Tracial Stability for C*-Algebras

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Abstract

We consider tracial stability, which requires that tuples of elements of a C*-algebra with a trace that nearly satisfy a 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 \(C^{*}\)-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 \(C^{*}\)-algebras if and only if X is approximately path-connected.

Keywords

Tracial ultraproduct Tracially stable Tracial norms Almost commuting matrices 

Mathematics Subject Classification

Primary 46Lxx Secondary 20Fxx 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.University of New HampshireDurhamUSA
  2. 2.Institute of Mathematics of the Polish Academy of SciencesWarsawPoland

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