This work relates several different notions of dimension for graph products of finite-dimensional *-algebras, and in the process introduces a notion of soficity for algebras.

Graph products of *-algebras generalize graph products of groups; given algebras indexed by the vertices of a graph, one takes a free product and imposes relations that $A_v$ and $A_w$ commute if $v$ is adjacent to $w$, and the algebra $A$ generates a von Neumann algebra in a canonical way by its left regular representation. Our goal is to show that if $A_v$ is finite-dimensional and the first $\ell^2$ Betti number of the graph product algebra $A$ vanishes, then its von Neumann algebra has vanishing $1$-bounded entropy as defined by Hayes (this is a kind of strong "1-dimensionality" condition). We draw on the work of Shlyakhtenko who showed a similar result for finitely presented sofic groups. and therefore we need an analog of soficity for algebras. While soficity produces approximations for a group by permutation matrices (which have integer entries) such that the proportion of fixed points tends to 0 or 1 in the limit, we generalize this to algebras by matrices with entries in a number ring, which have constant diagonal asymptotically. We show that this "algebraic soficity" property is preserved by graph products, using conjugation by random permutation matrices in a tensor product model. This also yields a new probabilistic proof that soficity of groups is preserved by graph products.

This is based on joint work with Ian Charlesworth, Rolando de Santiago, Ben Hayes, Srivatsav Kunnawalkam Elayavalli, Brent Nelson