Tracial stability for C*-algebras

TL;DR

This paper investigates tracial stability in C*-algebras, providing a complete characterization for nuclear cases, identifying obstructions for non-nuclear cases, and linking stability to topological properties of underlying spaces.

Contribution

It offers a full characterization of matricial tracial stability for nuclear C*-algebras and introduces new obstructions for non-nuclear cases based on free entropy dimension.

Findings

Characterizes nuclear C*-algebras' tracial stability via approximation properties.

Identifies obstructions to stability in non-nuclear C*-algebras using free entropy dimension.

Shows that $C(X)$ is tracially stable iff $X$ is approximately path-connected.

Abstract

We consider tracial stability, which requires that tuples of elements of a C*-algebra with a trace that nearly satisfy the relation are close to tuples that actually satisfy the relation. Here both "near" and "close" are in terms of the associated 2-norm from the trace, e.g., the Hilbert-Schmidt norm for matrices. Precise definitions are stated in terms of liftings from tracial ultraproducts of C*-algebras. We completely characterize matricial tracial stability for nuclear C*-algebras in terms of certain approximation properties for traces. For non-nuclear $C^{\ast}$-algebras we find new obstructions for stability by relating it to Voiculescu's free entropy dimension. We show that the class of C*-algebras that are stable with respect to tracial norms on real-rank-zero C*-algebras is closed under tensoring with commutative C*-algebras. We show that $C(X)$ is tracially stable with respect to tracial norms on all $C^{\ast}$-algebras if and only if $X$ is approximately path-connected.