Naturality of Rieffel’s Morita Equivalence for Proper Actions

Huef, Astrid an; Kaliszewski, S.; Raeburn, Iain; Williams, Dana P.

doi:10.1007/s10468-009-9201-2

Cited by 9 publications

(52 citation statements)

References 36 publications

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“…The result generalizes the main result of [1] where the authors obtain a reduced version under the additional assumption that G acts freely on X . PROPOSITION 6.5.…”

Section: A Buss and S Echterhoffsupporting

confidence: 82%

Imprimitivity theorems for weakly proper actions of locally compact groups

Buss¹,

Echterhoff²

2014

Ergod. Th. Dynam. Sys.

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In a recent paper the authors introduced universal and exotic generalized fixed-point algebras for weakly proper group actions on C * -algebras. Here we extend the notion of weakly proper actions to actions on Hilbert modules. As a result we obtain several imprimitivity theorems establishing important Morita equivalences between universal, reduced, or exotic crossed products and appropriate universal, reduced, or exotic fixed-point algebras, respectively. In particular, we obtain an exotic version of Green's imprimitivity theorem and a very general version of the symmetric imprimitivity theorem by weakly proper actions of product groups G × H . In addition, we study functorial properties of generalized fixed-point algebras for equivariant categories of C * -algebras based on correspondences.

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“…The result generalizes the main result of [1] where the authors obtain a reduced version under the additional assumption that G acts freely on X . PROPOSITION 6.5.…”

Section: A Buss and S Echterhoffsupporting

confidence: 82%

Imprimitivity theorems for weakly proper actions of locally compact groups

Buss¹,

Echterhoff²

2014

Ergod. Th. Dynam. Sys.

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“…The weakest notion for proper actions is due to Rieffel (see [28], [29]), and his concept has been studied in a number of papers by several authors (e.g. see [2,4,5,18,23,28,29]). In [28] Rieffel says that an action α : G → Aut(A) is proper, if there exists a dense α-invariant subalgebra A c (playing the role of C c (X)) such that the following conditions are satisfied:…”

Section: Some Preliminaries On Proper Actionsmentioning

confidence: 99%

“…We should also note that weakly proper X ⋊ G-algebras have been studied in several papers (using different terminology) in the context of Rieffel-proper actions (e.g. see [2,3,18]).…”

Section: Some Preliminaries On Proper Actionsmentioning

confidence: 99%

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Universal and exotic generalized fixed-point algebras for weakly proper actions and duality

Buss¹,

Echterhoff²

2014

Indiana Univ. Math. J.

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Given a C * -dynamical system (A, G, α), we say that A is a weakly proper X ⋊ G-algebra if there exists a proper G-space X together with a nondegenerate G-equivariant * -homomorphism φ : C 0 (X) → M(A). Weakly proper G-algebras form a large subclass of the class of proper G-algebras in the sense of Rieffel. In this paper we show that weakly proper X ⋊ G-algebras allow the construction of full fixed-point algebras A G,α u corresponding to the full crossed product A ⋊α G, thus solving, in this setting, a problem stated by Rieffel in his 1988's original article on proper actions. As an application we obtain a general Landstad duality result for arbitrary coactions together with a new and functorial construction of maximalizations of coactions.The same methods also allow the construction of exotic generalized fixedpoint algebras associated to crossed-product norms lying between the reduced and universal ones. Using these, we give complete answers to some questions on duality theory for exotic crossed products recently raised by Kaliszewski, Landstad and Quigg. c

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“…The notation was chosen to indicate some parallel with the nondegenerate versions rt/Ac, K/C * , and K/Co. In fact, rt/Ac en is a semi-comma category in the sense of [HKRW11].…”

Section: Enchilada Categories and Functorsmentioning

confidence: 99%

Dualities for Maximal Coactions

Kaliszewski

Omland

Quigg

2016

J. Aust. Math. Soc.

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Abstract. We present a new construction of crossed-product duality for maximal coactions that uses Fischer's work on maximalizations. Given a group G and a coaction (A, δ) we define a generalized fixed-point algebra as a certain subalgebra of M (A δ G δ G), and recover the coaction via this double crossed product. Our goal is to formulate this duality in a category-theoretic context, and one advantage of our construction is that it breaks down into parts that are easy to handle in this regard. We first explain this for the category of nondegenerate *-homomorphisms, and then analogously for the category of C * -correspondences. Also, we outline partial results for the "outer" category, studied previously by the authors.

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Naturality of Rieffel’s Morita Equivalence for Proper Actions

Cited by 9 publications

References 36 publications

Imprimitivity theorems for weakly proper actions of locally compact groups

Imprimitivity theorems for weakly proper actions of locally compact groups

Universal and exotic generalized fixed-point algebras for weakly proper actions and duality

Dualities for Maximal Coactions

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