On fundamental groups of tensor product $\mathrm{I}{\mathrm{I}}_1$ factors

Abstract

Let $M$ be a $\mathrm{I}{\mathrm{I}}_1$ factor and let $\mathcal{F}(M)$ denote the fundamental group of $M$. In this article, we study the following property of $M$: for arbitrary $\mathrm{I}{\mathrm{I}}_1$ factor $B$, we have $\mathcal{F}(M\overline{\otimes }B)=\mathcal{F}(M)\mathcal{F}(B)$. We prove that for any subgroup $G\le {\mathbb{R}}_{+}^{\ast }$ which is realized as a fundamental group of a $\mathrm{I}{\mathrm{I}}_1$ factor, there exists a $\mathrm{I}{\mathrm{I}}_1$ factor $M$ which satisfies this property and whose fundamental group is $G$. Using this, we deduce that if $G,H\le {\mathbb{R}}_{+}^{\ast }$ are realized as fundamental groups of $\mathrm{I}{\mathrm{I}}_1$ factors (with separable predual), then so are groups $G\cdot H$ and $G\cap H$.

17 pages, final version, to appear in J. Inst. Math. Jussieu

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