Bunce–Deddens algebra

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:

${\displaystyle B(\{n_{k}\})=\varinjlim \cdots \rightarrow M_{n_{k}}(C(\mathbb {T} ))\;{\stackrel {\beta _{k}}{\rightarrow }}\;M_{n_{k+1}}(C(\mathbb {T} ))\rightarrow \cdots .}$

One needs to define the embeddings

${\displaystyle \beta _{k}:M_{n_{k}}(C(\mathbb {T} ))\;\rightarrow \;M_{n_{k+1}}(C(\mathbb {T} )).}$

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) embeds 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 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 n with periodic weights ½, …, ½, 1, ½, …, ½, 1, …. In the appropriate basis of H, T is represented by the n × n operator matrix

${\displaystyle T={\begin{bmatrix}0&\;&\cdots &T_{z}\\{\frac {1}{2}}I&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&{\frac {1}{2}}I&0\end{bmatrix}},}$

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,

${\displaystyle 0\rightarrow {\mathcal {K}}\;{\rightarrow }\;C^{*}(T_{z})\;{\rightarrow }\;C(\mathbb {T} )\rightarrow 0,}$

one has,

${\displaystyle 0\rightarrow M_{n}({\mathcal {K}})\;{\stackrel {i}{\hookrightarrow }}\;M_{n}(C^{*}(T_{z}))\;{\stackrel {j}{\rightarrow }}\;M_{n}(C(\mathbb {T} ))\rightarrow 0,}$

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

${\displaystyle {\tilde {T}}={\begin{bmatrix}0&\;&\cdots &z\\{\frac {1}{2}}&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&{\frac {1}{2}}&0\end{bmatrix}},}$

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_kdivides 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):

${\displaystyle T\mapsto {\begin{bmatrix}0&\;&&&&\;&&T_{z}\\{\frac {1}{2}}I&\ddots &&&&&&0\\\;&\ddots &\ddots &&&&&\vdots \\\;&\;&{\frac {1}{2}}I&0&&\;&&\\&\;&&I&0&&&\\&&&\;&{\frac {1}{2}}I&\ddots &&\;\\\;&&&&\;&\ddots &\ddots &\vdots \\\;&\;&&&\;&\;&{\frac {1}{2}}I&0\end{bmatrix}}.}$

Therefore, the action of the β_k on the generator is

${\displaystyle \beta _{k}({\tilde {T}})={\begin{bmatrix}0&\;&&&&\;&&z\\{\frac {1}{2}}&\ddots &&&&&&0\\\;&\ddots &\ddots &&&&&\vdots \\\;&\;&{\frac {1}{2}}&0&&\;&&\\&\;&&1&0&&&\\&&&\;&{\frac {1}{2}}&\ddots &&\;\\\;&&&&\;&\ddots &\ddots &\vdots \\\;&\;&&&\;&\;&{\frac {1}{2}}&0\end{bmatrix}}.}$

A computation with matrix units yields that

${\displaystyle \beta _{k}(E_{ij})=E_{ij}\otimes I_{2}}$

and

${\displaystyle \beta _{k}(zE_{11})=E_{11}\otimes \mathrm {Z} _{2},}$

where

${\displaystyle \mathrm {Z} _{2}={\begin{bmatrix}0&z\\1&0\end{bmatrix}}\in M_{2}(C(\mathbb {T} )).}$

So

${\displaystyle \beta _{k}(f_{ij}(z))=f_{ij}(\mathrm {Z} _{2}).\;}$

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.

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 ${\displaystyle \mapsto }$ U_t of G, on the same Hilbert space, such that

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

for all a, t.

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

${\displaystyle A\rtimes _{\sigma }G,}$

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

${\displaystyle A\rtimes _{\sigma }G.}$

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,

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

with the product topology. Define a homeomorphism

${\displaystyle \alpha :X\rightarrow X}$

by

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

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.