SOME SUBSEMIGROUP OF AN EXTENSION SEMIGROUP OF C*-ALGEBRAS

YUTAKA KATABAMI AND MASARU NAGISA

Abstract. In this paper we investigate the structure of the subsemigroup generated by the inner automorphisms in Ext(𝕂, Q). As an application, we give a new point of view to the example of J. Plastiras, which are two C*-algebras 𝔄 and 𝔅 satisfying 𝔄 ≅ 𝔅̸ and 𝕄_{2} ⊗ 𝔄 ≅ 𝕄_{2} ⊗ 𝔅.

0. Introduction

J. Plastiras exhibited an example which is a pair of C*-algebras such that 𝔄 ≅ 𝔅̸ and 𝕄_{2} ⊗ 𝔄 ≅ 𝕄_{2} ⊗ 𝔅 ([5], [6]). They are constructed as extensions of 𝕂 by Q, where 𝕂 is the C*-algebra of compact operators and Q is the quotient C*-algebra of all the bounded linear operators 𝔹 by 𝕂. So they are not nuclear. For a class of nuclear C*-algebras, we can construct such a pair of C*-algebras using the classification result for them by K-theory([2], [3]). In [7], T. Sakamoto construct such a pair of non-nuclear C*-algebras.

In this paper, we consider a family of special extensions of 𝕂 by Q which contains Plastiras’ examples. Our aim is to investigate their semigroup structure and to show that the datum for this semigroup is the useful invariant for them as like as K-thoretic datum for some nuclear C*-algebras.

1. Preliminaries and Main result

Here we give fundamental facts of extension theory along [1] and [8]. Let H be a separable infinite dimensional Hilbert space. We denote by 𝔹 (resp. 𝕂) a C*-algebra 𝔹(H) (resp. 𝕂(H)) of bounded linear operators (resp. compact operators) on H. We also denote by Q a C*-algebra 𝔹(H)/𝕂(H). Let A, B and C be C*-algebras and α (resp. β) a *-homomorphism from A to B (resp. from B to C). We call a short exact sequence E as below an extension of A by C:

E : 0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0 ,

that is, α is injective, β is surjective and Imα = Kerβ. Then there exists a *-homomorphism σ from B to the multiplier C*-algebra M(A) of A with σ ∘ α = ι, i.e.,

\begin{matrix}A & ⟶^{α} & B \\ ∥ & & ↓^{σ} \\ A & ⟶_{ι} & M(A) \end{matrix},

where ι is the canonical inclusion map from A to M(A). The Busby invariant for this extension E is defined as the *-homomorphism τ_{E} from C to M(A)/A given by

τ_{E}(c) = π ∘ σ(b) ,

where b is a lift of c through β and π is the quotient map from M(A) to M(A)/A. It is known that τ_{E} is characterized by the following commutative diagram:

\begin{aligned}0 & ⟶ & A & ⟶^{α} & B & ⟶^{β} & C & ⟶ & 0 \\ & & ∥ & & ↓^{σ} & & ↓^{τ_{E}} \\ 0 & ⟶ & A & ⟶^{ι} & M(A) & ⟶^{π} & M(A)/A & ⟶ & 0 \end{aligned} .

We remark that, if we define the pull-back C*-algebra P B and the map ψ as follows:

P B = {(x, c) ∈ M(A) ⊕ C | π(x) = τ_{E}(c)}
ψ : B ∋ b ⟼ (σ(b), β(b)) ∈ P B,

then B is isomorphic to P B for the isomorphism ψ making the following diagram commutative:

\begin{aligned}0 & ⟶ & A & ⟶^{α} & B & ⟶^{β} & C & ⟶ & 0 \\ & & ∥ & & ↓^{ψ} & & ∥ \\ 0 & ⟶ & A & ⟶ & P B & ⟶ & C & ⟶ & 0 \end{aligned} .

