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In C*-algebras, a Bunce–Deddens algebra is a certain type of direct limit of matrix algebras over the continuous functions on the circle. They are therefore examples of simple unital AT algebras. In the inductive system defining these algebras, the connecting maps between each stage are given by embeddings between families of shift operators with periodic weights.

Each inductive system defining a Bunce–Deddens algebra is associated with a supernatural number, which is a complete invariant for these algebras. In the language of K-theory, the supernatural number correspond to the K₀ group of the algebra. Also, Bunce–Deddens algebras can be expressed as the C*-crossed product of the Cantor set with a certain natural minimal action, so-called odometer action. They also admit a unique tracial state. Together with the fact that they are AT, this implies they have real rank zero.

In a broader context of the classification program for simple separable nuclear C*-algebras, AT-algebras of real rank zero were shown to be completely classified by their K-theory, the Choquet simplex of tracial states, and the natural pairing between K₀ and traces. The classification of Bunce–Deddens algebras is thus a precursor to the general result.

It is also known that, in general, crossed products arising from minimal homeomorphism on the Cantor set are simple AT-algebras of real rank zero.

Definition and basic properties

Definition

Let C( T ) denote continuous functions on the circle and M_r(C(T)) be the C*-algebra of r × r matrices with entries in C(T). For a supernatural number {n_k}, the corresponding Bunce–Deddens algebra B({n_k}) is the direct limit:

$B(\{n_k\})=\underrightarrow{\lim }\cdots \to M_{n_k}(C(\mathbb{𝕋}))\stackrel{{\beta }_k}{\to }{M}_{n_{k+1}}(C(\mathbb{𝕋}))\to \cdots .$

One needs to define the embeddings

${\beta }_k:M_{n_k}(C(\mathbb{𝕋}))\to M_{n_{k+1}}(C(\mathbb{𝕋})).$

These imbedding maps arise from the natural embeddings between C*-algebras generated by shifts with periodic weights. For integers n and m, we define an embedding β : M_n(C(T)) → M_nm(C(T)) as follows. On a separable Hilbert space H, consider the C*-algebra W(n) generated by weighted shifts of fixed period n with respect to a fixed basis. W(n) embedds into W(nm) in the obvious way; any n-periodic weighted shift is also a nm-periodic weighted shift. W(n) is isomorphic to M_n(C*(T_z)), where C*(T_z) denotes the Toeplitz algebra. Therefore W contains the compact operators as an ideal, and modulo this ideal it is M_n(C(T)). Because the map from W(n) into W(nm) preserves the compact operators, it descends into an embedding β : M_n(C(T)) → M_nm(C(T)). It is this embedding that is used in the definition of Bunce–Deddens algebras.

The connecting maps

The β_k's can be computed more explicitly and we now sketch this computation. This will be useful in obtaining an alternative characterization description of the Bunce–Deddens algebras, and also the classification of these algebras.

The C*-algebra W(n) is in fact singly generated. A particular generator of W(n) is the weighted shift T of period with periodic weights ½, …, ½, 1, ½, …, ½, 1, …. In the appropriate basis of H, T is represented by the n × n operator matrix

$T=\left[\begin{array}{cccc}0& & \cdots & T_z\\ \frac{1}{2}I& \ddots & \ddots & \\ & \ddots & \ddots & ⋮\\ & & \frac{1}{2}I& 0\end{array}\right],$

where T_z is the unilateral shift. A direct calculation using functional calculus shows that the C*-algebra generated by T is M_n(C*(T_z)), where C*(T_z) denotes the Toeplitz algebra, the C*-algebra generated by the unilateral shift. Since it is clear that M_n(C*(T_z)) contains W(n), this shows W(n) = M_n(C*(T_z)).

From the Toeplitz short exact sequence,

$0\to \mathcal{𝒦}\to C^{*}(T_z)\to C(\mathbb{𝕋})\to 0,$

one has,

$0\to M_n(\mathcal{𝒦})\stackrel{i}{\hookrightarrow }{M}_n(C^{*}(T_z))\stackrel{j}{\to }{M}_n(C(\mathbb{𝕋}))\to 0,$

where i is the entrywise embedding map and j the entrywise quotient map on the Toeplitz algebra. So the C*-algebra M _nk (C (T)) is singly generated by

$\tilde{T}=\left[\begin{array}{cccc}0& & \cdots & z\\ \frac{1}{2}& \ddots & \ddots & \\ & \ddots & \ddots & ⋮\\ & & \frac{1}{2}& 0\end{array}\right],$

where the scalar entries denote constant functions on the circle and z is the identity function.

