Some results on tracial stability and graph products

TL;DR

This paper proves tracial stability for certain graph products of C*-algebras, introduces the 'pincushion class' of graphs, and applies these results to almost commuting operators and properties of right-angled Artin groups.

Contribution

It develops the 'pincushion class' of graphs and demonstrates tracial stability for associated graph products, with applications to operator approximation and group C*-algebras.

Findings

Full C*-algebras of right-angled Artin groups are quasidiagonal.

Tracial stability applies to a new class of graph products.

A selective version of Lin's Theorem is established.

Abstract

We establish the tracial stability of a certain class of graph products of C*-algebras. This result involves the development of the "pincushion class" of finite graphs. We then apply this result in two ways. The first application yields a selective version of Lin's Theorem for almost commuting operators. The second application addresses some approximation properties of right-angled Artin groups. In particular, we show that the full C*-algebra of any right-angled Artin group is quasidiagonal and thus has a non-trivial amenable trace, and then we apply tracial stability to show when these amenable traces are in fact locally finite dimensional.