Let

\begin{matrix}E_{1} : 0 & ⟶ & A & ⟶ & B_{1} & ⟶ & C & ⟶ & 0 \\ E_{2} : 0 & ⟶ & A & ⟶ & B_{2} & ⟶ & C & ⟶ & 0 \end{matrix}

be extensions and τ_{i} the Busby invariant for E_{i} (i = 1, 2). We call E_{1} and E_{2} strongly equivalent when there is a unitary u ∈ M(A) such that τ_{2}(c) = π(u)τ_{1}(c)π(u)^{∗} for all c ∈ C, equivalently there are a unitary v ∈ M(A) and a *-isomorphism γ such that the diagram

0 ⟶ A ⟶ B_{1} ⟶ C ⟶ 0
↓^{Ad(v)}  ↓^{γ}  ∥
0 ⟶ A ⟶ B_{2} ⟶ C ⟶ 0

is commutative. Then we denote E_{1} ≈ E_{2} or τ_{1} ≈ τ_{2}. Let Ext(A, C) be a set of extensions of A by C. We denote by Ext(A, C) the set Ext(A, C)/ ≈ of strongly equivalent classes of Ext(A, C) . When C is stable (i.e., C ≅ 𝕄_{2}(C)), Ext(A, C) becomes an abelian semigroup. The addition of [E_{1}] and [E_{2}] ∈ Ext(A, C) is defined by the equivalent class of the extension which is corresponding to the Busby invariant

τ_{1} ⊕ τ_{2} : C ⟶ C ⊕ C ↪ C.

In this paper, we consider the extension semigroup Ext(𝕂, Q). We denote by π the canonical quotient map from 𝔹 onto Q. Let α be an inner *-automorphism of Q. Then we can see that α is the Busby invariant for an extension E ∈ Ext(𝕂, Q). We denote by G a subsemigroup of Ext(𝕂, Q) generated by extensions corresponding to all the inner *-automorphisms of Q.

The inner automorphism α has the form α(·) = u^{∗} · u for some unitary u ∈ Q. Let V ∈ 𝔹 be a lift of u, that is, π(V) = u. Then V is a Fredholm operator, and we put n = Index V ∈ ℤ. Let S(∈ 𝔹) be a unilateral shift. We remark Index S = −1. We define a *- automorphism τ(n) of Q by

τ(n)(x) = π(S)^{n}xπ(S^{∗})^{n}, x ∈ Q.

Then there exists a unitary U ∈ 𝔹 such that V S^{n} = U|V S^{n}|, i.e., uπ(S)^{n} = π(U). So we have that α is strongly equivalent to τ(n), that is, [α] = [τ(n)].

Let G be a restricted direct product of non-negative integers ℤ_{≥0} except 0, i.e.,

G = ∐_{ℤ}ℤ_{≥0} \ {0}

={g = (m(k))_{k∈ℤ} | m(k) ∈ ℤ_{≥0}, 0 < ♯{k ∈ ℤ | m(k) ≠ 0} < ∞},

where ♯ denotes the cardinal number of set. By the above fact, we can define the surjective semigroup homomorphism τ from G to G as follows:

τ(g) = [⨁_{k∈ℤ} ⨁_{m(k)} τ(k)] = ∑_{k∈ℤ}m(k)[τ(k)],

where g = (m(k))_{k∈ℤ} ∈ G.

We define a map φ from G to ℕ and a map ψ from G to ℤ as follows: for g = (m(k))_{k∈ℤ} ∈ G,

φ(g) = ∑_{k∈ℤ}m(k),
ψ(g) = ∑_{k∈ℤ}km(k).

We introduce two notations as follows: for l ∈ ℤ and g = (m(k))_{k∈ℤ} ∈ G,

l · g = (m(lk))_{k∈ℤ},
l + g = (m(l + k))_{k∈ℤ}.

Then we can easily get

φ(l · g) = φ(g), ψ(l · g) = lψ(g),
φ(l + g) = φ(g) and ψ(l + g) = ψ(g) + lφ(g).

