Free monotone transport for infinite variables

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 $X_n^Q=l_n+l_n^{\ast },n\in \mathbb{N}$ and $(l_n)$ are the Fock space representations of the commutation relation $l_i^{\ast }{l}_j-q_{ij}{l}_j{l}_i^{\ast }={\delta }_{i=j},i,j\in \mathbb{N}$, $-1<q_{ij}=q_{ji}<1$.

References in corpus (3)

• Non injectivity of the q-deformed von Neumann algebra
• Combinatorial aspects of matrix models
• Free monotone transport without a trace

Cited by in corpus (1)

• Free transport for interpolated free group factors