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 $(M, \tau)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $\Gamma$), one can find a diffuse, hyperfinite subalgebra $N \subseteq (W^*(X))^{\omega}$ (often in $W^*(X)$ itself), such that $W^*(N,X+\sqrt{t}S) = W^*(N,X,S)$ 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.