Establishing strong 1-boundedness via non-microstates free entropy techniques
Benjamin Major, Dimitri Shlyakhtenko
TL;DR
The paper develops a non-microstates framework to establish strong $1$-boundedness for broad classes of finitely generated diffuse von Neumann algebras by embedding a diffuse abelian subalgebra $N$ into $W^*(X)^\omega$ and showing that the deformation by a free semicircular family $\mathbf{S}$ satisfies $N\vee W^*(\mathbf{X}+\sqrt{t}\mathbf{S})=N\vee W^*(\mathbf{X},\mathbf{S})$ for all $t>0$; this leads to precise identities for the relative free Fisher information $\Phi^*(\cdot:N)$ and non-microstates entropy $\chi^*(\cdot:N)$, namely $\Phi^*(\mathbf{X}+\sqrt{\epsilon}\mathbf{S}:N)=\frac{d}{\epsilon}$ and $\chi^*(\mathbf{X}+\sqrt{\epsilon}\mathbf{S}:N)=\frac{d}{2}[\log(2\pi e)+\log\epsilon]$. By connecting conditional microstates entropy to microstates entropy and applying JP24, the authors derive upper bounds on entropy in this relative setting, culminating in $h(W^*(\mathbf{X}))\le 0$, i.e. strong $1$-boundedness, under Cartan, $\Gamma$, tensor decomposition, or Property (T) hypotheses; they also show obstructions for L$(\mathbb{F}_r)$ factors and free products. The work thus provides a concise non-microstates route to strong $1$-boundedness for a wide range of algebras and clarifies the role of diffuse subalgebras in entropy methods.
Abstract
We show that, for many choices of finite tuples of generators $X = (x_1, \dots , x_d)$ of a tracial von Neumann algebra $(M, τ)$ satisfying certain decomposition properties (non-primeness, possessing a Cartan subalgebra, or property $Γ$), one can find a diffuse, hyperfinite subalgebra $N \subseteq (W^*(X))^ω$ (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.