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. [ABSTRACT FROM AUTHOR]