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Invariant means on a class of von Neumann algebras related to ultraspherical hypergroups

Volume 225 / 2014

Nageswaran Shravan Kumar Studia Mathematica 225 (2014), 235-247 MSC: Primary 43A62, 43A15; Secondary 43A30, 46J10. DOI: 10.4064/sm225-3-4

Abstract

Let $K$ be an ultraspherical hypergroup associated to a locally compact group $G$ and a spherical projector $\pi$ and let VN$(K)$ denote the dual of the Fourier algebra $A(K)$ corresponding to $K.$ In this note, invariant means on VN$(K)$ are defined and studied. We show that the set of invariant means on VN$(K)$ is nonempty. Also, we prove that, if $H$ is an open subhypergroup of $K,$ then the number of invariant means on VN$(H)$ is equal to the number of invariant means on VN$(K).$ We also show that a unique topological invariant mean exists precisely when $K$ is discrete. Finally, we show that the set TIM$(\widehat{K})$ becomes uncountable if $K$ is nondiscrete.

Authors

  • Nageswaran Shravan KumarDepartment of Mathematics
    Indian Institute of Technology Delhi
    Delhi 110016, India
    e-mail

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