Rotation numbers for automorphisms of \(C^ *\)-algebras
type: Item
instance of: scholarly article
title: Rotation numbers for automorphisms of \(C^ *\)-algebras
zbMATH Open document ID: 0579.46039
author: Ruy Exel
published in: Pacific Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: We characterize the class of traced \(C^*\) algebras to which a suitable notion of determinant can be extended, and we study the behavior of such determinants under the actions of groups of automorphisms. We also define, based on our theory of determinants, the rotation number for an automorphism of a \(C^*\) algebra, generalizing the notion of rotation number for homeomorphisms of the circle. Applying our theory of rotation numbers we derive a simple and effective formula for computing the range of a trace on the \(K_ 0\)-group of a crossed product algebra by \({\mathbb{Z}}\). We also give conditions for deciding when the range of the trace on \(K_ 0\) of such algebras has only integer values. Applications include a study of almost periodic automorphisms of \(C^*\) algebras, as well as automorphisms and affine homeomorphisms of connected groups. Based on our methods, we obtain a simple proof of a result of Rieffel, Pimsner and Voiculescu on the range of the trace on \(K_ 0\) of irrational rotation \(C^*\) algebras. We also prove that the \(C^*\) algebras of the discrete Heisenberg group and related groups contain no proper projections.
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L55
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 46L40
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 3927668
zbMATH Keywords: traced \(C^ *\) algebras
zbMATH Keywords: determinant
zbMATH Keywords: actions of groups of automorphisms
zbMATH Keywords: rotation number for an automorphism of a \(C^ *\) algebra
zbMATH Keywords: range of a trace on the \(K_ 0\)-group of a crossed product algebra by \({\mathbb Z}\)
zbMATH Keywords: almost periodic automorphisms of \(C^ *\) algebras
zbMATH Keywords: affine homeomorphisms of connected groups
zbMATH Keywords: discrete Heisenberg group
MaRDI profile type: Publication
full work available at URL: pjm.1987.127.31
OpenAlex ID: W2031965066
DOI: 10.2140/PJM.1987.127.31
description: scientific article; zbMATH DE number 3927668