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Description

The free group factor $L(\mathbb{F}_d)$ can realized as the von Neumann algebra generated by $d$ freely independent semicircular varables $S_1, \ldots, S_d$, the free probabilistic analogue of an independent family of Gaussians. We consider non-commutative random variables $X_1, \ldots, X_m$ whose non-commutative distribution satisfy the integration-by-parts relation $\tau(D_{X_j} V(X) p(X)) = \tau \otimes \tau(\partial_{X_j} p(X))$, where $V$ is a suitably regular convex function. By studying conditional transport of measure" for the associated $N \times N$ random matrix models in the large $N$ limit, we show that there is an isomorphism $W^*(X_1, \ldots, X_d) \to W^*(S_1, \ldots, S_d)$ such that the $W^*$-algebra of the first $k$ generators is mapped to the $W^*$-algebra of the first $k$ generators for every $k$. In particular, $W^*(X_1)$ is sent to the generator MASA in $L(F_d)$, so it is freely complemented. As an application, we deduce that for every non-commutative polynomial $p$, the algebra $W^*(S_1 + \epsilon p(S))$ is a freely complemented MASA in $L(F_d)$ for sufficiently small $\epsilon$.