For integers n_k and n_{k + 1}, where n_k divides n_{k + 1}, the natural embedding of W(n_k) into W(n_{k + 1}) descends into an (unital) embedding from M_nk(C(T)) into M _{nk + 1}(C(T)). This is the connecting map β_k from the definition of the Bunce–Deddens algebra that we need to analyze.

For simplicity, assume n_k = n and n_{k + 1} = 2n_k. The image of the above operator T ∈ W(n) under the natural embedding is the following 2n × 2n operator matrix in W(2n):

$T\mapsto \left[\begin{array}{cccccccc}0& & & & & & & T_z\\ \frac{1}{2}I& \ddots & & & & & & 0\\ & \ddots & \ddots & & & & & ⋮\\ & & \frac{1}{2}I& 0& & & & \\ & & & I& 0& & & \\ & & & & \frac{1}{2}I& \ddots & & \\ & & & & & \ddots & \ddots & ⋮\\ & & & & & & \frac{1}{2}I& 0\end{array}\right].$

Therefore the action of the β_k on the generator is

${\beta }_k(\tilde{T})=\left[\begin{array}{cccccccc}0& & & & & & & z\\ \frac{1}{2}& \ddots & & & & & & 0\\ & \ddots & \ddots & & & & & ⋮\\ & & \frac{1}{2}& 0& & & & \\ & & & 1& 0& & & \\ & & & & \frac{1}{2}& \ddots & & \\ & & & & & \ddots & \ddots & ⋮\\ & & & & & & \frac{1}{2}& 0\end{array}\right].$

A computation with matrix units yields that

${\beta }_k(E_{ij})=E_{ij}\otimes I_2$

and

${\beta }_k(z{E}_{11})=E_{11}\otimes {\mathrm{Z}}_2,$

where

${\mathrm{Z}}_2=\left[\begin{array}{cc}0& z\\ 1& 0\end{array}\right]\in M_2(C(\mathbb{𝕋})).$

So

${\beta }_k(f_{ij}(z))=f_{ij}({\mathrm{Z}}_2).$

In this particular instance, β_k is called a twice-around embedding. The reason for the terminology is as follows: as z varies on the circle, the eigenvalues of Z₂ traces out the two disjoint arcs connecting 1 and -1. An explicit computation of eigenvectors shows that the circle of unitaries implementing the diagonalization of Z₂ in fact connect the beginning and end points of each arc. So in this sense the circle gets wrap around twice by Z₂. In general, when n_{k + 1} = m·n_k, one has a similar m-times around embedding.

K-theory and classification

Bunce–Deddens algebras are classified by their K₀ groups. Because all finite dimensional vector bundles over the circle are homotopically trivial, the K₀ of M_r(C(T)), as an ordered abelian group, is the integers Z with canonical ordered unit r. According to the above calculation of the connecting maps, given a supernatural number {n_k}, the K₀ of the corresponding Bunce–Deddens algebra is precisely the corresponding dense subgroup of the rationals Q.

As it follows from the definition that two Bunce–Deddens algebras with the same supernatural number, in the sense that the two supernatural numbers formally divide each other, are isomorphic, K₀ is a complete invariant of these algebras.

It also follows from the previous section that the K₁ group of any Bunce–Deddens algebra is Z.

As a crossed product

C*-crossed product

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A C*-dynamical system is a triple (A, G, σ), where A is a C*-algebra, G a group, and σ an action of G on A via C*-automorphisms. A covariant representation of (A, G, σ) is a representation π of A, and a unitary representation t $\mapsto$ U_t of G, on the same Hilbert space, such that

$U_t\pi (a)U_t^{*}=\pi (\sigma (t)(a)),$

for all a, t.

Assume now A is unital and G is discrete. The (C*-)crossed product given by (A, G, σ), denoted by

$A{⋊}_{\sigma }G,$

is defined to be the C*-algebra with the following universal property: for any covariant representation (π, U), the C*-algebra generated by its image is a quotient of

$A{⋊}_{\sigma }G.$

Odometer action on Cantor set

The Bunce–Deddens algebras in fact are crossed products of the Cantor sets with a natural action by the integers Z. Consider, for example, the Bunce–Deddens algebra of type 2^∞. Write the Cantor set X as sequences of 0's and 1's,

$X=\prod \{0,1\},$

with the product topology. Define a homeomorphism

$\alpha :X\to X$

by

$\alpha (x)=x+(\cdots ,0,0,1)$

where + denotes addition with carryover. This is called the odometer action. The homeomorphism α induces an action on C(X) by pre-composition with α. The Bunce–Deddens algebra of type 2^∞ is isomorphic to the resulting crossed product.

References

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