##### Department of Mathematics,

University of California San Diego

****************************

### Math 243, Functional Analysis

## Dr. Ian Charlesworth & Dr. David Jekel

#### Cardiff University/Fields Institute for Research in Mathematical Sciences

## Algebraic soficity and graph products

##### Abstract:

** **We show that a graph product of tracial von Neumann algebras is strongly $1$-bounded if the first $\ell^2$-Betti number vanishes for an associated dense $*$-subalgebra. Graph products of tracial von Neumann algebras were studied by Caspers and Fima, and generalize Green's graph product of groups. Given groups $G_v$ for each vertex of a graph $\Gamma$, the graph product is the free product modulo the relations that $G_v$ and $G_w$ commute when $v \sim w$; for von Neumann algebras, graph products are described by a certain moment relation. In our paper, the crux of the argument is a generalization to tracial von Neumann algebras of the statement that soficity of groups is preserved by graph products. We replace soficity for groups with a more general notion of *algebraic soficity* for a $*$-algebra $A$, which means the existence of certain approximations for the generators of $A$ by matrices with algebraic integer entries and approximately constant diagonal. We show algebraic soficity is preserved under graph products through a random permutation construction, inspired by previous work of Charlesworth and Collins as well as Au-C{\'e}bron-Dahlqvist-

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

Host: Priyanga Ganesan

### April 16, 2024

### 11:00 AM

APM 7218 and** **Zoom (meeting ID: 94246284235)

****************************