The atoms of graph product von Neumann algebras

TL;DR

This paper classifies the atomic components in graph product von Neumann algebras, providing explicit criteria based on the graph and component algebras for their existence and structure.

Contribution

It offers a complete classification of atomic summands in graph product von Neumann algebras, linking their structure to the underlying graph and component algebras.

Findings

Type I summands are tensor products of component summands.

Existence and weight of summands are determined by explicit graph-associated polynomials.

Provides a complete classification of atomic parts in the algebra.

Abstract

We completely classify the atomic summands in a graph product $(M,\varphi) = *_{v \in \mathcal{G}} (M_v,\varphi_v)$ of von Neumann algebras with faithful normal states. Each type I factor summand $(N,\psi)$ is a tensor product of type I factor summands $(N_v,\psi_v)$ in the individual algebras. The existence of such a summand and its weight in the direct sum can be determined from the $(N_v,\psi_v)$'s using explicit polynomials associated to the graph.