New York Journal of Mathematics

New York J. Math. 13 (2007) 199–213.

Iterating the Pimsner construction

Valentin Deaconu

Abstract. For A a C^{∗}-algebra, E_{1}, E_{2} two Hilbert bimodules over A, and a fixed isomorphism χ : E_{1} ⊗_{A} E_{2} → E_{2} ⊗_{A} E_{1}, we consider the problem of computing the K-theory of the Cuntz–Pimsner algebra O_{E_{2}⊗_{A}O_{E1}} obtained by extending the scalars and by iterating the Pimsner construction.

The motivating examples are a commutative diagram of Douglas and Howe for the Toeplitz operators on the quarter plane, and the Toeplitz extensions associated by Pimsner and Voiculescu to compute the K-theory of a crossed product. The applications are for Hilbert bimodules arising from rank two graphs and from commuting endomorphisms of abelian C^{∗}-algebras.

Contents

1. Introduction 199 2. Preliminaries 200 3. Extending the scalars 204 4. Iterating the Pimsner construction 205 5. Exact sequences in K-theory 208 References 212

1. Introduction

In his seminal paper [Pi], Pimsner introduced a large class of C^{∗}-algebras generalizing both the Cuntz–Krieger algebras and the crossed products by an automorphism. The central notion is that of a Hilbert bimodule or C^{∗}-correspondence, which appeared also in the theory of subfactors. His construction was modified by Katsura (see [Ka1]) to include bimodules defined from graphs with sinks, or more generally, from topological graphs.

In the same way that a crossed product by ℤ^{2} could be thought as an iterated crossed product by ℤ, we iterate the Pimsner construction for two “commuting” Hilbert bimodules over a C^{∗}-algebra. A basic example comes from a rank two graph of Kumjian and Pask (see [KP]), where the two bimodules correspond to the horizontal and vertical edges, and the commutation relation is given by the unique

199^{ISSN 1076-9803/07}

Received January 2, 2007. Revised July 11, 2007.

Mathematics Subject Classification. Primary 46L05; Secondary 46L55, 46L80.

Key words and phrases. C^{∗}-algebra, Hilbert bimodule, K-theory.

factorization property. We get a particular case of the product systems defined by Fowler (see [F2]), but our approach has the advantage that it generates some exact sequences of K-theory.

After some preliminaries about the algebras T_{E} and O_{E}, we describe how from a Hilbert module E over A and a map A → B we can get a Hilbert module over B, using a tensor product. Given two Hilbert bimodules E_{1}, E_{2} over A and an isomorphism χ : E_{1} ⊗_{A} E_{2} → E_{2} ⊗_{A} E_{1} we consider E_{2} ⊗_{A} T_{E_{1}} and E_{2} ⊗_{A} O_{E_{1}} as Hilbert bimodules over T_{E_{1}} and O_{E_{1}} and repeat the Pimsner construction. The last section deals with a 3 × 3 diagram involving the iterated Cuntz–Pimsner algebras, inspired from the work of Douglas and Howe (see [DH]) and Pimsner and Voiculescu (see [PV]). As a corollary, we obtain some exact sequences of K-theory, including information about the K-groups of the C^{∗}-algebra O_{E_{2}⊗_{A}O_{E_{1}}} . Several examples are considered, including commuting endomorphisms of abelian C^{∗}-algebras.

Since this paper was completed, we discovered that J. Lindiarni and I. Raeburn ([LR]) already used the diagram which appears in our Lemma 5.2.

2. Preliminaries

Recall that a (right) Hilbert A-module is a Banach space E with a right action of a C^{∗}-algebra A and an A-valued inner product ⟨·, ·⟩ : E × E → A linear in the second variable such that

⟨ξ, ηa⟩ = ⟨ξ, η⟩a, ⟨ξ, η⟩ = ⟨η, ξ⟩^{∗}, ⟨ξ, ξ⟩ ≥ 0, ∥ξ∥ = ∥⟨ξ, ξ⟩∥^{1/2}.

A Hilbert module is called full if the closed linear span of the inner products coincides with A. We denote by L(E) the C^{∗}-algebra of adjointable operators on E, and by θ_{ξ,η} ∈ L(E) the rank one operator

θ_{ξ,η}(ζ) = ξ⟨η, ζ⟩.

The closed linear span of rank one operators is the ideal K(E). We have L(E) ≅ M(K(E)), the multiplier algebra. Also, K(E) can be identified with the balanced tensor product E ⊗_{A} E^{∗}, where E^{∗} is the dual of E, a left Hilbert A-module.

A Hilbert bimodule over A (sometimes called a C^{∗}-correspondence from A to A) is a Hilbert A-module with a left action of A given by a homomorphism φ : A → L(E). A Hilbert bimodule is called faithful if φ is injective. The left action is nondegenerate if \overline{φ(A)E} = E. For n ≥ 0 we denote by E^{⊗n} the Hilbert bimodule obtained by taking the tensor product of n copies of E, balanced over A (for n = 0, E^{⊗0} = A). Recall that for n = 2, the inner product is given by

⟨ξ ⊗ η, ξ′ ⊗ η′⟩ = ⟨η, φ(⟨ξ, ξ′⟩)η′⟩,

and it is inductively defined for general n.

Definition 2.1. A Toeplitz representation of a Hilbert bimodule E over A in a C^{∗}-algebra C is a pair (τ, π) with τ : E → C a linear map and π : A → C a *-homomorphism, such that

τ(ξa) = τ(ξ)π(a), τ(ξ)^{∗}τ(η) = π(⟨ξ, η⟩), τ(φ(a)ξ) = π(a)τ(ξ).

Note that the first property actually follows from the second. Indeed,

∥τ(ξa) − τ(ξ)π(a)∥^{2}
= (τ(ξa) − τ(ξ)π(a))^{∗}(τ(ξa) − τ(ξ)π(a))
= π(⟨ξa, ξa⟩) − π(a)^{∗}π(⟨ξ, ξa⟩) − π(⟨ξa, ξ⟩)π(a) + π(a)^{∗}π(⟨ξ, ξ⟩)π(a) = 0.

The corresponding universal C^{∗}-algebra is called the Toeplitz algebra of E, denoted by T_{E}. If E is full, then T_{E} is generated by elements τ^{n}(ξ)τ^{m}(η)^{∗}, m, n ≥ 0, where τ^{0} = π and for n ≥ 1, τ^{n}(ξ_{1} ⊗ ⋯ ⊗ ξ_{n}) = τ(ξ_{1}) … τ(ξ_{n}) is the extension of τ to E^{⊗n}. If E is also faithful, then A ⊂ T_{E}.