For g = (m(k))_{k∈ℤ} ∈ G, we define a C*-subalgebra A(g) of 𝔹(⊕_{φ(g)}H) ≅ 𝕄_{φ(g)}(𝔹) as follows:

A(g) = \{⨁_{k∈ℤ}(\overbrace{S^{k}T S^{∗}^{k} ⊕ ⋯ ⊕ S^{k}T S^{∗}^{k}}^{m(k)}) | T ∈ 𝔹\} + 𝕂(⊕_{φ(g)}H),

where S^{k} (resp. (S^{∗})^{k}) means (S^{∗})^{−k} (resp. S^{−k}) for a negative integer k. Let ι(g) be a injective *-homomorphism from 𝕂 to A(g) which is obtained by a composition of a natural isomorphism of 𝕂 to 𝕂(⊕_{φ(g)}H) and the canonical inclusion map of 𝕂(⊕_{φ(g)}H) into A(g). We define a surjective *-homomorphism π(g) from A(g) to Q as follows:

π(g)(⨁_{k∈ℤ}(\overbrace{S^{k}T S^{∗}^{k} ⊕ ⋯ ⊕ S^{k}T S^{∗}^{k}}^{m(k)}) + K) = π(T),

where K ∈ 𝕂(⊕_{φ(g)}H). Then we have the following extension:

E(g) : 0 ⟶ 𝕂 ⟶^{ι(g)} A(g) ⟶^{π(g)} Q ⟶ 0,

and its Busby invariant coincides with

⨁_{k∈ℤ} ⨁_{m(k)} τ(k).

Then we have the following statement and this is our main result:

Theorem 1.1. For g, h ∈ G and n ∈ ℕ, we have the following:

(1) τ(g) = τ(h) ⟺ φ(g) = φ(h) and ψ(g) = ψ(h), that is,

G ∋ τ(g) ⟼ (φ(g), ψ(g)) ∈ ℕ × ℤ

gives a semigroup isomorphism from G onto ℕ × ℤ.

(2) A(g) ≅ A(h) ⟺ φ(g) = φ(h) and ψ(g) ≡ ψ(h) mod φ(g).

(3) A(g) ⊗ 𝕄_{n} ≅ A(n · g).

We give the proof of theorem in the next section.

Corollary 1.2. For any n ∈ ℕ and n ≥ 2, there exist g, h ∈ G such that A(g) ⊗ 𝕄_{k} is not isomorphic to A(h) ⊗ 𝕄_{k} for any 1 ≤ k ≤ n − 1 and A(g) ⊗ 𝕄_{n} is isomorphic to A(h) ⊗ 𝕄_{n}.

Proof. We choose g and h such that

φ(g) = φ(h) = n, ϕ(g) = 0 and ψ(h) = 1.

Then we have ψ(k·g) = 0 < ψ(k·h) = k < n for any k = 1, 2, …, n−1 and φ(n · g) = φ(n · h) = n, ψ(n · g) ≡ ψ(n · h) ≡ 0 mod n. This implies that A(g) and A(h) satisfy the required property.

For g = (m(k))_{k∈ℤ}, h = (n(k))_{k∈ℤ} ∈ G with

m(k) = \begin{cases}2 & k = 0 \\ 0 & otherwise \end{cases} and n(k) = \begin{cases}1 & k = 0, 1 \\ 0 & otherwise \end{cases} ,

we have φ(g) = φ(h) = 2, ψ(g) = 0, ψ(h) = 1. It follows that A(g) ⊗ 𝕄_{2} ≅ A(h) ⊗ 𝕄_{2}, but A(g) is not isomorphic to A(h). This example is the same one given by J. Plastiras.

2. Proof of Theorem

Lemma 2.1. The K_{0}-group K_{0}(A(g)) for A(g) is isomorphic to ℤ/φ(g)ℤ.

