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| Format: | Preprint |
| Published: |
2007
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/0704.3502 |
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Table of Contents:
- In the context of Free Probability Theory, we study two different constructions that provide new examples of factors of type ${\rm II_1}$ 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 $S \subset \R^*_+$, we show that there exists an action of $G$ on $L(\F_\infty)$ such that $L(\F_\infty) \rtimes G$ is a type ${\rm II_1}$ factor and its fundamental group is $S$. The second construction is based on a free product. Take $(B(H), ψ)$ any factor of type ${\rm I}$ endowed with a faithful normal state and denote by $Γ\subset \R^*_+$ the subgroup generated by the point spectrum of $ψ$. We show that the centralizer $(L(G) \ast B(H))^{τ\ast ψ}$ is a type ${\rm II_1}$ factor and its fundamental group is $Γ$. Our proofs rely on Popa's deformation/rigidity strategy using his intertwining-by-bimodules technique.