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Article Dans Une Revue Indagationes Mathematicae Année : 2021

THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS

Bachir Bekka
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Résumé

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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Dates et versions

hal-02928504 , version 1 (02-09-2020)
hal-02928504 , version 2 (26-10-2020)

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Bachir Bekka. THE PLANCHEREL FORMULA FOR COUNTABLE GROUPS. Indagationes Mathematicae, 2021, 32 (3), pp.619-638. ⟨10.1016/j.indag.2021.01.002⟩. ⟨hal-02928504v2⟩
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