Generalized $q$-gaussian von Neumann algebras with coefficients, I: Relative strong solidity

Abstract

We define ${\mathrm{\Gamma }}_q\left(B,S\otimes H\right)$, the generalized $q$-gaussian von Neumann algebras associated to a sequence of symmetric independent copies $\left({\pi }_j,B,A,D\right)$ and to a subset $1\in S=S^{\ast }\subset A$ and, under certain assumptions, prove their strong solidity relative to $B$. We provide many examples of strongly solid generalized $q$-gaussian von Neumann algebras. We also obtain nonisomorphism and nonembedability results about some of these von Neumann algebras.