Generalized $q$-Gaussian von Neumann algebras with coefficients, I. Relative strong solidity

Abstract

We define ${\Gamma }_q(B,S\otimes H)$, the generalized $q$-gaussian von Neumann algebras associated to a sequence of symmetric independent copies $({\pi }_j,B,A,D)$ 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 non-isomorphism and non-embedability results about some of these von Neumann algebras.

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