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 &times; r matrices with entries in C(S1). For a supernatural number {nk}, the corresponding Bunce-Deddens algebra B({nk}) is the direct limit:



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 &beta;k. For integers n and m, we specify an embedding &beta; : Mn(C(S1)) &rarr; 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 &times; n operator matrix acting on H


 * $$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, &hellip;, ½, 1, ½, &hellip;, ½, 1, &hellip;.

From the short exact sequence,


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

one has,


 * $$0 \rightarrow M_n(\mathcal{K}) \; \stackrel{i}{\hookrightarrow} \; M_n(C^*(T_z)) \; \stackrel{j}{\rightarrow} \; M_n(C(S^1)) \rightarrow 0,$$

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 &times; 2n operator matrix


 * $$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

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}, $$



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 &beta; : Mn(C(S1)) &rarr; 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.