Three versions of categorical crossed-product duality

TL;DR

This paper compares three categorical frameworks for crossed-product duality in C*-algebras, analyzing their fixed-point functors and inversion properties, with new generalizations of Pedersen's theorem for actions and coactions.

Contribution

It introduces a formal comparison of three categorical approaches to crossed-product duality and extends Pedersen's theorem to outer-conjugacy categories.

Findings

Good inversion for outer-conjugacy categories with Pedersen's theorem

Partial fixed-point functor for coactions, open quasi-inverse question

Formal framework for fixed-point functors as inversions of crossed products

Abstract

In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be formulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomorphisms, imprimitivity bimodules, and outer conjugacies, respectively. In each case, a variation of the fixed-point functor that arises from classical Landstad duality is used to obtain a quasi-inverse for a crossed-product functor. To compare the various cases, we describe in a formal way our view of the fixed-point functor as an "inversion" of the process of forming a crossed product. In some cases, we obtain what we call "good" inversions, while in others we do not. For the outer-conjugacy categories, we generalize a theorem of Pedersen to obtain a fixed-point functor that is quasi-inverse to the reduced-crossed-product functor for actions, and we show that this gives a good inversion. For coactions, we prove a partial version of Pedersen's theorem that allows us to define a fixed-point functor, but the question of whether it is a quasi-inverse for the crossed-product functor remains open.