transactions of the american mathematical society Volume 215, 1976 SOME C*-ALGEBRAS WITHA SINGLE GENERATOR*1 ) BY CATHERINE L. OLSEN AND WILLIAM R. ZAME ABSTRACT. This paper grew out of the following question: If X is a com- pact subset of Cn, is C(X) ® M„ (the C*-algebra of n x n matrices with entries from C(X)) singly generated? It is shown that the answer is affirmative; in fact, A ® M„ is singly generated whenever A is a C*-algebra with identity, generated by a set of n(n + l)/2 elements of which n(n - l)/2 are selfadjoint. If A is a separable C*-algebra with identity, then A ® K and A ® U are shown to be sing- ly generated, where K is the algebra of compact operators in a separable, infinite- dimensional Hubert space, and U is any UHF algebra. In all these cases, the gen- erator is explicitly constructed. 1. Introduction. This paper grew out of a question raised by Claude Scho- chet and communicated to us by J. A. Deddens: If X is a compact subset of C, is C(X) ® M„ (the C*-algebra ofnxn matrices with entries from C(X)) singly generated? We show that the answer is affirmative; in fact, A ® Mn is singly gen- erated whenever A is a C*-algebra with identity, generated by a set of n(n + l)/2 elements of which n(n - l)/2 are selfadjoint. Working towards a converse, we show that A ® M2 need not be singly generated if A is generated by a set con- sisting of four elements. If A lacks an identity, our results are weaker, and we obtain them only in the commutative case. Informally, one might say that there are enough degrees of freedom in M„ to allow a small generating set for A to be combined into a single generator for A ® M„. For countably generated A we prove two natural infinite analogs: If A is any separable C*-algebra with identity, then A ® K and A ® U are singly gen- erated, where K is the algebra of compact operators on a separable, infinite-di- mensional Hubert space and where U is any UHF algebra. In all these cases, we explicitly construct a generator. Single generators for C*-algebrasand for von Neumann algebras have been studiedby R. G. Douglas, C. Pearcy, T. Saitô, N. Suzuki, D. Topping, W.Wogen Received by the editors August 22, 1974. AMS (MOS) subject classifications (1970). Primary 46L05; Secondary 46M05, 46L10. Key words and phrases. C* -algebras, generators, matrix algebras, tensor products, com- pact operators, UHF algebra. (!) Research supported in part by National Science Foundation Grants P037621-001 and PO37961-001. „._ Copyright © 1976. American Mathematical Society License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use