Fixed-points in the cone of traces on a 𝐶*-algebra

Rørdam, Mikael

doi:10.1090/tran/7797

Fixed-points in the cone of traces on a $C^{\ast }$-algebra
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Trans. Amer. Math. Soc. 371 (2019), 8879-8906 Request permission

Abstract:

Nicolas Monod introduced the class of groups with the fixed-point property for cones, characterized by always admitting a nonzero fixed-point when acting (suitably) on proper weakly complete cones. He proved that his class of groups contains the class of groups with subexponential growth and is contained in the class of supramenable groups. In this paper we investigate what Monod’s results say about the existence of invariant traces on (typically nonunital) $C^*$-algebras equipped with an action of a group with the fixed-point property for cones. As an application of these results, we provide results on the existence (and nonexistence) of traces on the (nonuniform) Roe algebra.

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