Free monotone transport for infinite variables

TL;DR

This paper extends the free monotone transport theorem to infinite variables and provides criteria for when certain mixed $q$-Gaussian algebras are isomorphic to the free group factor, based on the decay of their structure array entries.

Contribution

It generalizes the free monotone transport theorem to infinite variables and characterizes isomorphisms of mixed $q$-Gaussian algebras to $L(F__)$ using decay conditions on the structure array.

Findings

Extended free monotone transport to infinite variables.

Provided a criterion for isomorphism to $L(F__)$ based on structure array decay.

Characterized mixed $q$-Gaussian algebras with small, rapidly decaying entries.

Abstract

We extend the free monotone transport theorem of Guionnet and Shlyakhtenko to the case of infinite variables. As a first application, we provide a criterion for when mixed $q$-Gaussian algebras are isomorphic to $L(\mathbb{F}_\infty)$; namely, when the structure array $Q$ of a mixed $q$-Gaussian algebra has uniformly small entries that decay sufficiently rapidly. Here a mixed $q$-Gaussian algebra with structure array $Q=(q_{ij})_{i,j\in\mathbb{N}}$ is the von Neumann algebra generated by $X_n^Q=l_n+l_n^*, n\in\mathbb{N}$ and $(l_n)$ are the Fock space representations of the commutation relation $l_i^*l_j-q_{ij}l_jl_i^*=\delta_{i=j}, i,j\in\mathbb{N}$, $-1<q_{ij}=q_{ji}<1$.