K-theoretic invariants for \(C^*\)-algebras associated to transformations and induced flows
A technique of studying the K-theory of a unital \(C^*\)-algebra associated to a homeomorphism on a compact metric space (Y,\({\mathbb{Z}})\) by examining the non-unital \(C^*\)-algebra associated to the induced topological flow \((Ind^{{\mathbb{R}}}_{{\mathbb{Z}}}(Y),{\mathbb{R}})\) is discussed. 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 \(C^*(X,{\mathbb{R}})\) for a continuous flow on a compact metric space (X,\({\mathbb{R}})\). Under certain conditions projections in \(C^*(X,{\mathbb{R}})\) 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 \(K_ 0(C^*(Y,{\mathbb{Z}}))\) are given. In particular, this invariant is calculated for minimal affine actions of \({\mathbb{Z}}\) on n-tori which have quasi-discrete spectrum, and for minimal actions of \({\mathbb{Z}}\) on compact abelian groups with topologically discrete spectrum (Riedel's Theorem). In both cases this tracial invariant is shown to be the preimage of the eigenvalues for (Y,\({\mathbb{Z}})\) under the natural projection \(\phi\) : \({\mathbb{R}}\to {\mathbb{R}}/{\mathbb{Z}}=S^ 1.\) In the Appendix to the article, an alternative derivation of Pimsner's formula for the range of the trace on the \(K_ 0\)-group of a reduced crossed product is obtained for the \(C^*\)-algebras \(C^*(Y,{\mathbb{Z}})\) mentioned above, using the methods set forth in earlier sections and the Wang exact sequence for the cohomology of mapping tori. Applications of this formula to general affine actions on n-tori are mentioned and the relationship to flow equivalence is discussed.
Pimsner's formula for the range of the trace on the \(K_ 0\)-group of
Wang exact sequence for the cohomology of mapping tori
Pimsner's formula for the range of the trace on the \(K_ 0\)-group of a reduced crossed product
scientific article; zbMATH DE number 3951329
scientific article