Construction of type II1 factors with prescribed countable fundamental group

Cyril Houdayer 1
  • 1 UMPA, ENS Lyon, 46, Allée d'Italie, 69364 Lyon Cedex 07, France. e-mail: cyril.houdayer@umpa.ens-lyon.fr

Abstract

In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type II1 with prescribed countable fundamental group. First we investigate state-preserving group actions on the almost periodic free Araki-Woods factors satisfying both a condition of mixing and a condition of free malleability in the sense of Popa. Typical examples are given by the free Bogoliubov shifts. Take an ICC w-rigid group G such that ℱ(L(G)) = {1} (e.g., G = ℤ2 ⋊ SL(2, ℤ)). For any countable subgroup S, we construct an action of G on L(𝔽) such that the associated crossed product L(𝔽) ⋊ G is a type II1 factor and its fundamental group is S. The second construction is based on a free product. Take (B(H),ψ) any factor of type I endowed with a faithful normal state and denote by S the subgroup generated by the point spectrum of ψ. We show that the centralizer (L(G) ∗ B(H))τ*ψ is a type II1 factor and its fundamental group is S. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.

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The Journal für die reine und angewandte Mathematik is the oldest mathematics periodical still in existence. Founded in 1826 by August Leopold Crelle and edited by him until his death in 1855, it soon became widely known under the name of Crelle’s Journal. In the 190 years of its existence, Crelle’s Journal has developed to an outstanding scholarly periodical with one of the worldwide largest circulations among mathematics journals.

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