K-theoretic invariants for \(C^*\)-algebras associated to transformations and induced flows
type: Item
instance of: scholarly article
title: K-theoretic invariants for \(C^*\)-algebras associated to transformations and induced flows
zbMATH Open document ID: 0592.46060
DOI: 10.1016/0022-1236(86)90042-X
author: Judith A. Packer
published in: Journal of Functional Analysis
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 46L05
zbMATH DE Number: 3951329
zbMATH Keywords: K-theory of a unital \(C^*\)-algebra
zbMATH Keywords: homeomorphism on a compact metric space
zbMATH Keywords: induced topological flow
zbMATH Keywords: Thom isomorphism of Connes
zbMATH Keywords: Schwartzman asymptotic cycle
zbMATH Keywords: range of the trace
zbMATH Keywords: compact metric space
zbMATH Keywords: minimal affine actions
zbMATH Keywords: tracial invariant
zbMATH Keywords: Pimsner's formula for the range of the trace on the \(K_ 0\)-group of
zbMATH Keywords: a reduced crossed product
zbMATH Keywords: Wang exact sequence for the cohomology of mapping tori
zbMATH Keywords: Pimsner's formula for the range of the trace on the \(K_ 0\)-group of a reduced crossed product
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3951329