Establishing strong 1-boundedness via non-microstates free entropy techniques

Abstract

We show that, for many choices of finite tuples of generators $X=(x_1,\dots ,x_d)$ of a tracial von Neumann algebra $(M,\ddot{I}„)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $\hat{I}“$), one can find a diffuse, hyperfinite subalgebra $N\subseteq (W^{\ast }(X){)}^{\ddot{I}}‰$ (often in $W^{\ast }(X)$ itself), such that $W^{\ast }(N,X+\sqrt{t}S)=W^{\ast }(N,X,S)$ for all $t>0$. (Here $S$ is a free semicircular family, free from $\{X\}\cup N$). This gives a short non-microstates proof of strong 1-boundedness for such algebras.

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