Simple equivariant C*-algebras whose full and reduced crossed products coincide

Abstract

For any second countable locally compact group G, we construct a simple G-C *algebra whose full and reduced crossed product norms coincide. We then construct its Gequivariant representation on another simple G-C * -algebra without the coincidence condition. This settles two problems posed by Anantharaman-Delaroche in 2002. Some constructions involve the Baire category theorem.2000 Mathematics Subject Classification. Primary 46L55, Secondary 46L05, 54H20.

Injectivity, crossed products, and amenable group actions

“…In [25], Yuhei Suzuki produces (amongst other things) a very striking class of examples. Let G be a countable, exact, non-amenable group.…”

“…Let G be a countable, exact, non-amenable group. Suzuki shows in [25,Proposition B] that there exists a simple, unital, separable, nuclear G-algebra A such that A¸r G " A¸m ax G. As A is simple and unital, its center is just scalar multiples of the unit, so A cannot be strongly amenable: if it were, G would necessarily be amenable. It is well-known that strong amenability implies equality of the maximal and reduced crossed product C˚-algebras, but While we guess Suzuki (and others) are aware of this, it does not seem to have been explicitly recorded in his paper.…”

“…Since pη i,n q is uniformly bounded (by 1) with respect to the ℓ 2 -norm, it follows that (ii) and (iii) hold for all a P A. Now, let us briefly describe Suzuki's examples as given in [25,Proposition B] in order to see how they fit into the above discussion. Let G be any countable exact group.…”

“…We close this section with a brief discussion on the amenability of some other examples of G-algebras A with A¸G " A¸r G constructed by Suzuki in [25]. In [25,Theorem A] he produces examples of actions α : G Ñ AutpAq of nonamenable second countable locally compact groups on simple C˚-algebras A such that the full and reduced crossed products coincide. Since A is simple, we have…”

“…‚ recent interesting examples of Suzuki [25] showing a disconnect between notions of amenable actions and the equality A¸m ax G " A¸r G; ‚ Matsumura's work [20] relating the property A¸m ax G " A¸r G to amenability (at least in the presence of exactness of the acting group); ‚ the extensive work of Anantharaman-Delaroche on amenable actions, most relevantly for this paper in [4] and [5]; ‚ the seminal theorem of Guentner-Kaminker [12] and Ozawa [21] relating exactness to existence of amenable actions; ‚ our study of the so-called maximal injective crossed product [9] and the connections to equivariant versions of injectivity; ‚ Hamana's classical study of equivariant injective envelopes [15] and the recent important exploitation of these ideas by Kalantar and Kennedy in their work on the Furstenberg boundary [16]. Our goal was to try to bridge connections between some of this in a way that we hope systematises some of the existing literature a little better, as well as solving some open problems.…”