Proof. Let g = (m(k))_{k∈ℤ} ∈ G. From the short exact sequence of C*-algebras

0 ⟶ 𝕂 ⟶^{ι(g)} A(g) ⟶^{π(g)} Q ⟶ 0,

we can get the exact sequence of K-groups of C*-algebras as follows:

K_{0}(𝕂) ⟶^{ι(g)_{∗}} K_{0}(A(g)) ⟶^{π(g)_{∗}} K_{0}(Q)
δ_{1}↑  ↓
K_{1}(Q) ⟵_{π(g)_{∗}} K_{1}(A(g)) ⟵_{ι(g)_{∗}} K_{1}(𝕂).

Since K_{0}(Q) = {0}, we have

K_{0}(A(g)) ≅ K_{0}(𝕂)/δ_{1}(K_{1}(Q)).

It is known that K_{0}(𝕂) ≅ K_{1}(Q) ≅ ℤ and the class of P (resp. π(S)) is a generator of K_{0}(𝕂) (resp. K_{1}(Q)), where P ∈ 𝔹 is a projection of rank one and S ∈ 𝔹 is a unilateral shift.

We put P_{n} = 1 − S^{n}S^{∗}^{n} (n = 1, 2, …) and define a unitary W (k) ∈ 𝕄_{2}(𝔹) as follows: for k ≥ 0,

W (k) = (\begin{matrix}S(1 − P_{k}) & P_{k+1} \\ −P_{k} & (1 − P_{k})S^{∗} \end{matrix})

= (\begin{matrix}S & 0 \\ 0 & S \end{matrix})^{k} (\begin{matrix}S & P_{1} \\ 0 & S^{∗} \end{matrix}) (\begin{matrix}S^{∗} & 0 \\ 0 & S^{∗} \end{matrix})^{k} + (\begin{matrix}0 & P_{k} \\ −P_{k} & 0 \end{matrix}) ,

and for k < 0,

W (k) = (\begin{matrix}S & P_{1} \\ 0 & S^{∗} \end{matrix}) = (\begin{matrix}S & 0 \\ 0 & S \end{matrix})^{k} (\begin{matrix}S & P_{1} \\ 0 & S^{∗} \end{matrix}) (\begin{matrix}S^{∗} & 0 \\ 0 & S^{∗} \end{matrix})^{k} + (\begin{matrix}0 & P_{1} \\ 0 & 0 \end{matrix}) .

Then we have

W = ⨁_{k∈ℤ} \overbrace{W (k) ⊕ ⋯ ⊕ W (k)}^{m(k)}

is unitary in 𝕄_{2}(A(g)) and

π(g) ⊗ id_{2}(W) = (\begin{matrix}π(S) & 0 \\ 0 & π(S^{∗}) \end{matrix}) .

By definition of δ_{1}, we have

δ_{1}([π(S)]) = [W^{∗}(1_{A(g)} ⊕ 0_{A(g)})W] − [1_{A(g)} ⊕ 0_{A(g)}].

AN EXTENSION SEMIGROUP OF C*-ALGEBRAS 7

By the calculation

[(\begin{matrix}S(1 − P_{k}) & P_{k+1} \\ −P_{k} & (1 − P_{k})S^{∗} \end{matrix})^{∗} (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}S(1 − P_{k}) & P_{k+1} \\ −P_{k} & (1 − P_{k})S^{∗} \end{matrix})]

− [(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})]

= [(\begin{matrix}1 − P_{k} & 0 \\ 0 & P_{k+1} \end{matrix})] − [(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})] = [P]

and

[(\begin{matrix}S & P_{1} \\ 0 & S^{∗} \end{matrix})^{∗} (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) (\begin{matrix}S & P_{1} \\ 0 & S^{∗} \end{matrix})] − [(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})]

=[(\begin{matrix}1 & 0 \\ 0 & P_{1} \end{matrix})] − [(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})] = [P],

it follows that

δ_{1}([π(S)]) = φ(g)[P].

This means that

K_{0}(A(g)) ≅ ℤ/φ(g)ℤ.