There is a homomorphism ψ : K(E) → C such that ψ(θ_{ξ,η}) = τ(ξ)τ(η)^{∗}. A representation (τ, π) is Cuntz–Pimsner covariant if π(a) = ψ(φ(a)) for all a in the ideal

I_{E} = φ^{−1}(K(E)) ∩ (ker φ)^{⊥}.

The Cuntz–Pimsner algebra O_{E} is universal with respect to the covariant representations, and it is a quotient of T_{E}.

There is a gauge action of 𝕋 on T_{E} and O_{E} defined by

z · (τ^{n}(ξ)τ^{m}(η)^{∗}) = z^{n−m}τ^{n}(ξ)τ^{m}(η)^{∗}, z ∈ 𝕋,

and using the universal properties. The core F_{E} is the fixed point algebra O_{E}^{𝕋}, generated by the union of the algebras K(E^{⊗n}).

The Toeplitz algebra T_{E} can be represented by creation operators T_{ξ}(η) = ξ ⊗ η on the Fock bimodule

ℓ^{2}(E) = ⨁_{n≥0} E^{⊗n},

and there is an ideal I = K(ℓ^{2}(E)I_{E}) in L(ℓ^{2}(E)) such that

0 → I → T_{E} → O_{E} → 0

is exact. In particular, if L(E) = K(E) or φ(A) ⊂ K(E), then

0 → K(ℓ^{2}(E)) → T_{E} → O_{E} → 0.

For more details about the algebras F_{E}, T_{E}, and O_{E} we refer to the original paper of Pimsner ([Pi]) and to [Ka1].

If E is finitely generated, then it has a basis {u_{i}}, in the sense that for all ξ ∈ E,

ξ = ∑_{i=1}^{n} u_{i}⟨u_{i}, ξ⟩.

In this case, L(E) = K(E), and the Cuntz–Pimsner algebra O_{E} is generated by S_{i} = S_{u_{i}} with relations

∑_{i=1}^{n} S_{i}S_{i}^{∗} = 1, S_{i}^{∗}S_{j} = ⟨u_{i}, u_{j}⟩, a · S_{j} = ∑_{i=1}^{n} S_{i}⟨u_{i}, a · u_{j}⟩.

The Toeplitz algebra T_{E} is generated by T_{i} = T_{u_{i}} which satisfy only the last two relations (see [KPW]).

Examples 2.2. (1) For A = ℂ and E = H a Hilbert space with orthonormal basis {ξ_{i}}_{i∈I}, the Toeplitz algebra T_{H} is generated by {S_{i}}_{i∈I} satisfying S_{i}^{∗}S_{j} = δ_{ij}·1, i, j ∈ I. In O_{H} we also have ∑_{i∈I} S_{i}S_{i}^{∗} = 1 if the dimension of H

is finite. In particular, for E = ℂ = A we get T_{E} ≅ T , the classical Toeplitz algebra generated by the unilateral shift, O_{E} ≅ C(𝕋), the continuous functions on the unit circle, and F_{E} ≅ ℂ. For E = ℂ^{n}, we get T_{E} ≅ E_{n}, the Cuntz–Toeplitz algebra, O_{E} ≅ O_{n}, the Cuntz algebra, and F_{E} ≅ UHF (n^{∞}). For H infinite-dimensional and separable, T_{H} ≅ O_{H} ≅ O_{∞}.

(2) Let E = A^{n} with the usual structure:

⟨(a_{1}, …, a_{n}), (b_{1}, …, b_{n})⟩ = ∑_{i=1}^{n} a_{i}^{∗}b_{i},

(a_{1}, …, a_{n}) · a = (a_{1}a, …, a_{n}a), a · (a_{1}, …, a_{n}) = (aa_{1}, …, aa_{n}). We get T_{E} ≅ A ⊗ E_{n}, O_{E} ≅ A ⊗ O_{n}, F_{E} ≅ A ⊗ UHF (n^{∞}).

(3) Let α : A → A be an automorphism of a unital C^{∗}-algebra, and let E = A(α) be the Hilbert bimodule obtained from A with the usual inner product and right multiplication, and with left action φ(a)x = α(a)x. Then T_{E} is isomorphic to the Toeplitz extension T_{α} used by Pimsner and Voiculescu in [PV], and O_{E} is isomorphic to the crossed product A ⋊_{α} ℤ. Indeed, let â denote the element in E obtained from a ∈ A. Then S = τ(ˆ1) is an isometry in any unital Toeplitz representation (τ, π), since ⟨ˆ1, ˆ1⟩ = 1, and it is an unitary in any unital covariant representation, since the rank one operator θ_{ˆ1,ˆ1} is the identity. We also have

π(α(a)) = π(⟨ˆ1, α̂(a)⟩) = τ(ˆ1)^{∗}τ(α̂(a)) = τ(ˆ1)^{∗}τ(φ(a)ˆ1)
= τ(ˆ1)^{∗}π(a)τ(ˆ1) = S^{∗}π(a)S.

Therefore O_{E} ≅ A ⋊_{α} ℤ. In the paper mentioned above, the Toeplitz extension T_{α} was defined as the C^{∗}-subalgebra of (A ⋊_{α} ℤ) ⊗ T generated by A ⊗ 1 and u ⊗ S_{+}, where α(a) = uau^{∗} and S_{+} is the unilateral shift. It is easy to see that T_{α} ≅ T_{E} by the map which takes a ⊗ 1 into a and u ⊗ S_{+} into S. Since K(ℓ^{2}(E)) ≅ A ⊗ K, we recover the short exact sequence

0 → A ⊗ K → T_{α} → A ⋊_{α} ℤ → 0.

(4) Graph C^{∗}-algebras. For an oriented countable graph G = (G^{0}, G^{1}, r, s), C^{∗}(G) is defined as the universal C^{∗}-algebra generated by mutually orthogonal projections {p_{v}}_{v∈G^{0}} and partial isometries {s_{e}}_{e∈G^{1}} with orthogonal ranges such that s_{e}^{∗}s_{e} = p_{r(e)}, s_{e}s_{e}^{∗} ≤ p_{s(e)} and

(∗)  p_{v} = ∑_{s(e)=v} s_{e}s_{e}^{∗} if 0 < |s^{−1}(v)| < ∞.

