Notes on C₀-representations and the Haagerup property

2013/04/23 by Paul Jolissaint, Jolissaint, Paul · 1 citation
Mathematics · #22D10 #22D25 #46L10 #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:22D10 #msc:22D25 #msc:46L10

paper · pdf · doi:10.48550/arxiv.1304.6193

Notes partly of an expository nature; 11 pages; a few typos fixed; final version to appear in the Bulletin of the Belgian Mathematical Society - Simon Stevin

arxiv created 2013/08/26 · arxiv updated 2013/08/27

Abstract

For any locally compact group G, we show the existence and uniqueness up to quasi-equivalence of a unitary C₀-representation π₀ of G such that all coefficient functions of C₀-representations of G are coefficient functions of π₀. The present work, strongly influenced by the work of N. Brown and E. Guentner (which dealt exclusively with discrete groups), leads to new characterizations of the Haagerup property: if G is second countable, then it has that property if and only if the representation π₀ induces a *-isomorphism of C^*(G) onto C^*_π0(G). When G is discrete, we also relate the Haagerup property to relative strong mixing properties of the group von Neumann algebra L(G) into finite von Neumann algebras.

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