Circle Actions on C*-Algebras, Partial Automorphisms and a Generalized Pimsner-Voiculescu Exact Sequence

TL;DR

This paper develops a framework for analyzing C*-algebras with circle group actions, generalizing crossed products via partial automorphisms and extending the Pimsner-Voiculescu exact sequence to this broader context.

Contribution

It introduces a structure theorem for circle actions on C*-algebras and extends the Pimsner-Voiculescu sequence to partial automorphisms, broadening the understanding of their K-theory.

Findings

Generalized crossed product construction for partial automorphisms

Extension of Pimsner-Voiculescu exact sequence to these constructions

Representation theory parallels for the new algebraic structures

Abstract

We introduce a method to study C*-algebras possessing an action of the circle group, from the point of view of its internal structure and its K-theory. Under relatively mild conditions our structure Theorem shows that any C*-algebra, where an action of the circle is given, arises as the result of a construction that generalizes crossed products by the group of integers. Such a generalized crossed product construction is carried out for any partial automorphism of a C*-algebra, where by a partial automorphism we mean an isomorphism between two ideals of the given algebra. Our second main result is an extension to crossed products by partial automorphisms, of the celebrated Pimsner-Voiculescu exact sequence for K-groups. The representation theory of the algebra arising from our construction is shown to parallel the representation theory for C*-dynamical systems. In particular, we generalize several of the main results relating to regular and covariant representations of crossed products.