A new proof of the Tikuisis–White–Winter theorem

Christopher Schafhauser

doi:10.1515/crelle-2017-0056

Published by De Gruyter March 8, 2018

A new proof of the Tikuisis–White–Winter theorem

Christopher Schafhauser

From the journal Journal für die reine und angewandte Mathematik (Crelles Journal)

https://doi.org/10.1515/crelle-2017-0056

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Abstract

A trace on a C*-algebra is amenable (resp. quasidiagonal) if it admits a net of completely positive, contractive maps into matrix algebras which approximately preserve the trace and are approximately multiplicative in the 2-norm (resp. operator norm). Using that the double commutant of a nuclear C*-algebra is hyperfinite, it is easy to see that traces on nuclear C*-algebras are amenable. A recent result of Tikuisis, White, and Winter shows that faithful traces on separable, nuclear C*-algebras in the UCT class are quasidiagonal. We give a new proof of this result using the extension theory of C*-algebras and, in particular, using a version of the Weyl–von Neumann Theorem due to Elliott and Kucerovsky.

Acknowledgements

In the first version of this paper posted on the arXiv, Theorem 1.4 was proven (although not explicitly stated) when A is nuclear and φ is non-unital. This is enough to prove Theorem 1.2 in the case when A is nuclear as stated in [24] by using the 2×2-matrix argument used in the proof of Theorem 1.4 to reduce to the non-unital setting. The modifications needed to extend the proof to the exact setting were suggested by Jamie Gabe. The idea of using the classification of injective von Neumann algebras to produce a unital version of Theorem 1.4 grew out of a conversation with Aaron Tikuisis and Stuart White. I am also grateful to Tim Rainone for several helpful conversation related to this work and helpful comments on earlier versions of this paper. Finally, I would like to thank the referee for many valuable comments which improved the exposition of this paper.

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Received: 2017-03-29

Revised: 2017-12-05

Published Online: 2018-03-08

Published in Print: 2020-02-01

A new proof of the Tikuisis–White–Winter theorem

Abstract

Acknowledgements

References

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