Graph products of von Neumann algebras are an interpolation between the tensor product and the free product, first studied in this setting by M\l{}otkowski who was interested in viewing the $q$-deformed Gaussian distribution as the result of a central limit-like theorem. They are an object of study whose properties are not yet well understood.

I will speak on some recent work looking into when graph product algebras are strongly 1-bounded. This is a free entropy condition first introduced by Jung which corresponds to a paucity of matricial microstates. Unlike many free entropy properties, it has shown to be a property of the von Neumann algebra rather than of a chosen generating set. We show that in our setting, strong 1-boundedness follows from the vanishing of the first $L^2$-Betti number. The argument follows from two main insights: a notion of algebraic soficity, and a random matrix model for graph independence using conjugation by permutation matrices which preserve nice algebraic properties.

This is joint work with de Santiago, Hayes, Jekel, Kunnawalkam Elayavalli, and Nelson.