Tracial stability for C*-algebras

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.