We can set A = C_{0}(G^{0}), and denote by E the Hilbert module we obtain after we complete C_{c}(G^{1}) in the norm given by the inner product

⟨ξ, η⟩(v) = ∑_{r(e)=v} \overline{ξ(e)}η(e)

with the right action defined by (ξf)(e) = ξ(e)f(r(e)). The left action is defined by

φ : A → L(E), φ(f)ξ(e) = f(s(e))ξ(e), ξ ∈ C_{c}(G^{1}) ⊂ E.

We have

φ^{−1}(K(E)) = C_{0}({v ∈ G^{0} : |s^{−1}(v)| < ∞}),
ker(φ) = C_{0}({v ∈ G^{0} : |s^{−1}(v)| = 0}),

hence

I_{E} = C_{0}({v ∈ G^{0} : 0 < |s^{−1}(v)| < ∞}).

Define

π(f) = ∑_{v∈G^{0}} f(v)p_{v}, f ∈ C_{0}(G^{0}),
τ(ξ) = ∑_{e∈G^{1}} ξ(e)s_{e}, ξ ∈ C_{c}(G^{1}) ⊂ E.

The pair (τ, π) is a Toeplitz representation into C^{∗}(G) iff s_{e}^{∗}s_{e} = p_{r(e)} and s_{e}s_{e}^{∗} ≤ p_{s(e)}, which is covariant iff (∗) is satisfied. This proves that C^{∗}(G) ≅ O_{E} (see [Ka1]). The core F_{E} is an AF-algebra. For an irreducible oriented finite graph with no sinks, we obtain the Cuntz–Krieger algebras O_{A} as Cuntz–Pimsner algebras.

(5) For a C^{∗}-algebra A and an injective unital endomorphism α ∈ End(A) such that there is a conditional expectation P onto the range α(A), one can define a Hilbert bimodule E = A(α, P), using the transfer operator L = α^{−1} ∘ P : A → A as in [EV]. We complete the vector space A with respect to the inner product ⟨ξ, η⟩ = L(ξ^{∗}η), and define the right and left multiplications by ξ · a = ξα(a), a · ξ = aξ. We have

⟨ξ, η · a⟩ = ⟨ξ, ηα(a)⟩ = L(ξ^{∗}ηα(a)) = α^{−1}(P (ξ^{∗}ηα(a)))
= α^{−1}(P (ξ^{∗}η)α(a)) = α^{−1}(P (ξ^{∗}η))a = ⟨ξ, η⟩a.

For α an automorphism and P = id, the corresponding C^{∗}-algebra O_{E} is isomorphic to the crossed product A ⋊_{α^{−1}} ℤ. Indeed, let F be the Hilbert bimodule A(α^{−1}) with the structure as in Example 2.2.3. The map α^{−1} : A → A induces an isomorphism of Hilbert bimodules h : E → F. For A = C(X) with X compact, and α induced by a surjective local homeomorphism σ : X → X, we take

(P f)(x) = \frac{1}{ν(x)} ∑_{σ(y)=σ(x)} f(y),

where ν(x) is the number of elements in the fiber σ^{−1}(x). It was proved in [D] that the corresponding algebra O_{A(α,P)} is isomorphic to C^{∗}(Γ(σ)), where Γ(σ) is the Renault groupoid

Γ(σ) = {(x, p − q, y) ∈ X × ℤ × X | σ^{p}(x) = σ^{q}(y)}.

3. Extending the scalars

Let E be a Hilbert module over A and let ρ : A → B be a C^{∗}-algebra homomorphism (we will be interested mostly in the case when ρ is an inclusion). Then B is a left A-module with multiplication a · b = ρ(a)b, and E ⊗_{A} B becomes a Hilbert module over B, with the inner product given by

⟨ξ_{1} ⊗ b_{1}, ξ_{2} ⊗ b_{2}⟩ = b_{1}^{∗}ρ(⟨ξ_{1}, ξ_{2}⟩)b_{2}

and right multiplication (ξ ⊗ b_{1}) · b_{2} = ξ ⊗ b_{1}b_{2}. We have

K(E ⊗_{A} B) ≅ (E ⊗_{A} B) ⊗_{B} (E ⊗_{A} B)^{∗} ≅ E ⊗_{A} B ⊗_{A} E^{∗},

which is strongly Morita equivalent to B in the case E is full. Also, we have an inclusion K(E) ⊂ K(E ⊗_{A} B) for B unital, given by ξ ⊗ η^{∗} ↦ ξ ⊗ 1 ⊗ η^{∗}. If E is a Hilbert bimodule over A, and if there is a *-morphism B → L(E) which extends the left multiplication of A on E, then E ⊗_{A} B becomes a Hilbert bimodule over B, and one can form the tensor powers (E ⊗_{A} B)^{⊗n}. Assuming that the left action of B on E ⊗_{A} B is nondegenerate, we get

(E ⊗_{A} B)^{⊗n} ≅ E^{⊗n} ⊗_{A} B.

In particular, the Toeplitz algebra T_{E⊗_{A}B} is represented on

ℓ^{2}(E ⊗_{A} B) ≅ ℓ^{2}(E) ⊗_{A} B,

and depends on the left multiplication of B. An interesting question is to relate the C^{∗}-algebras T_{E⊗_{A}B} and O_{E⊗_{A}B} to T_{E}, O_{E}, and B.

Example 3.1. Let A = ℂ, let E = H be a separable infinite-dimensional Hilbert space, and let B be a separable unital C^{∗}-algebra. Then H ⊗B is a Hilbert module over B, and K(H ⊗B) ≅ K(H)⊗B. If B is faithfully represented on H, then H ⊗B becomes a Hilbert bimodule over B. Assuming, in addition, that the intersection of B with K(H) is trivial, Kumjian (see [K]) showed that T_{H⊗B} ≅ O_{H⊗B} is simple and purely infinite, with the same K-theory as B.

Example 3.2. Let A = C_{0}(X) and let E be a Hilbert module given by a continuous field of elementary C^{∗}-algebras over X. Then for an abelian C^{∗}-algebra B containing C_{0}(X), the tensor product E ⊗_{A} B is obtained by a pullback. In particular, let (G^{0}, G^{1}, r, s) be a (topological) graph, and consider a covering map p : G̃^{0} → G^{0} which gives an inclusion A = C_{0}(G^{0}) ⊂ C_{0}(G̃^{0}) = B. The Hilbert module E ⊗_{A} B, where E is obtained from C_{c}(G^{1}) as in Example 2.2.4, is associated to a“pullback graph” in which the set of vertices is G̃^{0} and the set of edges is G̃^{1} := {(x, e, y) ∈ G̃^{0} × G^{1} × G̃^{0} | s(e) = p(x), r(e) = p(y)}. The new range and source maps are s(x, e, y) = x, r(x, e, y) = y. It is known that K(E) is isomorphic to C^{∗}(R), where R is the equivalence relation

R = {(e_{1}, e_{2}) ∈ G^{1} × G^{1} | r(e_{1}) = r(e_{2})}.

