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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Some $C^{\ast }$-alegebras with a single generator

Authors: Catherine L. Olsen and William R. Zame
Journal: Trans. Amer. Math. Soc. 215 (1976), 205-217
MSC: Primary 46L05
MathSciNet review: 0388114
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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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Keywords: <IMG WIDTH="31" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${C^\ast }$">-algebras, generators, matrix algebras, tensor products, compact operators, UHF algebra
Article copyright: © Copyright 1976 American Mathematical Society