Continuous fields ofC *-algebras coming from group cocycles and actions

Abstract

Recently I have been attempting to formulate a suitable C*-algebraic framework for the subject of deformation quantization [-3, 19]. Continuous fields of C*-algebras provide one of the key elements for this framework. The main examples of deformation quantizations which I have constructed up to now in this C*-algebra framework come from letting either cocycles on groups, or actions of groups, vary. It has thus become necessary to show that one obtains in this way fields of C*-aigebras that are indeed continuous. Since this material is of a general nature, and can be useful in other situations [4-6, 11-13] it has seemed appropriate to give a separate exposition of it, in the present article. Section 1 of this article contains a review of the published results on continuous fields which we will need, as well as a discussion of the fact that the approach which we will take involves treating upper and lower semi-continuity separately. In Sect. 2 we discuss the continuity of fields of C*-algebras which arise from varying cocycles on groups, while in Sect. 3 we do the same for actions which vary.