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Invariant states and a conditional fixed point property for affine actions. (English) Zbl 0842.46042

Let \(G\) be a locally compact group acting on a von Neumann algebra \(M\). We prove that the existence of a (not necessarily normal) \(G\)-invariant state on \(M\) is equivalent to a conditional fixed point property for affine actions of \(G\). This answers a question of M. Bekka for amenable unitary representations of \(G\), and our fixed point property generalizes a condition studied by R. J. Zimmer for standard measure \(G\)-spaces.
We also establish a new condition for inner amenability of discrete groups, and we study an approximation property by completely bounded maps for crossed products by discrete groups introduced by M. Cowling and R. J. Zimmer.

MSC:

46L55 Noncommutative dynamical systems
46L30 States of selfadjoint operator algebras
46L10 General theory of von Neumann algebras
43A07 Means on groups, semigroups, etc.; amenable groups
22D10 Unitary representations of locally compact groups
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References:

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