### 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) |
---|---|

Pages (from-to) | 293-339 |

Number of pages | 47 |

Journal | New York Journal of Mathematics |

Volume | 22 |

State | Published - 2016 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*New York Journal of Mathematics*,

*22*, 293-339.

**Three versions of categorical crossed-product duality.** / Kaliszewski, Steven; Omland, Tron; Quigg, John.

Research output: Contribution to journal › Article

*New York Journal of Mathematics*, vol. 22, pp. 293-339.

}

TY - JOUR

T1 - Three versions of categorical crossed-product duality

AU - Kaliszewski, Steven

AU - Omland, Tron

AU - Quigg, John

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - Action

KW - Category equivalence

KW - Coaction

KW - Crossed-product duality

KW - Exterior equivalence

KW - Outer conjugacy

UR - http://www.scopus.com/inward/record.url?scp=84961837248&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84961837248&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84961837248

VL - 22

SP - 293

EP - 339

JO - New York Journal of Mathematics

JF - New York Journal of Mathematics

SN - 1076-9803

ER -