Physics @ by Springer-Verlag 1977 Simple C*-Algebras Generated by Isometries

Abstract. We consider the C*-algebra (9, generated by n ~ 2 isometries $1..... S, on an infinite-dimensional Hilbert space, with the property that StS ~ +... + S,S * = 1. It turns out that (9, has the structure of a crossed product of a finite simple C*-algebra ~ by a single endomorphism scaling the trace of~- by 1/n. Thus, (9, is a separable C*-algebra sharing many of the properties of a factor of t)23e 1II~. with 2 = 1/n. As a consequence we show that (9, is simple and that its isomorphism type does not depend on the choice of S t...., S,. A C*-algebra is simple if it contains no non-trivial closed two-sided ideals. We call a simple C*-algebra with unit infinite if it contains an element X such that X*X =! and XX * +1. While non-separable algebras of this type are well known (e.g. the Calkin algebra or type III factors on a separable Hilbert space) there is to my knowledge no explicit example of a separable simple infinite C*-algebra. The existence of such algebras was proved by Dixmier in [9, 2.1] by the following argument. Let $1, S z be two isometries (S*S i = 1, i = 1, 2) on an infinite-dimensional Hilbert space ~ such that $IS*+$2S*=1. Since the C*-algebra C*(St, $2