Then K(E ⊗_{A} B) is isomorphic to C^{∗}(p^{∗}(R)), where

p^{∗}(R) = {((x_{1}, e_{1}, y_{1}), (x_{2}, e_{2}, y_{2})) ∈ G̃^{1} × G̃^{1} | p(y_{1}) = p(y_{2})}.

Example 3.3. Let α : A → A be an automorphism of a C^{∗}-algebra A, which extends to α̃ : B → B, where A ⊂ B. Consider E = A(α) as in Example 2.2.3.

Then E ⊗_{A} B ≅ B(α̃), T_{E⊗_{A}B} ≅ Tα̃ and O_{E⊗_{A}B} ≅ B ⋊α̃ ℤ, which contains O_{E} ≅ A ⋊_{α} ℤ.

Example 3.4. For a Hilbert bimodule E over A, Pimsner used the Hilbert module E_{∞} = E ⊗_{A} F_{E} (see [Pi], Section 2) in order to get an inclusion T_{E} ⊂ T_{E_{∞}}, an isomorphism O_{E} ≅ O_{E_{∞}}, and a completely positive map ϕ : O_{E} → T_{E_{∞}} which is a cross-section to the quotient map T_{E_{∞}} → O_{E}.

4. Iterating the Pimsner construction

Consider now two full finitely generated Hilbert A-bimodules E_{1} and E_{2} such that A is unital and the left actions φ_{i} : A → L(E_{i}) are injective and nondegenerate. We assume that there is an isomorphism of Hilbert A-bimodules χ : E_{1} ⊗_{A} E_{2} → E_{2} ⊗_{A} E_{1}. This isomorphism should be understood as a kind of commutation relation. The most interesting cases are when E_{1} and E_{2} are independent, in the sense that no tensor power of one is isomorphic to the other.

Note that the isomorphism χ induces and isomorphism E_{1}^{∗} ⊗_{A} E_{2}^{∗} → E_{2}^{∗} ⊗_{A} E_{1}^{∗} because (E_{1} ⊗_{A} E_{2})^{∗} ≅ E_{2}^{∗} ⊗_{A} E_{1}^{∗}.

Since A ⊂ T_{E_{1}}, the tensor product E_{2} ⊗_{A} T_{E_{1}} becomes a Hilbert module over T_{E_{1}} as in Section 3, with the inner product given by ⟨ξ ⊗ x, η ⊗ y⟩ = x^{∗}⟨ξ, η⟩y, and the right multiplication by (ξ ⊗ x)y = ξ ⊗ xy. Since T_{E_{1}} is generated by E_{1}, in order to define a left multiplication of T_{E_{1}} on E_{2} ⊗_{A} T_{E_{1}}, it is sufficient to define the left multiplication by elements in E_{1}. This is done via the composition

E_{1} ⊗_{A} E_{2} ⊗_{A} T_{E_{1}} → E_{2} ⊗_{A} E_{1} ⊗_{A} T_{E_{1}} → E_{2} ⊗_{A} T_{E_{1}},

where the first map is χ ⊗ id, and the second is given by absorbing E_{1} into T_{E_{1}}. For the adjoint, note that there is a map

E_{1}^{∗} ⊗_{A} E_{2} ⊗_{A} E_{1} → E_{1}^{∗} ⊗_{A} E_{1} ⊗_{A} E_{2} → E_{2},

where the first map is id ⊗ χ^{−1}, and the second is given by left multiplication with the inner product in E_{1}. We will denote by E_{2} ⊗_{A}^{χ} T_{E_{1}} the bimodule obtained using this left multiplication, when χ is not understood. In the same way, we may consider the bimodule E_{1} ⊗_{A} T_{E_{2}} with the left multiplication induced by χ^{−1}.

Lemma 4.1. With the above structure, E_{2} ⊗_{A} T_{E_{1}} is a Hilbert bimodule over T_{E_{1}}, and we can consider the Toeplitz algebra T_{E_{2}⊗_{A}T_{E_{1}}} . We have

T_{E_{2}⊗_{A}T_{E_{1}}} ≅ T_{E_{1}⊗_{A}T_{E_{2}}} .

Proof. Both algebras T_{E_{2}⊗_{A}T_{E_{1}}} , T_{E_{1}⊗_{A}T_{E_{2}}} are represented on the Fock space

ℓ^{2}(E_{2}) ⊗_{A} ℓ^{2}(E_{1}) ≅ ℓ^{2}(E_{1}) ⊗_{A} ℓ^{2}(E_{2}),

and are generated by T_{E_{1}} and T_{E_{2}} with the commutation relation given by the isomorphism χ. □

Similarly, we can construct the Hilbert bimodules E_{2} ⊗_{A} O_{E_{1}} and E_{1} ⊗_{A} O_{E_{2}}. We get

Lemma 4.2. With the above notation,

T_{E_{2}⊗_{A}O_{E_{1}}} ≅ O_{E_{1}⊗_{A}T_{E_{2}}} , O_{E_{2}⊗_{A}O_{E_{1}}} ≅ O_{E_{1}⊗_{A}O_{E_{2}}} .

Proof. The first two algebras are quotients of T_{E_{2}⊗_{A}T_{E_{1}}} ≅ T_{E_{1}⊗_{A}T_{E_{2}}} by the ideal generated by K(ℓ^{2}(E_{1})), and the last two are quotients by the ideal generated by K(ℓ^{2}(E_{1})) and K(ℓ^{2}(E_{2})). □

Note that there is a gauge action of 𝕋^{2} on O_{E_{2}⊗_{A}O_{E_{1}}} ≃ O_{E_{1}⊗_{A}O_{E_{2}}} .

Remark 4.3. Given E_{1}, E_{2} as above, we can define a product system E = E^{χ} of Hilbert bimodules over the semigroup (ℕ^{2}, +) (see [F2]), as follows. Define the fibers by E_{(m,n)} = E_{1}^{⊗m} ⊗_{A} E_{2}^{⊗n} for (m, n) ∈ ℕ^{2}, and the multiplication induced by the isomorphism χ. It is easy to see that we get associativity, and therefore we may consider the C^{∗}-algebras T_{E}, O_{E} as defined by Fowler. Note that if we change χ, the product system also changes. In particular, a path of isomorphisms χ_{t} will determine a family of product systems E^{t} = E^{χ_{t}}.

