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

We introduce a method to study C*-algebras possessing an action of the circle group, from the point of view of their internal structure and their 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 o...

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Bibliographic Details
Published inJournal of functional analysis Vol. 122; no. 2; pp. 361 - 401
Main Author Exel, R.
Format Journal Article
LanguageEnglish
Published Orlando, FL Elsevier Inc 1994
San Diego, CA Academic Press
Brugge
Subjects
Exact sciences and technology
Functional analysis
Mathematical analysis
Mathematics
Sciences and techniques of general use
K theory
Representation
Dynamical system
C star algebra
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Summary:We introduce a method to study C*-algebras possessing an action of the circle group, from the point of view of their internal structure and their 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.
ISSN:0022-1236
1096-0783
DOI:10.1006/jfan.1994.1073