The maximal injective crossed product

Abstract

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 Þ Ñ A¸i nj 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˚-algebras; this is a sort of a '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¸i nj has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.2010 Mathematics Subject Classification. 46L55, 46L80.

Topological boundaries of covariant representations

“…One could add a rather methodological comment to these by reminding that recently an "injective" crossed product is introduced (c.f. [14], [15], [16]) which could eventually be related to the boundary in this more general sense.…”

Partial tensor-product functors and crossed-product functors

“…General exotic crossed-product functors and their properties were studied systematically by Buss, Echterhoff and Willett [BEW17,BEW18a,BEW18b,BEW20a,BEW20b]. They introduced the minimal exact crossed-product functor − ⋊ E G, the minimal exact correspondence crossed-product functor − ⋊ E Corr G (which agrees with the minimal exact Morita compatible crossed-product functor of [BGW16] for separable G-C * -algebras [BEW18a, Cor.…”

The Baum–Connes conjecture localised at the unit element of a discrete group

“…The left-hand side is a topological object, based on the K-homology of proper Γ-spaces with compact quotient; the right-hand side is analytic, the K-theory of the reduced C * -algebra of Γ. We refer to the vast literature on the subject (see [BC00,BCH93], the book [Val02] and the recent articles [BGW16, BEW18,GJV19] and the references therein). One of the main motivations for the conjecture is that the injectivity of μ implies the Novikov conjecture.…”