Invariant states and a conditional fixed point property for affine actions
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.
compact group acting on a von Neumann algebra
conditional fixed point property for affine actions
scientific article; zbMATH DE number 861926
scientific article