A short proof of Thoma's theorem on type I groups

Fabio Elio Tonti; Asger Törnquist

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arXiv · 1904.08313

A short proof of Thoma's theorem on type I groups

Fabio Elio Tonti·Asger Törnquist

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Abstract

In the theory of unitary group representations, a group is called type I if all factor representations are of type I, and by a celebrated theorem of James Glimm [Gli61b], the type I groups are precisely those groups for which the irreducible unitary representations are what descriptive set theorists now call "concretely classifiable". Elmar Thoma [Tho64] proved the following surprising characterization of the countable discrete groups of type I: They are precisely those that contain a finite index abelian subgroup. In this paper we give a new, simpler proof of Thoma's theorem, which relies only on relatively elementary methods. [Gli61b] James Glimm, Type I $C^{\ast} $-algebras, Ann. of Math. (2) 73 (1961), 572--612. MR 0124756 [Tho64] Elmar Thoma, \"Uber unit\"are Darstellungen abz\"ahlbarer, diskreter Gruppen, Math. Ann. 153 (1964), 111--138. MR 0160118

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Fabio Elio Tonti, Asger Törnquist. 2019-04-16. A short proof of Thoma's theorem on type I groups. https://arxiv.org/abs/1904.08313

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