User:Mct mht/BD2

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Definition and basic properties[edit]

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:

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

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,

one has,

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

where

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.