Recall that a Toeplitz representation of a product system E in a C^{∗}-algebra is obtained from a family of Toeplitz representations of the fibers, compatible with the product. In this particular case, it is sufficient to consider two Toeplitz representations (τ_{1}, π), (τ_{2}, π) of the generators E_{1} and E_{2}, respectively, and to define τ(ξ_{1} ⊗ ⋯ ⊗ ξ_{m} ⊗ η_{1} ⊗ ⋯ ⊗ η_{n}) = τ_{1}(ξ_{1}) … τ_{1}(ξ_{m})τ_{2}(η_{1}) … τ_{2}(η_{n}) for ξ_{i} ∈ E_{1}, i = 1, …, m and η_{j} ∈ E_{2}, j = 1, …, n. The Toeplitz algebra T_{E} is represented on the Fock space ℓ^{2}(E) ≅ ℓ^{2}(E_{1}) ⊗_{A} ℓ^{2}(E_{2}) by creation operators. The covariance condition requires that each (τ_{i}, π) is covariant, i = 1, 2. This means that ψ_{i}(φ_{i}(a)) = π(a) for a ∈ φ_{i}^{−1}(K(E_{i})), i = 1, 2, where ψ_{i}(θ_{ξ,η}) = τ_{i}(ξ)τ_{i}(η)^{∗}. We get that

T_{E_{2}⊗_{A}T_{E_{1}}} ≅ T_{E_{1}⊗_{A}T_{E_{2}}} ≅ T_{E} and O_{E_{2}⊗_{A}O_{E_{1}}} ≅ O_{E_{1}⊗_{A}O_{E_{2}}} ≅ O_{E}.

Example 4.4. Let A be a unital C^{∗}-algebra, and α, β two commuting automorphisms of A. We assume that α and β generate ℤ^{2} as a subgroup of Aut(A). Denote by A(α), A(β) the Hilbert bimodules as in Example 2.2.3, which are independent.

We have A(α) ⊗_{A} A(β) ≅ A(β) ⊗_{A} A(α) by the map χ(â ⊗ b̂) = { \atop β(α^{−1}(a))} ⊗ b̂.

Indeed,

⟨χ(â_{1} ⊗ b̂_{1}), χ(â_{2} ⊗ b̂_{2})⟩ = ⟨{ \atop β(α^{−1}(a_{1}))} ⊗ b̂_{1}, { \atop β(α^{−1}(a_{2}))} ⊗ b̂_{2}⟩

= ⟨b̂_{1}, ⟨{ \atop β(α^{−1}(a_{1}))}, { \atop β(α^{−1}(a_{2}))}⟩ · b̂_{2}⟩

= b_{1}^{∗}α(β(α^{−1}(a_{1}))^{∗}β(α^{−1}(a_{2})))b_{2}

= b_{1}^{∗}β(a_{1}^{∗}a_{2})b_{2} = ⟨b̂_{1}, ⟨â_{1}, â_{2}⟩ · b̂_{2}⟩

= ⟨â_{1} ⊗ b̂_{1}, â_{2} ⊗ b̂_{2}⟩,

and

χ(a′ · (â ⊗ b̂) · b′) = χ({ \atop α(a′)a} ⊗ b̂b′) = { \atop β(α^{−1}(α(a′)a))} ⊗ b̂b′

= { \atop β(a′α^{−1}(a))} ⊗ b̂b′ = { \atop β(a′)β(α^{−1}(a))} ⊗ b̂b′

= a′ · χ(â ⊗ b̂) · b′.

Denote by T_{α}, T_{β} the corresponding Toeplitz algebras, each generated by A and an isometry as in Example 2.2.3. Then A(β) ⊗_{A} T_{α} becomes a Hilbert bimodule over T_{α}, isomorphic to T_{α}(β), where β is extended to T_{α} in the natural way, fixing the isometry. Similarly, A(α) ⊗_{A} T_{β} ≃ T_{β}(α). We have

T_{T_{α}(β)} ≅ T_{T_{β}(α)} ≅ T_{E},

where E is the product system over ℕ^{2} constructed from A(α), A(β), and the isomorphism χ. Also, we may consider the Hilbert bimodules A(β) ⊗_{A} (A ⋊_{α} ℤ) ≅ (A ⋊_{α} ℤ)(β), and A(α) ⊗_{A} (A ⋊_{β} ℤ) ≅ (A ⋊_{β} ℤ)(α), where again α and β are extended to A ⋊_{β} ℤ and A ⋊_{α} ℤ, respectively, by fixing the unitaries implementing the actions of ℤ. We have

T_{(A⋊_{α}ℤ)(β)} ≅ T_{β} ⋊_{α} ℤ,
T_{(A⋊_{β}ℤ)(α)} ≅ T_{α} ⋊_{β} ℤ,
O_{(A⋊_{α}ℤ)(β)} ≅ O_{(A⋊_{β}ℤ)(α)} ≅ O_{E} ≅ A ⋊_{α,β} ℤ^{2}.

Example 4.5. Let E = F = ℂ^{2} with the usual structures of Hilbert bimodules over A = ℂ. Then O_{E} ≅ O_{2}, and F ⊗ O_{2} becomes a Hilbert module over O_{2} with the usual operations. The left action will depend on a fixed isomorphism χ : E ⊗ F → F ⊗ E. If {e_{1}, e_{2}} and {f_{1}, f_{2}} are the canonical bases in E and F respectively, and χ_{1}(e_{i} ⊗ f_{j}) = f_{j} ⊗ e_{i}, i, j = 1, 2, then the left multiplication is given by S_{i}(f_{j} ⊗S_{k}) = f_{j} ⊗S_{i}S_{k}, where O_{2} has generators {S_{1}, S_{2}}, and O_{F ⊗^{χ_{1}} O_{2}} ≅ O_{2} ⊗ O_{2} ≅ O_{2}. On the other hand, if χ_{2}(e_{i} ⊗ f_{j}) = f_{i} ⊗ e_{j}, i, j = 1, 2, then the left action is given by S_{i}(f_{j} ⊗ S_{k}) = f_{i} ⊗ S_{j}S_{k}, and the O_{2}-bimodule F ⊗^{χ_{2}} O_{2} is degenerate. The corresponding Cuntz–Pimsner algebra O_{F ⊗^{χ_{2}} O_{2}} is isomorphic to C(𝕋) ⊗ O_{2}. Indeed, if we interpret E and F as being each associated to the 1-graph Γ defining O_{2} (Γ has two edges and one vertex), then the isomorphism χ_{2} is defining the rank 2 graph ϕ^{∗}(Γ) where ϕ : ℕ^{2} → ℕ, ϕ(m, n) = m + n , and the last assertion follows from Example 6.1 and Proposition 2.10 in [KP].