For g = (m(k))_{k∈ℤ}, we can choose integers k_{1} < k_{2} < ⋯ < k_{l} such that

{k ∈ ℤ | m(k) ≠ 0} = {k_{1}, k_{2}, …, k_{l}}.

We remark that, if we put

m_{1} = ⋯ = m_{m(k_{1})} = k_{1}, m_{m(k_{1})+1} = ⋯ = m_{m(k_{1})+m(k_{2})} = k_{2},

…, m_{m(k_{1})+···+m(k_{l−1})+1} = ⋯ = m_{φ(g)} = k_{l},

then we have ψ(g) = ∑_{j=1}^{φ(g)} m_{j} and

A(g) = \{⨁_{k∈ℤ}(\overbrace{S^{k}T S^{∗}^{k} ⊕ ⋯ ⊕ S^{k}T S^{∗}^{k}}^{m(k)}) | T ∈ 𝔹\} + 𝕂(⊕_{φ(g)}H)
= {⊕_{j=1}^{φ(g)}S^{m_{j}}T (S^{∗})^{m_{j}} | T ∈ 𝔹} + 𝕂(⊕_{φ(g)}H).

Lemma 2.2. For any n ∈ ℤ and g ∈ G, we have

A(g) ≅ A(n + g).

Proof. It is sufficient to show that A(g) ≅ A(1 + g). Using the above notation and φ(g) = φ(1 + g) and ψ(1 + g) = ψ(g) + φ(g), we have

A(1 + g) = {⊕_{j=1}^{φ(g)}S^{m_{j}+1}T (S^{∗})^{m_{j}+1} | T ∈ 𝔹} + 𝕂(⊕_{φ(g)}H)
= {⊕_{j=1}^{φ(g)}S^{m_{j}}ST S^{∗}(S^{∗})^{m_{j}} | T ∈ 𝔹} + 𝕂(⊕_{φ(g)}H).

Clearly A(1 + g) ⊂ A(g). Remarking the fact 𝔹 ⊂ S𝔹S^{∗} + 𝕂, we have A(1 + g) = A(g).

Lemma 2.3. The class of the unit of A(g) is equal to ψ(g)[P] in K_{0}(A(g)), where P is a minimal projection of A(g).

Proof. By the above lemma, we can see

A(g) = {⊕_{j=1}^{φ(g)}S^{m_{j}}T (S^{∗})^{m_{j}} | T ∈ 𝔹} + 𝕂(⊕_{φ(g)}H)
= {⊕_{j=1}^{φ(g)}S^{n+m_{j}}T (S^{∗})^{n+m_{j}} | T ∈ 𝔹} + 𝕂(⊕_{φ(g)}H),

for n ∈ ℕ with n+m_{1} > 0. Since 1 ∈ 𝔹 is equivalent to some orthogonal projections Q_{1}, Q_{2}, …, Q_{φ(g)} such that 1 = Q_{1}+Q_{2}+⋯+Q_{φ(g)}, S^{k}S^{∗}^{k} is equivalent to S^{k}Q_{i}S^{∗}^{k} for positive integer k. So we have

[1_{A(g)}] ∈ K_{0}(A(g))
=[⊕_{j=1}^{φ(g)}S^{n+m_{j}}(S^{∗})^{n+m_{j}}] + ∑_{j=1}^{φ(g)}(n + m_{j})[P]
=φ(g)[⊕_{j=1}^{φ(g)}S^{n+m_{j}}(S^{∗})^{n+m_{j}}] + (nφ(g) + ψ(g))[P]

This implies [1_{A(g)}] = ψ(g)[P] in K_{0}(A(g)).

Lemma 2.4. The commutant (Q⊗1_{n})′ of Q⊗1_{n} in Q⊗𝕄_{n} coincides with 1_{Q} ⊗ 𝕄_{n}. In particular, any unitary element in (Q ⊗ 1_{n})′ has a unitary lift in 𝔹 ⊗ 𝕄_{n}.

