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

Fabio Elio Tonti; Asger Törnquist

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

Group Theory 2019-04-18 v1 Logic Operator Algebras

Authors: Fabio Elio Tonti , Asger Törnquist

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

Keywords

type I groups unitary representations countable discrete groups abelian subgroup Thoma's theorem

Cite

@article{arxiv.1904.08313,
  title  = {A short proof of Thoma's theorem on type I groups},
  author = {Fabio Elio Tonti and Asger Törnquist},
  journal= {arXiv preprint arXiv:1904.08313},
  year   = {2019}
}

Comments

The statement of Lemma 7 has the following correction over earlier circulated drafts of the paper: The unnecessary assumption that H_0 and H_1 are non-abelian has been removed

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

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