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Transactions of the American Mathematical Society

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

 

 

Some $ C\sp{\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
DOI: https://doi.org/10.1090/S0002-9947-1976-0388114-7
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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DOI: https://doi.org/10.1090/S0002-9947-1976-0388114-7
Keywords: $ {C^\ast}$-algebras, generators, matrix algebras, tensor products, compact operators, UHF algebra
Article copyright: © Copyright 1976 American Mathematical Society