# THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS

Abstract : We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group Γ 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 of Γ (that is, the normal sugbroup of Γ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of the FC-center. 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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https://hal.archives-ouvertes.fr/hal-02928504
Contributor : Bachir Bekka <>
Submitted on : Monday, October 26, 2020 - 11:34:17 AM
Last modification on : Wednesday, October 28, 2020 - 3:35:09 AM

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PlancherelFormula-v2.pdf
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• HAL Id : hal-02928504, version 2

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Bachir Bekka. THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS. 2020. ⟨hal-02928504v2⟩

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