Note that an arbitrary isomorphism χ does not necessarily define a rank 2 graph. For example, χ could be given by the unitary matrix U = (u_{kl}), where

u_{11} = cos α, u_{12} = − sin α, u_{21} = sin α, u_{22} = cos α,
u_{33} = cos β, u_{34} = − sin β, u_{43} = sin β, u_{44} = cos β,

for some angles α and β, and the rest of the entries equal to zero. For other C^{∗}- algebras defined by a product system of finite-dimensional Hilbert spaces over the semigroup ℕ^{2}, see [F1].

Example 4.6. Consider two star-commuting onto local homeomorphisms σ_{1} and σ_{2} of a compact space X. By definition, that means that for every x, y ∈ X such that σ_{1}(x) = σ_{2}(y), there exists a unique z ∈ X such that σ_{2}(z) = x and σ_{1}(z) = y. This condition ensures that the associated conditional expectations P_{1}, P_{2} commute (see [ER]). Unfortunately, this condition was omitted in the Proposition on page 8 in [D2]. I would like to thank Ruy and Jean for pointing this to me.

We can define the Hilbert bimodules E_{i} = A(α_{i}, P_{i}), i = 1, 2 over A = C(X) as in Example 2.2.5. Then there is an isomorphism

h : A(α_{1}, P_{1}) ⊗_{A} A(α_{2}, P_{2}) → A(α_{1} ∘ α_{2}, P_{1} ∘ P_{2}),

given by h(â ⊗ b̂) = { \atop aα_{1}(b)}, with inverse h^{−1}(x̂) = x̂ ⊗ ˆ1, which induces an isomorphism χ : E_{1}⊗_{A}E_{2} → E_{2}⊗_{A}E_{1}. The resulting C^{∗}-algebra O_{E_{2}⊗_{A}O_{E_{1}}} ≅ O_{E_{1}⊗_{A}O_{E_{2}}} can be understood as a crossed product of C(X) by the semigroup ℕ^{2}. For other examples of semigroups of local homeomorphisms, see [ER].

5. Exact sequences in K-theory

To study the C^{∗}-algebra of Toeplitz operators on the quarter plane, Douglas and Howe (see [DH]) considered the commutative diagram with exact rows and columns, where j is the inclusion map, and π is the quotient map:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & K ⊗ K \ar[r]^{j⊗1} \ar[d]^{1⊗j} & T ⊗ K \ar[r]^{π⊗1} \ar[d]^{1⊗j} & C(𝕋) ⊗ K \ar[r] \ar[d]^{1⊗j} & 0 \\ 0 \ar[r] & K ⊗ T \ar[r]^{j⊗1} \ar[d]^{1⊗π} & T ⊗ T \ar[r]^{π⊗1} \ar[d]^{1⊗π} & C(𝕋) ⊗ T \ar[r] \ar[d]^{1⊗π} & 0 \\ 0 \ar[r] & K ⊗ C(𝕋) \ar[r]^{j⊗1} \ar[d] & T ⊗ C(𝕋) \ar[r]^{π⊗1} \ar[d] & C(𝕋) ⊗ C(𝕋) \ar[r] \ar[d] & 0. \\  & 0 & 0 & 0}

Corollary 5.1. We have the short exact sequences

0 → T ⊗ K + K ⊗ T ⟶^{1⊗j+j⊗1} T ⊗ T ⟶^{π⊗π} C(𝕋) ⊗ C(𝕋) → 0,

0 → K ⊗ K ⟶^{j⊗1+1⊗j} T ⊗ K + K ⊗ T ⟶^{π⊗1+1⊗π} C(𝕋) ⊗ K ⊕ K ⊗ C(𝕋) → 0.

The above 3 × 3 diagram and the exact sequences in the corollary are particular cases of a more general situation, for which we provide a proof.

Lemma 5.2. Let A be a C^{∗}-algebra and I, J two closed two-sided ideals of A. Then we have the commutative diagram with exact rows and columns, where the maps are the canonical ones:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & I ∩ J \ar[r]^{λ_{1}} \ar[d]^{λ_{2}} & I \ar[r]^{ω_{1}} \ar[d]^{ι_{1}} & I/(I ∩ J) \ar[r] \ar[d]^{σ_{1}} & 0 \\ 0 \ar[r] & J \ar[r]^{ι_{2}} \ar[d]^{ω_{2}} & A \ar[r]^{π_{2}} \ar[d]^{π_{1}} & A/J \ar[r] \ar[d]^{ρ_{2}} & 0 \\ 0 \ar[r] & J/(I ∩ J) \ar[r]^{σ_{2}} \ar[d] & A/I \ar[r]^{ρ_{1}} \ar[d] & A/(I + J) \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

From this we get the exact sequences

0 → I + J ⟶^{ι_{1}+ι_{2}} A ⟶^{π} A/(I + J) → 0,

where π = ρ_{1} ∘ π_{1} = ρ_{2} ∘ π_{2}, and

0 → I ∩ J ⟶^{λ_{1}+λ_{2}} I + J ⟶^{ω_{1}+ω_{2}} I/(I ∩ J) ⊕ J/(I ∩ J) → 0.

Applying the K-theory functor, we get

\xymatrix{K_{0}(I + J) \ar[r] & K_{0}(A) \ar[r] & K_{0}(A/(I + J)) \ar[d] \\ K_{1}(A/(I + J)) \ar[u] & K_{1}(A) \ar[l] & K_{1}(I + J), \ar[l]}

\xymatrix{K_{0}(I ∩ J) \ar[r] & K_{0}(I + J) \ar[r] & K_{0}(I/(I ∩ J)) ⊕ K_{0}(J/(I ∩ J)) \ar[d] \\ K_{1}(I/(I ∩ J)) ⊕ K_{1}(J/(I ∩ J)) \ar[u] & K_{1}(I + J) \ar[l] & K_{1}(I ∩ J). \ar[l]}

