Rotation numbers for automorphisms of \(C^ *\)-algebras
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.
rotation number for an automorphism of a \(C^ *\) algebra
range of a trace on the \(K_ 0\)-group of a crossed product algebra by \({\mathbb Z}\)
almost periodic automorphisms of \(C^ *\) algebras
affine homeomorphisms of connected groups
scientific article; zbMATH DE number 3927668