Proof. It is sufficient to show that Q′ ∩ Q = ℂ1_{Q}. This fact is well known as the following form:

{T ∈ 𝔹 | T X − XT ∈ 𝕂 for all X ∈ 𝔹} = ℂ1 + 𝕂.

For the convenience of readers, we give its proof.

We denote by EC(𝔹) the essential commutant for 𝔹

{T ∈ 𝔹 | T X − XT ∈ 𝕂 for all X ∈ 𝔹}.

It is trivial that EC(𝔹) ⊃ ℂ1 + 𝕂. So we have to show that the reverse inclusion holds. Since EC(𝔹) is a closed *-subalgebra of 𝔹, any element in EC(𝔹) is represented by a linear combination of self-adjoint elements. Let T be a self-adjoint element in EC(𝔹) and its spectral decomposition

T = ∫_{−∥T ∥}^{∥T ∥} λde(λ),

where {e(λ)} is the right continuous spectral family of projections for T.

For −∥T ∥ < a < b < ∥T ∥, we assume that two projections

∫_{−∥T ∥}^{a} de(λ) and ∫_{b}^{∥T ∥} de(λ)

are infinitely dimensional. Since H is separable (𝔹 = 𝔹(H)), there exists a partial isometry V such that

V^{∗}V = ∫_{b}^{∥T ∥} de(λ), V V^{∗} = ∫_{−∥T ∥}^{a} de(λ).

Then we have

(V T − T V)V^{∗} = V ∫_{b}^{∥T ∥} λde(λ)V^{∗} − ∫_{−∥T ∥}^{a} λde(λ)
≥ b ∫_{−∥T ∥}^{a} de(λ) − ∫_{−∥T ∥}^{a} λde(λ)
≥ (b − a) ∫_{−∥T ∥}^{a} de(λ) ∉ 𝕂.

This means that V T − T V ∉ 𝕂, i.e., T ∉ EC(𝔹).

This fact implies that σ(T) has at most one accumulation point. If an accumulation point c exists, then each λ ∈ σ(T)\{c} is an eigenvalue for T and its eigenprojection is finite dimensional. So we have

T − c1 ∈ 𝕂.

If an accumulation point does not exist, then σ(T) is a finite set of eigenvalues for T and their eigenprojections are finite dimensional except for one point c. Also we have

T − c1 ∈ 𝕂.

Proof of Theorem 1.1. (1) First we assume that τ_{g} = τ_{h}. Then the fact A(g) ≅ A(h) implies φ(g) = φ(h) by lemma2.1. We use the notation τ(g) = [⊕_{k∈ℤ} ⊕_{m(k)} τ_{k}], τ(h) = [⊕_{k∈ℤ} ⊕_{n(k)} τ_{k}] and S_{g} = ⊕_{k∈ℤ} ⊕_{m(k)} π(S)^{k}, S_{h} = ⊕_{k∈ℤ} ⊕_{n(k)} π(S)^{k} ∈ Q ⊗ 𝕄_{φ(g)}. Then τ(g) = τ(h) means that there exist a unitary U in 𝔹 ⊗ 𝕄_{φ(g)} such that

S_{g}x ⊗ 1_{φ(g)}S_{g}^{∗} = π ⊗ id_{φ(g)}(U)^{∗}S_{h}x ⊗ 1_{φ(g)}S_{h}^{∗}π ⊗ id_{φ(g)}(U)

