} |⟨π_{φ}(a)V_{i}π_{φ}(u)Ω_{φ}, π_{φ}(u)Ω_{φ}⟩|^{2}s(φ).

Let K be the closure of π_{φ}(C)Ω_{φ}. Then, (7.1) is equivalent to

∑_{i=1}^{m} |⟨π_{φ}(a)V_{i}ξ, ξ⟩|^{2} ≥ δ^{2}||ξ||^{4}, ξ ∈ K.

We introduce a map Φ : K × K ⟶ ℂ^{m} by

Φ(ξ, η) = (⟨π_{φ}(a)V_{1}ξ, η⟩, ⟨π_{φ}(a)V_{2}ξ, η⟩, ⋯, ⟨π_{φ}(a)V_{m}ξ, η⟩),

and set

δ_{0} = inf{||Φ(ξ, ξ)||; ξ ∈ K, ||ξ|| = 1} ≥ δ.

We take a sequence of unit vectors {ξ_{n}} ∈ K satisfying

lim_{n→∞} ||Φ(ξ_{n}, ξ_{n})|| = δ_{0}.

We fix a free ultrafilter ω ∈ βℕ \ ℕ. For a unit vector η ∈ K, we set

e = lim_{n→ω} Φ(ξ_{n}, ξ_{n}),

f = lim_{n→ω}(Φ(ξ_{n}, η) + Φ(η, ξ_{n})),

g = Φ(η, η),

h = lim_{n→ω}(⟨ξ_{n}, η⟩ + ⟨η, ξ_{n}⟩).

Note that since Φ is bounded on the unit ball of K, the above limits exist. For all t ∈ ℝ, we have

lim_{n→ω}(||Ψ(ξ_{n} + tη, ξ_{n} + tη)||^{2} − δ_{0}^{2}||ξ_{n} + tη||^{4})

= ||e + tf + t^{2}g||^{2} − δ_{0}^{2}(1 + th + t^{2})^{2}
= (||g||^{2} − δ_{0}^{2})t^{4} + (⟨f, g⟩ + ⟨g, f⟩ − δ_{0}^{2}(h + \overline{h}))t^{3}
+ (⟨e, g⟩ + ⟨g, e⟩ + ||f||^{2} − δ_{0}^{2}(2 + |h|^{2}))t^{2} + (⟨e, f⟩ + ⟨f, e⟩ − δ_{0}^{2}(h + \overline{h}))t
≥ 0.

This implies

⟨e, f⟩ + ⟨f, e⟩ = δ_{0}^{2}(h + \overline{h}),
⟨e, g⟩ + ⟨g, e⟩ + ||f||^{2} ≥ δ_{0}^{2}(2 + |h|^{2})

¿From the latter, we get

⟨e, g⟩ + ⟨g, e⟩ + 2 lim_{n→ω}(||Φ(ξ_{n}, η)||^{2} + ||Φ(η, ξ_{n})||^{2}) ≥ 2δ_{0}^{2}

Let V = ∑_{i=1}^{m} \overline{e_{i}}V_{i}, and p_{ξ_{n}} be the projection onto ℂξ_{n}. The above inequality is equivalent to

⟨(π_{φ}(a)V + V^{∗}π_{φ}(a)^{∗})η, η⟩ + 2⟨T η, η⟩ ≥ 2δ_{0}^{2},

where

w− lim_{n→ω} ∑_{i=1}^{m}(π_{φ}(a)V_{i}p_{ξ_{n}}V_{i}^{∗}π_{φ}(a)^{∗} + V_{i}^{∗}π_{φ}(a)^{∗}p_{ξ_{n}}π_{φ}(a)V_{i}).

Since C is a hereditary C^{∗}-subalgebra with s(φ) ∈ C^{∗∗}, K is actually the closure of π_{φ}(C)H_{φ}. Therefore, for any x ∈ C we get

π_{φ}(x)^{∗}(π_{φ}(a)V + V^{∗}π_{φ}(a)^{∗} + 2T)π_{φ}(x) ≥ δ_{0}^{2}π_{φ}(x^{∗}x).

To obtain contradiction applying Proposition 7.3, we need to get rid of T. For any orthonormal system {ζ_{j}}_{j=1}^{k} in H_{φ}, we have

∑_{j=1}^{k} ∑_{i=1}^{m}⟨(π_{φ}(a)V_{i}p_{ξ_{n}}V_{i}^{∗}π_{φ}(a)^{∗} + V_{i}^{∗}π_{φ}(a)^{∗}p_{ξ_{n}}π_{φ}(a)V_{i})ζ_{j}, ζ_{j}⟩

= ∑_{j=1}^{k} ∑_{i=1}^{m}(|⟨φ(a)V_{i}ξ_{n}, ζ_{j}⟩|^{2} + |⟨V_{i}^{∗}π_{φ}(a)^{∗}ξ_{n}, ζ_{j}⟩|^{2})

≤ ∑_{i=1}^{m}(||π_{φ}(a)V_{i}ξ_{n}||^{2} + ||V_{i}^{∗}π_{φ}(a)^{∗}ξ_{n}||^{2})
≤ 2m||a||^{2},

which implies that T is a trace class operator. Let π be the quotient map from 𝔹(H_{φ}) onto the Calkin algebra 𝔹(H_{φ})/𝕂(H_{φ}). Then, we get

π(π_{φ}(x)^{∗}(π_{φ}(a)V + V^{∗}π_{φ}(a)^{∗})π_{φ}(x)) ≥ δ_{0}^{2}π(π_{φ}(x^{∗}x)).

Let T be the finite closed system of irreducible endomorphisms in End(A)_{0} generated by ρ and \overline{ρ}, and D and E be as in Proposition 7.3, where we identify A with its image in 𝔹(H_{φ}). Then, thanks to Proposition 7.3, D is simple, and so either D ∩ 𝕂(H_{φ}) is trivial or D = 𝕂(H_{φ}). The latter does not occur because it would imply that π_{φ}(A) would be a proper C^{∗}-subalgebra of 𝕂(H_{φ}) acting irreducibly on H_{φ}. Therefore, the restriction of π to D is faithful, and so we get

π_{φ}(x)^{∗}(π_{φ}(a)V + V^{∗}π_{φ}(a)^{∗})π_{φ}(x) ≥ δ_{0}^{2}π_{φ}(x^{∗}x).

Applying E to the both sides, we obtain

0 ≥ δ_{0}^{2}π_{φ}(x^{∗}x),

which is contradiction. Therefore, we finally get the statement.

Corollary 7.6. Let A ⊃ B be an irreducible and finite depth inclusion of simple σunital C^{∗}-algebras, and E be a conditional expectation from A onto B of a finite index. Then, for every element x ∈ M(A) and every non-zero hereditary C^{∗}-subalgebra C of B,

inf{||c(x − E(x))c||; c ∈ C_{+}, ||c|| = 1} = 0.

Proof. It suffices to show the statement for (A ⊗ 𝕂, B ⊗ 𝕂, E ⊗ Id) as usual and we may assume that A and B are stable from the beginning. Then, as in the usual finite group crossed product case, the statement follows from repeated use of Theorem 7.5 and the “crossed product expression” discussed in Section 6.

Remark 7.7. It might be an interesting problem to seek for a statement corresponding to [2, Theorem 1] in our situation.

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Department of Mathematics, Graduate School of Science, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan

  E-mail address: izumi@kusm.kyoto-u.ac.jp