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The maximal injective crossed product

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  22 April 2019

ALCIDES BUSS ,
SIEGFRIED ECHTERHOFF  and
RUFUS WILLETT
ALCIDES BUSS
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900Florianópolis-SC, Brazil email alcides@mtm.ufsc.br
SIEGFRIED ECHTERHOFF
Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, 48149Münster, Germany email echters@uni-muenster.de
RUFUS WILLETT
Affiliation:
Mathematics Department, University of Hawai‘i at Mānoa, Keller 401A, 2565 McCarthy Mall, Honolulu, HI96822, USA email rufus@math.hawaii.edu
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Abstract

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A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^{\ast }$-algebras; this is a sort of ‘dual’ result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum–Connes conjecture. It turns out that $\rtimes _{\text{inj}}$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.

Keywords

exotic crossed productsinjective C∗ -algebrasexact groupslocal lifting propertyweak expectation property

MSC classification

Primary: 46L55: Noncommutative dynamical systems
Secondary: 46L80: $K$-theory and operator algebras (including cyclic theory)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 11 , November 2020 , pp. 2995 - 3014
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2019

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