An application of free transport to mixed $q$-Gaussian algebras

Abstract

We consider the mixed $q$-Gaussian algebras introduced by Speicher which are generated by the variables $X_i=l_i+l_i^{\ast },i=1,\dots ,N$, where $l_i^{\ast }{l}_j-q_{ij}{l}_j{l}_i^{\ast }={\delta }_{i,j}$ and $-1<q_{ij}=q_{ji}<1$. Using the free monotone transport theorem of Guionnet and Shlyakhtenko, we show that the mixed $q$-Gaussian von Neumann algebras are isomorphic to the free group von Neumann algebra $L({\mathbb{F}}_N)$, provided that ${\max }_{i,j}|q_{ij}|$ is small enough. The proof relies on some estimates which are generalizations of Dabrowski's results for the special case $q_{ij}\equiv q$.

Typos and imprecision (mainly in the Introduction) are corrected. Type III case is erased for further more detailed investigation

References in corpus (2)

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

Cited by in corpus (1)

• Fock representations of $Q$-deformed commutation relations