Toeplitz Algebras and Rotational Automorphisms Associated to Polydiscs

Abstract

Introduction. Given a bounded symmetric domain $D in Cm (with $D being normalized to be circular and to contain the origin), Upmeier [22] showed that there is a natural way to associate a C*-algebra to it. This C*algebra, called the Toeplitz C*-algebra of $D and denoted 35,, is a solvable C*-algebra with a composition series {Jk)}O<k<n+l of length n, where n is the rank of the symmetric domain [22; Theorem 3.12]. We observe that there is a natural group action on 3 , by the compact group G of all biholomorphic automorphisms of $D which lie in the connected component of the identity and which fix the origin (Proposition 1.1). Every element 0 in G preserves the solvable series {Jk}O<k?<n+l of 3%.D We thereby obtain a composition series {Jk X0 Z } for the crossed product C*-algebra 3%, Xo Z. In this paper, we show (by introducing the notion of stratification of composition series-see section 2) that the series {Jk X0 Z} is exactly the maximal radical series of 3%) X0 Z (section 3), and thus obtain an isomorphism invariant for the latter C*-algebra. When $ is the polydisc D , we determine the K-theory of the series {Jk Xo Z} and their subquotients (section 4). Techniques from K-theory (using strong Morita equivalence) yield a complete classification of the C*algebras 3%) Xo Z, where D is the n-dimensional polydisc, up to *-isomorphism, for all 0 in G = Tn: