The maximal injective crossed product

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 funct...

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Veröffentlicht in:Ergodic theory and dynamical systems 2020-11, Vol.40 (11), p.2995-3014, Article 2995
Hauptverfasser: BUSS, ALCIDES, ECHTERHOFF, SIEGFRIED, WILLETT, RUFUS
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Sprache:eng
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Zusammenfassung: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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2019.25