Rigid Graph Products

TL;DR

This paper establishes rigidity and uniqueness properties for von Neumann algebraic graph products, introducing rigid graphs and a class of II$_1$-factors, with applications to prime factorization and graph radius recovery.

Contribution

It introduces the notion of rigid graphs and a new class of II$_1$-factors, proving unique decomposition results and characterizing properties like nuclearity and solidity for graph products.

Findings

Unique prime and free product decompositions for certain von Neumann algebras.

Ability to recover graph radius from many graph products of II$_1$-factors.

Conditions for nuclearity, primeness, and strong solidity of graph products.

Abstract

We prove rigidity properties for von Neumann algebraic graph products. We introduce the notion of rigid graphs and define a class of II$_1$-factors named $\mathcal{C}_{\rm Rigid}$. For von Neumann algebras in this class we show a unique rigid graph product decomposition. In particular, we obtain unique prime factorization results and unique free product decomposition results for new classes of von Neumann algebras. Furthermore, we show that for many graph products of II$_1$-factors, including the hyperfinite II$_1$-factor, we can, up to a constant 2, retrieve the radius of the graph from the graph product. We also prove several technical results concerning relative amenability and embeddings of (quasi)-normalizers in graph products. Furthermore, we give sufficient conditions for a graph product to be nuclear and characterize strong solidity, primeness and free-indecomposability for graph products.