K-theoretic invariants for C∗-algebras associated to transformations and induced flows

Abstract

We discuss a technique of studying the K-theory of a unital C ∗-algebra associated to a homomorphism on a compact metric space ( Y Z ) by examining the non-unital C ∗-algebra associated to the induced topological flow (Ind Z R ( Y), R ). The Thom isomorphism of Connes and the Schwartzman asymptotic cycle are used to calculate the range of the trace corresponding to an invariant measure on the K 0 group of C ∗( X, R ) for a continuous flow on a compact metric space ( X, R ). Under certain conditions projections in C ∗( X, R ) with trace r corresponding to cross sections to the flow can be constructed for every positive real number r in this range, again by combining techniques of Connes and Schwartzman. Applications to the calculation of the tracial range of K 0( C ∗( X, R )) are discussed. In particular, this invariant is calculated for minimal affine actions of Z on n-tori which have quasi-discrete spectrum, and for minimal actions of Z on compact abelian groups with topologically discrete spectrum. In both cases this tracial invariant is shown to be the preimage of the eigenvalues for ( Y, Z ) under the natural projection ø: R → R / Z = S 1.