Construction of type $\mathrm{I}{\mathrm{I}}_1$ factors with prescribed countable fundamental group

Abstract

In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type $\mathrm{I}{\mathrm{I}}_1$ with prescribed 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 $\mathcal{F}(L(G))=\{1\}$ (e.g. $G = \Z^2 \rtimes \SL(2, \Z)$). For any countable subgroup $S\subset {\mathbb{R}}_{+}^{\ast }$, we show that there exists an action of $G$ on $L(\F_\infty)$ such that $L(\F_\infty) \rtimes G$ is a type $\mathrm{I}{\mathrm{I}}_1$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H),\psi )$ any factor of type $\mathrm{I}$ endowed with a faithful normal state and denote by $\Gamma \subset {\mathbb{R}}_{+}^{\ast }$ the subgroup generated by the point spectrum of $\psi$. We show that the centralizer $(L(G)\ast B(H){)}^{\tau \ast \psi }$ is a type $\mathrm{I}{\mathrm{I}}_1$ factor and its fundamental group is $\Gamma$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.

33 pages

References in corpus (1)

• On some free products of von Neumann algebras which are free Araki-Woods factors

Cited by in corpus (12)

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• Structure of $\mathrm{I}{\mathrm{I}}_1$ factors arising from free Bogoljubov actions of arbitrary groups
• Strongly solid $\mathrm{I}{\mathrm{I}}_1$ factors with an exotic MASA
• Strongly solid group factors which are not interpolated free group factors
• Structural results for free Araki-Woods factors and their continuous cores
• On fundamental groups of tensor product $\mathrm{I}{\mathrm{I}}_1$ factors
• Thin II_1 factors with no Cartan subalgebras
• Explicit computations of all finite index bimodules for a family of II_1 factors
• II${}_1$ factors and equivalence relations with distinct fundamental groups
• Every compact group arises as the outer automorphism group of a II_1 factor
• From Unitarity to Irreversibility: The Role of Infinite Tensor Products and Nested Wigner's Friends