DUALITY OF COMPACT GROUPS AND HILBERT C*-SYSTEMS FOR C*-ALGEBRAS WITH A NONTRIVIAL CENTER

Abstract

In this paper we present duality theory for compact groups in the case when the C*-algebra [Formula: see text], the fixed point algebra of the corresponding Hilbert C*-system [Formula: see text], has a nontrivial center [Formula: see text] and the relative commutant satisfies the minimality condition [Formula: see text] as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*-categories [Formula: see text], where [Formula: see text] is a suitable DR-category and [Formula: see text] a full subcategory of the category of endomorphisms of [Formula: see text]. Both categories have the same objects and the arrows of [Formula: see text] can be generated from the arrows of [Formula: see text] and the center [Formula: see text]. A crucial new element that appears in the present analysis is an abelian group [Formula: see text], which we call the chain group of [Formula: see text], and that can be constructed from certain equivalence relation defined on [Formula: see text], the dual object of [Formula: see text]. The chain group, which is isomorphic to the character group of the center of [Formula: see text], determines the action of irreducible endomorphisms of [Formula: see text] when restricted to [Formula: see text]. Moreover, [Formula: see text] encodes the possibility of defining a symmetry ∊ also for the larger category [Formula: see text] of the previous inclusion.