### Abstract

Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C_{0} (X). We consider a category in which the objects are C^{*}-dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C_{0} (X), γ) into the multiplier algebra M (A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra A^{α} which is Morita equivalent to A ×_{α, r} G. We show that the assignment (A, α) {mapping} A^{α} is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.

Original language | English (US) |
---|---|

Pages (from-to) | 2949-2968 |

Number of pages | 20 |

Journal | Journal of Functional Analysis |

Volume | 254 |

Issue number | 12 |

DOIs | |

State | Published - Jun 15 2008 |

### Fingerprint

### Keywords

- Coaction
- Comma category
- Crossed product
- Fixed-point algebra
- Landstad duality
- Proper actions

### ASJC Scopus subject areas

- Analysis

### Cite this

*Journal of Functional Analysis*,

*254*(12), 2949-2968. https://doi.org/10.1016/j.jfa.2008.03.010

**Proper actions, fixed-point algebras and naturality in nonabelian duality.** / Kaliszewski, Steven; Quigg, John; Raeburn, Iain.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 254, no. 12, pp. 2949-2968. https://doi.org/10.1016/j.jfa.2008.03.010

}

TY - JOUR

T1 - Proper actions, fixed-point algebras and naturality in nonabelian duality

AU - Kaliszewski, Steven

AU - Quigg, John

AU - Raeburn, Iain

PY - 2008/6/15

Y1 - 2008/6/15

N2 - Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C0 (X). We consider a category in which the objects are C*-dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C0 (X), γ) into the multiplier algebra M (A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A ×α, r G. We show that the assignment (A, α) {mapping} Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.

AB - Suppose a locally compact group G acts freely and properly on a locally compact Hausdorff space X, and let γ be the induced action on C0 (X). We consider a category in which the objects are C*-dynamical systems (A, G, α) for which there is an equivariant homomorphism of (C0 (X), γ) into the multiplier algebra M (A). Rieffel has shown that such systems are proper and saturated, and hence have a generalized fixed-point algebra Aα which is Morita equivalent to A ×α, r G. We show that the assignment (A, α) {mapping} Aα is functorial, and that Rieffel's Morita equivalence is natural in a suitable sense. We then use our results to prove a categorical version of Landstad duality which characterizes crossed products by coactions, and to prove that Mansfield imprimitivity for crossed products by homogeneous spaces is natural.

KW - Coaction

KW - Comma category

KW - Crossed product

KW - Fixed-point algebra

KW - Landstad duality

KW - Proper actions

UR - http://www.scopus.com/inward/record.url?scp=43049098926&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=43049098926&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2008.03.010

DO - 10.1016/j.jfa.2008.03.010

M3 - Article

AN - SCOPUS:43049098926

VL - 254

SP - 2949

EP - 2968

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 12

ER -