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The conjugation representation and inner amenability of discrete groups.

The conjugation representation yG of a discrete group G on the Hubert space l (G) is defined by yG(x)f(y) = f(x~yx), /e/(G), x,yeG. Apart from being of interest in group representation theory, certain properties of the support of yG in the dual space ö of G are important since they are related to problems on inner invariant means on /°°(G). For instance, an ICC group G fails to be inner amenable if and only if the trivial representation 1G of G is an isolated point in suppyG. Inner invariant means on discrete groups have been a topic of interest for some time ([1], [2], [3], [4], [6], [14], [15], [19]) and are related to questions on C*and von Neumann algebras associated to the conjugation and the left regulär representation.