Exotic Crossed Products

TL;DR

This paper surveys exotic crossed products, generalizing universal and reduced crossed products, and explores their role in K-theory and reformulations of the Baum-Connes conjecture, providing new results and insights.

Contribution

It introduces a large class of exotic crossed products called correspondence functors with properties similar to maximal and reduced products, and applies them to K-theory and Baum-Connes conjecture reformulations.

Findings

Computed K-theory for many exotic group algebras.

Established properties of correspondence functors in KK-theory.

Connected exotic crossed products with the reformulated Baum-Connes conjecture.

Abstract

An exotic crossed product is a way of associating a C*-algebra to each C*-dynamical system that generalizes the well-known universal and reduced crossed products. Exotic crossed products provide natural generalizations of, and tools to study, exotic group C*-algebras as recently considered by Brown-Guentner and others. They also form an essential part of a recent program to reformulate the Baum-Connes conjecture with coefficients so as to mollify the counterexamples caused by failures of exactness. In this paper, we survey some constructions of exotic group algebras and exotic crossed products. Summarising our earlier work, we single out a large class of crossed products --- the correspondence functors --- that have many properties known for the maximal and reduced crossed products: for example, they extend to categories of equivariant correspondences, and have a compatible descent morphism in KK-theory. Combined with known results on K-amenability and the Baum-Connes conjecture, this allows us to compute the K-theory of many exotic group algebras. It also gives new information about the reformulation of the Baum-Connes Conjecture mentioned above. Finally, we present some new results relating exotic crossed products for a group and its closed subgroups, and discuss connections with the reformulated Baum-Connes conjecture.