Proof. Note that the first two rows and the first two columns are obviously exact. For the third row, the map π_{1} ∘ ι_{2} has kernel I ∩ J. This defines a map σ_{2} : J/(I ∩ J) → A/I such that σ_{2} ∘ ω_{2} = π_{1} ∘ ι_{2}. By the second isomorphism theorem, (I +J)/I ≅ J/(I ∩J), and by the third isomorphism theorem, (A/I)/((I +J)/I) ≅ A/(I +J), hence the third row is exact. The exactness of the third column is proved similarly. Consider now the diagonal morphism π = ρ_{1} ∘ π_{1} = ρ_{2} ∘ π_{2} : A → A/(I + J). If a ∈ ker π, then π_{2}(a) ∈ ker ρ_{2} = σ_{1}(I/(I ∩ J)) = σ_{1}(ω_{1}(I)), hence there is b ∈ I with π_{2}(a) = σ_{1}(ω_{1}(b)) = π_{2}(ι_{1}(b)). It follows that a − ι_{1}(b) ∈ ker π_{2} = ι_{2}(J), and there is c ∈ J with a = ι_{1}(b) + ι_{2}(c). This gives the first exact sequence. For the second, we use the map ω_{1} + ω_{2} : I + J → I/(I ∩ J) ⊕ J/(I ∩ J) which has kernel I ∩ J. □

We generalize the above diagram of Douglas and Howe to certain iterated Toeplitz and Cuntz–Pimsner algebras. By the lemma, we get some exact sequences which we hope will help to do K-theory computations in some particular cases.

Theorem 5.3. Consider a C^{∗}-algebra A and two finitely generated Hilbert bimodules E_{1}, E_{2} with a fixed isomorphism χ : E_{1} ⊗_{A} E_{2} → E_{2} ⊗_{A} E_{1}. We assume that E_{2} ⊗_{A} T_{E_{1}}, E_{2} ⊗_{A} O_{E_{1}}, E_{1} ⊗_{A} T_{E_{2}}, E_{1} ⊗_{A} O_{E_{2}} are nondegenerate as Hilbert bimodules, with the structure described in Section 3. Then we have the following commuting diagram (depending on χ), with exact rows and columns, where the maps are canonical:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & K(ℓ^{2}(E_{1} ⊗_{A} E_{2})) \ar[r] \ar[d] & K(ℓ^{2}(E_{2} ⊗_{A} T_{E_{1}})) \ar[r] \ar[d] & K(ℓ^{2}(E_{2} ⊗_{A} O_{E_{1}})) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K(ℓ^{2}(E_{1} ⊗_{A} T_{E_{2}})) \ar[r] \ar[d] & T_{E_{1}⊗_{A}T_{E_{2}}} ≅ T_{E_{2}⊗_{A}T_{E_{1}}} \ar[r] \ar[d] & O_{E_{1}⊗_{A}T_{E_{2}}} ≅ T_{E_{2}⊗_{A}O_{E_{1}}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K(ℓ^{2}(E_{1} ⊗_{A} O_{E_{2}})) \ar[r] \ar[d] & T_{E_{1}⊗_{A}O_{E_{2}}} ≅ O_{E_{2}⊗_{A}T_{E_{1}}} \ar[r] \ar[d] & O_{E_{1}⊗_{A}O_{E_{2}}} ≅ O_{E_{2}⊗_{A}O_{E_{1}}} \ar[r] \ar[d] & 0. \\  & 0 & 0 & 0}

Proof. Recall that the algebra T_{E_{i}} is represented on the Fock space ℓ^{2}(E_{i}) and we have a short exact sequence

0 → K(ℓ^{2}(E_{i})) → T_{E_{i}} → O_{E_{i}} → 0,

for i = 1, 2. We apply the above lemma for the C^{∗}-algebra T_{E_{1}⊗_{A}T_{E_{2}}} ≅ T_{E_{2}⊗_{A}T_{E_{1}}} with ideals I = K(ℓ^{2}(E_{2} ⊗_{A} T_{E_{1}})) and J = K(ℓ^{2}(E_{1} ⊗_{A} T_{E_{2}})). The nondegeneracy assumption implies that ℓ^{2}(E_{2} ⊗_{A} T_{E_{1}}) ≅ ℓ^{2}(E_{2}) ⊗_{A} T_{E_{1}} and ℓ^{2}(E_{2} ⊗_{A} O_{E_{1}}) ≅ ℓ^{2}(E_{2})⊗_{A} O_{E_{1}}. Note that the map χ : E_{1} ⊗_{A} E_{2} → E_{2} ⊗_{A} E_{1} induces isomorphisms K(ℓ^{2}(E_{2} ⊗_{A} K(ℓ^{2}(E_{1})))) ≅ K(ℓ^{2}(E_{1} ⊗_{A} K(ℓ^{2}(E_{2})))) ≅ K(ℓ^{2}(E_{1} ⊗ E_{2})), therefore I ∩ J = K(ℓ^{2}(E_{1} ⊗_{A} E_{2})). □

Corollary 5.4. Under the same assumptions as in the theorem, we get the short exact sequences

0 → K(ℓ^{2}(E_{2} ⊗_{A} T_{E_{1}})) + K(ℓ^{2}(E_{1} ⊗_{A} T_{E_{2}})) → T_{E_{1}⊗_{A}T_{E_{2}}} → O_{E_{1}⊗_{A}O_{E_{2}}} → 0

and

0 → K(ℓ^{2}(E_{1} ⊗_{A} E_{2})) → K(ℓ^{2}(E_{2} ⊗_{A} T_{E_{1}})) + K(ℓ^{2}(E_{1} ⊗_{A} T_{E_{2}})) →

→ K(ℓ^{2}(E_{2} ⊗_{A} O_{E_{1}})) ⊕ K(ℓ^{2}(E_{1} ⊗_{A} O_{E_{2}})) → 0.

The corresponding six-term exact sequences of K-theory give us information about the K-theory of O_{E_{1}⊗_{A}O_{E_{2}}} , once we identify the maps between the various K-groups. Note that the Toeplitz algebras are KK-equivalent to the C^{∗}-algebra A.

