# User:Mct mht/BD2

## Definition and basic properties

Let C(S1) denote continuous functions on the circle and Mr(C(S1)) be the C*-algebra of r × r matrices with entries in C(S1). For a supernatural number {nk}, the corresponding Bunce-Deddens algebra B({nk}) is the direct limit:

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

One needs to define the embeddings βk. For integers n and m, we specify an embedding β : Mn(C(S1)) → Mnm(C(S1)) 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.

Lemma

1. W(n) is singly generated
2. W(n) is isomorphic to Mn(C*(Tz)), where C*(Tz) denotes the Toeplitz algebra.

This can be shown by identifying the following particular generator of W(n): the n × n operator matrix acting on H

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

where Tz is the unilateral shift. The operator T is the weighted shift of period with periodic weights ½, …, ½, 1, ½, …, ½, 1, … .

From the short exact sequence,

${\displaystyle 0\rightarrow {\mathcal {K}}\;{\rightarrow }\;C^{*}(T_{z})\;{\rightarrow }\;C(S^{1})\rightarrow 0,}$

one has,

${\displaystyle 0\rightarrow M_{n}({\mathcal {K}})\;{\stackrel {i}{\hookrightarrow }}\;M_{n}(C^{*}(T_{z}))\;{\stackrel {j}{\rightarrow }}\;M_{n}(C(S^{1}))\rightarrow 0,}$

where i is the entrywise embedding map and j the entrywise quotient map on the Toeplitz algebra.

For integers n and m, W(n) embedds naturally into W(nm); any n-periodic weighted shift is also a nm-periodic weighted shift. It is usedful to identify explicitly the image of the above generator of W(n) under this natural embedding. For simplicity, assume m = 2. The the above operator T gets mapped to a 2n × 2n operator matrix

${\displaystyle T\mapsto \left[{\begin{array}{c | c}{\begin{bmatrix}0&\;&\cdots &0\\{\frac {1}{2}}I&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&{\frac {1}{2}}I&0\end{bmatrix}}&{\begin{bmatrix}0&\;&\cdots &T_{z}\\0&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&0&0\end{bmatrix}}\\\hline {\begin{bmatrix}0&\;&\cdots &1\\0&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&0&0\end{bmatrix}}&T_{22}\end{array}}\right],}$

where

${\displaystyle T_{11}={\begin{bmatrix}0&\;&\cdots &0\\{\frac {1}{2}}I&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&{\frac {1}{2}}I&0\end{bmatrix}},T_{12}={\begin{bmatrix}0&\;&\cdots &T_{z}\\0&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&0&0\end{bmatrix}},}$
${\displaystyle T_{21}={\begin{bmatrix}0&\;&\cdots &1\\0&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&0&0\end{bmatrix}},T_{22}={\begin{bmatrix}0&\;&\cdots &0\\{\frac {1}{2}}I&\ddots &\ddots &\;\\\;&\ddots &\ddots &\vdots \\\;&\;&{\frac {1}{2}}I&0\end{bmatrix}}.}$

This embedding descends to a (unital) embedding β : Mn(C(S1)) → Mnm(C(S1)), and this is the embedding used in the definition of the Bunce-Deddens algebra.

So Mn(C(S1)) is also singly generated with a generator given by.