Factoriality of Mixed $q$-Gaussian Algebras at the Boundary

Abstract

We determine the center and characterize factoriality for mixed $q$-Gaussian von Neumann algebras associated with real symmetric matrices $Q=(q_{ij})_{i,j\in I}$ satisfying $|q_{ij}|\le1$. We prove that every central element is supported on the singleton components of the strict graph of $Q$ and on its two-vertex components $\{a,b\}$ with $q_{aa}=q_{bb}=-1$ in terms of the graph-theoretic language. We also determine the spectral laws of some central elements.