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

We consider the mixed $q$-Gaussian algebras introduced by Speicher which are generated by the variables $X_i=l_i+l_i^*,i=1,\ldots,N$, where $l_i^* l_j-q_{ij}l_j l_i^*=δ_{i,j}$ 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 $q_{ij}\equiv q$.