Indagationes Mathematicae ( IF 0.882 ) Pub Date : 2021-01-13 , DOI: 10.1016/j.indag.2021.01.002
Bachir Bekka

We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group $\Gamma$ into a direct integral of factor representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center ${\Gamma }_{\mathrm{fc}}$ of $\Gamma$ (that is, the normal sugbroup of $\Gamma$ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of ${\Gamma }_{\mathrm{fc}}$. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma’s and Kaniuth’s theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II.

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