Brief communications

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

D. Hadwin^a, T. V. Shulman<sup>b

a</sup> University of New Hampshire
^b Institute of Mathematics of the Polish Academy of Sciences

Abstract: 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.

Keywords: tracial ultraproduct, tracial stability, tracial norms, almost commuting matrices.

UDC: 917.98

Received: 24.05.2017

DOI: 10.4213/faa3479

 English version: Functional Analysis and Its Applications, 2018, 52:3, 236–240

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