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

Abstract

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

Exotic Coactions

“…We can now settle [11, Conjecture 6.14] affirmatively (again, see [5,Theorem 5.1] for an alternative proof).…”

“…The aforementioned counterexample of [5] shows that not all large quotients of (A, δ) arise this way; nevertheless, we feel that this tool deserves to become more widely known. Actually, our original motivation in writing this paper involves crossed-product duality; everything we need can be found in, for example, [7, Appendix A], [1] and [6], and in the following few sentences we very briefly recall the essential facts.…”

Imprimitivity theorems for weakly proper actions of locally compact groups

“…This notion substantially generalizes the notion of (centrally) proper C 0 (X )-algebras, in which the homomorphism φ : C 0 (X ) → M(A) is assumed to take values in the center Z M(A) of the multiplier algebra M(A). On the other hand, it follows from [29,Proposition 5.9] that weakly proper actions are always proper in the sense of Rieffel (see [28,29]), who showed in [28] that all proper actions in his sense allow the construction of a generalized fixedpoint algebra A G r which is Morita equivalent to an ideal in the reduced crossed product A α,r G. But already in his paper [28], Rieffel discussed whether it is possible to obtain similar constructions which involve the universal crossed product A α,u G. We show in [7] that such theory exists in the case of weakly proper actions. In that paper we also construct a universal version of the generalized fixed-point algebras A G u , which is Morita equivalent to an ideal in the universal crossed product A α,u G. It can be obtained as a completion of the fixed-point algebra with compact supports…”

“…Such norms are most interesting if they satisfy f r ≤ f µ ≤ f u and if they are coming from a crossedproduct functor (B, G, β) → B β,µ G. Such functors have recently been studied by several authors (see e.g. [6,7,15,23]). Although some of our results could possibly be stated for more general crossed-product norms, in this paper we shall work exclusively with the crossed-product norms as introduced by Kaliszewski, Quigg, and Landstad in [15].…”

“…Existence of the map φ µ G is given by [7,Proposition 5.2] and it is then easy to check that conditions (i) and (ii) hold. In order to check (iv) we first note that β • (φ u G) = (φ u G ⊗ id G ) • α, which follows from the fact that both are given by the covariant homomorphism (i B • φ ⊗ 1, i G ⊗ u) of (A, G, α).…”