for all x ∈ Q. Using lemma2.4, we have S_{h}^{∗}(π ⊗ id_{φ(g)})(U)S_{g} ∈ 1_{Q} ⊗ 𝕄_{φ(g)}. So S_{h}^{∗}(π ⊗ id_{φ(g)})(U)S_{g} have a unitary lift in 1_{𝔹} ⊗ 𝕄_{φ(g)}. This means 0 = Index(⊕_{k∈ℤ} ⊕_{n(k)} S^{k})^{∗}U(⊕_{k∈ℤ} ⊕_{m(k)} S^{k}) = −ψ(h) + ψ(g), that is, ψ(g) = ψ(h). Conversely we assume that φ(g) = φ(h) and ψ(g) = ψ(h). Then we have Index(⊕_{k∈ℤ}⊕_{n(k)} S^{k})(⊕_{k∈ℤ}⊕_{m(k)} S^{k})^{∗} = 0. So there exists a unitary U in 𝔹⊗𝕄_{φ(g)} such that S_{h}S_{g}^{∗} = π⊗id_{φ(g)}(U). This implies that τ(g) = τ(h).

(2) First we assume that A_{g} ≅ A_{h}. By lemma2.1 and lemma2.3, it is immediately found that φ(g) = φ(h) and ψ(g) ≡ ψ(h) mod φ(g). Conversely we assume that φ(g) = φ(h) and ψ(g) = ψ(h) + nφ(g) for n ∈ ℤ. Then we have τ(g) = τ(n+g). This implies A(g) ≅ A(n+g) ≅ A(h) by lemma2.2.

(3) Suppose that A(g) is the following form:

A(g) = {⊕_{k∈ℤ} ⊕_{m(k)} S^{k}T S^{∗}^{k} | T ∈ 𝔹} + 𝕂(⊕_{φ(g)}H).

Therefore we can regard A(g) ⊗ 𝕄_{n} as the following:

A(g) ⊗ 𝕄_{n} = {⊕_{k∈ℤ} ⊕_{m(k)} (S^{k} ⊗ 1_{n})T′(S^{∗}^{k} ⊗ 1_{n})|
T′ ∈ 𝔹(⊕_{n}H)} + 𝕂(⊕_{nφ(g)}H).

This means that A(g) ⊗ 𝕄_{n} ≅ A(n · g).

For g ∈ G, we define the C*-algebra

A(g) = {⊕_{k∈ℤ} ⊕_{m(k)} S^{k}T S^{∗}^{k} | T ∈ 𝔹(H)} + 𝕂(⊕_{φ(g)}H).

Then we can see that the essential commutant EC(A(g)) of A(g) becomes an AF-algebra and π⊗id_{φ(g)}(EC(A(g)) is isomorphic to 𝕄_{φ(g)}(ℂ). Since A(g) and A(h) contain the algebra of compact operators, the isomorphism from A(g) to A(h) deduces the isomorphism from EC(A(g)) to EC(A(h)). It is known that isomorphism classes of AF-algebras are classified up by the K-theoretic datum. In this case, we can see

(K_{0}(EC(A(g))), K_{0}(EC(A(g)))_{+}, [1]_{K_{0}})
= (ℤ ⊕ ℤ, ({0} ⊕ ℤ_{≥0}) ∪ (ℤ_{>0} ⊕ ℤ), (φ(g), ψ(g))).

We remark that, for any integer k ∈ ℤ, the following groups are order isomorphic (and preserving the order unit):

(ℤ ⊕ ℤ, ({0} ⊕ ℤ_{≥0}) ∪ (ℤ_{>0} ⊕ ℤ), (φ(g), ψ(g))),
(ℤ ⊕ ℤ, ({0} ⊕ ℤ_{≥0}) ∪ (ℤ_{>0} ⊕ ℤ), (φ(g), ψ(g) + kφ(g))).

This means that the K-theoretic datum is complete invariant for a family {A(g)|g ∈ G} of non-nuclear C*-algebras.

Acknowledgement. The authors would like to express their thanks to Professor K. Kodaka for the meaningful discussion using the Space Collaboration System of National Institute of Multimedia Education in Japan.

References

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∗Department of Mathematics and Information, Graduate School of Science and Technology Chiba University, 1-33, Yayoi-Cho, Inage-Ku, Chiba 263-8522 Japan

 E-mail address: 99um0102@g.math.s.chiba-u.ac.jp
 E-mail address: nagisa@math.s.chiba-u.ac.jp