Bunce–Deddens algebra

In mathematics, a Bunce–Deddens algebra, named after John W. Bunce and James A. Deddens, is a certain type of AT algebra, a direct limit of matrix algebras over the continuous functions on the circle, in which the connecting maps 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 Template:Math group of the algebra. Also, Bunce–Deddens algebras can be expressed as the Template:Math-crossed product of the Cantor set with a certain natural minimal action known as an 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 Template:Math 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 Template:Math denote continuous functions on the circle and Template:Math be the Template:Math-algebra of Template:Math matrices with entries in Template:Math. For a supernatural number Template:Math, the corresponding Bunce–Deddens algebra Template:Math is the direct limit:

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

One needs to define the embeddings

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

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

The connecting maps

The Template:Math'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 Template:Math-algebra Template:Math is in fact singly generated. A particular generator of Template:Math is the weighted shift Template:Math of period Template:Math with periodic weights Template:Math. In the appropriate basis of Template:Math, Template:Math is represented by the Template:Math 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 Template:Math is the unilateral shift. A direct calculation using functional calculus shows that the Template:Math-algebra generated by Template:Math is Template:Math, where Template:Math denotes the Toeplitz algebra, the Template:Math-algebra generated by the unilateral shift. Since it is clear that Template:Math contains Template:Math, this shows Template:Math.

From the Toeplitz short exact sequence,

$0\to 𝒦\to C^{*}(T_z)\to C(𝕋)\to 0,$

one has,

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

where Template:Math is the entrywise embedding map and Template:Math the entrywise quotient map on the Toeplitz algebra. So the Template:Math-algebra Template:Math 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 Template:Math is the identity function.

For integers Template:Math and Template:Math, where Template:Mathdivides Template:Math, the natural embedding of Template:Math into Template:Math descends into an (unital) embedding from Template:Math into Template:Math. This is the connecting map Template:Math from the definition of the Bunce–Deddens algebra that we need to analyze.

For simplicity, assume Template:Math and Template:Math. The image of the above operator Template:Math under the natural embedding is the following Template:Math operator matrix in Template:Math:

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

Therefore, the action of the Template:Math 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

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

So

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

In this particular instance, Template:Math is called a twice-around embedding. The reason for the terminology is as follows: as Template:Math varies on the circle, the eigenvalues of Template:Math 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 Template:Math in fact connect the beginning and end points of each arc. So in this sense the circle gets wrap around twice by Template:Math. In general, when Template:Math, one has a similar Template:Math-times around embedding.

K-theory and classification

Bunce–Deddens algebras are classified by their Template:Math groups. Because all finite-dimensional vector bundles over the circle are homotopically trivial, the Template:Math of Template:Math, as an ordered abelian group, is the integers Template:Math with canonical ordered unit Template:Math. According to the above calculation of the connecting maps, given a supernatural number Template:Math, the Template:Math of the corresponding Bunce–Deddens algebra is precisely the corresponding dense subgroup of the rationals Template:Math.

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, Template:Math is a complete invariant of these algebras.

It also follows from the previous section that the Template:Math group of any Bunce–Deddens algebra is Template:Math.

As a crossed product

Template:Math-crossed product

A Template:Math-dynamical system is a triple Template:Math, where Template:Math is a Template:Math-algebra, Template:Math a group, and Template:Math an action of Template:Math on Template:Math via Template:Math-automorphisms. A covariant representation of Template:Math is a representation Template:Math of Template:Math, and a unitary representation Template:Math $\mapsto$ Template:Math of Template:Math, on the same Hilbert space, such that

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

for all Template:Math, Template:Math.

Assume now Template:Math is unital and Template:Math is discrete. The Template:Mathcrossed product given by Template:Math, denoted by

$A{⋊}_{\sigma }G,$

is defined to be the Template:Math-algebra with the following universal property: for any covariant representation Template:Math, the Template:Math-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 Template:Math. Consider, for example, the Bunce–Deddens algebra of type Template:Math. Write the Cantor set Template:Math 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 Template:Math denotes addition with carryover. This is called the odometer action. The homeomorphism Template:Math induces an action on Template:Math by pre-composition with Template:Math. The Bunce–Deddens algebra of type Template:Math is isomorphic to the resulting crossed product.

References

• Template:Citation