Example 5.5. For α_{1}, α_{2} two commuting independent automorphisms of a C^{∗}- algebra A, the diagram is

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & K ⊗ A ⊗ K \ar[r] \ar[d] & T_{α_{1}} ⊗ K \ar[r] \ar[d] & A ⋊_{α_{1}} ℤ ⊗ K \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K ⊗ T_{α_{2}} \ar[r] \ar[d] & T_{T_{α_{2}} (α_{1})} ≅ T_{T_{α_{1}} (α_{2})} \ar[r] \ar[d] & T_{α_{2}} ⋊_{α_{1}} ℤ ≅ T_{(A⋊_{α_{1}} ℤ)(α_{2})} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K ⊗ A ⋊_{α_{2}} ℤ \ar[r] \ar[d] & T_{α_{1}} ⋊_{α_{2}} ℤ ≅ T_{(A⋊_{α_{2}} ℤ)(α_{1})} \ar[r] \ar[d] & A ⋊_{α_{1},α_{2}} ℤ^{2} \ar[r] \ar[d] & 0. \\  & 0 & 0 & 0}

We get the exact sequences

\xymatrix{K_{0}(A) \ar[r] & K_{0}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}) \ar[d] \\ K_{1}(A ⋊_{α_{1}} ℤ) ⊕ K_{1}(A ⋊_{α_{2}} ℤ) \ar[u]^{∂^{1}+∂^{2}} & K_{0}(A ⋊_{α_{1}} ℤ) ⊕ K_{0}(A ⋊_{α_{2}} ℤ) \ar[d]_{∂^{1}+∂^{2}} \\ K_{1}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}) \ar[u] & K_{1}(A) \ar[l]}

\xymatrix{K_{0}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}) \ar[r] & K_{0}(A) \ar[d] \\ K_{1}(A ⋊_{α_{1},α_{2}} ℤ^{2}) \ar[u] & K_{0}(A ⋊_{α_{1},α_{2}} ℤ^{2}) \ar[d] \\ K_{1}(A) \ar[u] & K_{1}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}). \ar[l]}

Let now A = C(X) with X a Cantor set, and let σ, τ : X → X be two commuting local homeomorphisms. Denote by α, β the induced endomorphisms of A, and assume that the associated conditional expectations commute. Applying the theorem for the Hilbert bimodules E = A(α, P), F = A(β, Q) and the canonical isomorphism E ⊗_{A} F ≅ F ⊗_{A} E, we get

\xymatrix{C(X, ℤ) \ar[r] & K_{0}(T_{E} ⊗ K + K ⊗ T_{F}) \ar[d] \\ ker(1 − α_{∗}) ⊕ ker(1 − β_{∗}) \ar[u] & C(X, ℤ)/im(1 − α_{∗}) ⊕ C(X, ℤ)/im(1 − β_{∗}) \ar[d] \\ K_{1}(T_{E} ⊗ K + K ⊗ T_{F}) \ar[u] & 0 \ar[l]}

\xymatrix{K_{0}(T_{E} ⊗ K + K ⊗ T_{F}) \ar[r] & C(X, ℤ) \ar[d] \\ K_{1}(O_{F ⊗_{A}O_{E}}) \ar[u] & K_{0}(O_{F ⊗_{A}O_{E}}) \ar[d] \\ 0 \ar[u] & K_{1}(T_{E} ⊗ K + K ⊗ T_{F}). \ar[l]}

Here we used the fact that K_{0}(A) = C(X, ℤ) and that K_{1}(A) = 0.

Example 5.6. Let E = ℂ^{m}, F = ℂ^{n} and fix χ : E ⊗ F → F ⊗ E an isomorphism. The corresponding diagram (depending on χ) is

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & K ⊗ K \ar[r] \ar[d] & E_{m} ⊗ K \ar[r] \ar[d] & O_{m} ⊗ K \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K ⊗ E_{n} \ar[r] \ar[d] & T_{ℂ^{m}⊗E_{n}} ≅ T_{ℂ^{n}⊗E_{m}} \ar[r] \ar[d] & O_{ℂ^{m}⊗E_{n}} ≅ T_{ℂ^{n}⊗O_{m}} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & K ⊗ O_{n} \ar[r] \ar[d] & T_{ℂ^{m}⊗O_{n}} ≅ O_{ℂ^{n}⊗E_{m}} \ar[r] \ar[d] & O_{ℂ^{m}⊗O_{n}} ≅ O_{ℂ^{n}⊗O_{m}} \ar[r] \ar[d] & 0. \\  & 0 & 0 & 0}

212 Valentin Deaconu

The exact sequences (depending on χ) are

\xymatrix{ℤ \ar[r] & K_{0}(E_{m} ⊗ K + K ⊗ E_{n}) \ar[r] & ℤ_{m−1} ⊕ ℤ_{n−1} \ar[d] \\ 0 \ar[u] & K_{1}(E_{m} ⊗ K + K ⊗ E_{n}) \ar[l] & 0 \ar[l]}

and

\xymatrix{K_{0}(E_{m} ⊗ K + K ⊗ E_{n}) \ar[r] & ℤ \ar[r] & K_{0}(O_{ℂ^{m}⊗O_{n}}) \ar[d] \\ K_{1}(O_{ℂ^{m}⊗O_{n}}) \ar[u] & 0 \ar[l] & 0. \ar[l]}

In particular, if χ is just the flip, we have

T_{ℂ^{m}⊗E_{n}} ≅ T_{ℂ^{n}⊗E_{m}} ≅ E_{n} ⊗ E_{m}, O_{ℂ^{m}⊗E_{n}} ≅ T_{ℂ^{n}⊗O_{m}} ≅ O_{m} ⊗ E_{n},

T_{ℂ^{m}⊗O_{n}} ≅ O_{ℂ^{n}⊗E_{m}} ≅ E_{m} ⊗ O_{n}, O_{ℂ^{m}⊗O_{n}} ≅ O_{ℂ^{n}⊗O_{m}} ≅ O_{m} ⊗ O_{n}, and we recover the K-theory of O_{m} ⊗ O_{n}.

Example 5.7. Let A = C(𝕋) and σ_{i}(x) = x^{p_{i}}, i = 1, 2 with p_{1}, p_{2} relatively prime. Then the associated conditional expectations commute, and using the K-theory computations in [D1], we get

\xymatrix{ℤ \ar[r] & K_{0}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}) \ar[r] & ℤ^{2} ⊕ ℤ_{p_{1}−1} ⊕ ℤ_{p_{2}−1} \ar[d] \\ ℤ ⊕ ℤ \ar[u] & K_{1}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}) \ar[l] & ℤ \ar[l]}

\xymatrix{K_{0}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}) \ar[r] & ℤ \ar[r] & K_{0}(O_{E_{2}⊗_{A}O_{E_{1}}} ) \ar[d] \\ K_{1}(O_{E_{2}⊗_{A}O_{E_{1}}} ) \ar[u] & ℤ \ar[l] & K_{1}(T_{α_{1}} ⊗ K + K ⊗ T_{α_{2}}). \ar[l]}

More generally, we may consider coverings of the n-torus 𝕋^{n}.

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Department of Mathematics and Statistics, University of Nevada, Reno NV 89557-0084, USA vdeaconu@unr.edu

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