Geodesic
Stability under Small Hilbert-Schmidt Perturbations for $C^*$-Algebras
Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 92-97.

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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. 
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D. Hadwin; T. V. Shulman. Stability under Small Hilbert-Schmidt Perturbations for $C^*$-Algebras. Funkcionalʹnyj analiz i ego priloženiâ, Tome 52 (2018) no. 3, pp. 92-97. http://geodesic.mathdoc.fr/item/FAA_2018_52_3_a9/

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