# Three versions of categorical crossed-product duality

Research output: Contribution to journalArticle

4 Citations (Scopus)

### Abstract

In this partly expository paper we compare three different categories of C*-algebras in which crossed-product duality can be for mulated, both for actions and for coactions of locally compact groups. In these categories, the isomorphisms correspond to C*-algebra isomor phisms, 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 Peder sen 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 the orem 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.

Original language English (US) 293-339 47 New York Journal of Mathematics 22 Published - 2016

### Fingerprint

Crossed Product
Categorical
Functor
Duality
Fixed point
Coaction
Inversion
C*-algebra
Bimodule
Conjugacy
Locally Compact Group
Isomorphism
Partial
Generalise
Theorem

### Keywords

• Action
• Category equivalence
• Coaction
• Crossed-product duality
• Exterior equivalence
• Outer conjugacy

### ASJC Scopus subject areas

• Mathematics(all)

### Cite this

In: New York Journal of Mathematics, Vol. 22, 2016, p. 293-339.

Research output: Contribution to journalArticle

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