Some 𝐶*-alegebras with a single generator

Olsen, Catherine L.; Zame, William R.

doi:10.1090/S0002-9947-1976-0388114-7

Some $C^{\ast }$-alegebras with a single generator
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by Catherine L. Olsen and William R. Zame PDF

Trans. Amer. Math. Soc. 215 (1976), 205-217 Request permission

Abstract:

This paper grew out of the following question: If X is a compact subset of ${C^n}$, is $C(X) \otimes {{\mathbf {M}}_n}$ (the ${C^\ast }$-algebra of $n \times n$ matrices with entries from $C(X)$) singly generated? It is shown that the answer is affirmative; in fact, $A \otimes {{\mathbf {M}}_n}$ is singly generated whenever A is a ${C^\ast }$-algebra with identity, generated by a set of $n(n + 1)/2$ elements of which $n(n - 1)/2$ are selfadjoint. If A is a separable ${C^\ast }$-algebra with identity, then $A \otimes K$ and $A \otimes U$ are shown to be singly generated, where K is the algebra of compact operators in a separable, infinite-dimensional Hilbert space, and U is any UHF algebra. In all these cases, the generator is explicitly constructed.

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