Stability under Small Hilbert-Schmidt Perturbations for $C^*$-Algebras

This paper studies the tracial stability of $C^*$-algebras, which is a general property of stability of relations in a Hilbert–Schmidt-type norm defined by a trace on a $C^*$-algebra. Precise definitions are formulated in terms of tracial ultraproducts. For nuclear $C^*$-algebras, a characterization of matricial tracial stability in terms of approximation of tracial states by traces of finite-dimensional representations is obtained. For the nonnuclear case, new obstructions and counterexamples are constructed in terms of free entropy theory.