Tracial Stability for C*-Algebras

Tracial Stability for C*-Algebras 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 $$C^{*}$$ 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 $$C^{*}$$ C ∗ -algebras if and only if X is approximately path-connected. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

Tracial Stability for C*-Algebras

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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-018-2430-1
Publisher site
See Article on Publisher Site

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 $$C^{*}$$ 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 $$C^{*}$$ C ∗ -algebras if and only if X is approximately path-connected.

Journal

Integral Equations and Operator TheorySpringer Journals

Published: Feb 24, 2018

References

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