Additive C*-categories and K-theory

Abstract

We review the notions of a multiplier category and the $W^{\ast }$-envelope of a $C^{\ast }$-category. We then consider the notion of an orthogonal sum of a (possibly infinite) family of objects in a $C^{\ast }$-category. Furthermore, we construct reduced crossed products of $C^{\ast }$-categories with groups. We axiomatize the basic properties of the $K$-theory for $C^{\ast }$-categories in the notion of a homological functor. We then study various rigidity properties of homological functors in general, and special additional features of the $K$-theory of $C^{\ast }$-categories. As an application we construct and study interesting functors on the orbit category of a group from $C^{\ast }$-categorical data.

173 pages, minor revision