Quasidiagonality of nuclear C*-algebras

Aaron Tikuisis; Stuart  White; Wilhelm Winter

doi:10.4007/annals.2017.185.1.4

Quasidiagonality of nuclear C*-algebras

Aaron Tikuisis
, Stuart White
, Wilhelm Winter

Mathematical Science

Research output: Contribution to journal › Article › peer-review

206 Citations (Scopus)

17 Downloads (Pure)

Abstract

Abstract. We prove that faithful traces on separable and nuclear C∗-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C∗-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms–Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C∗-algebras.

Original language	English
Pages (from-to)	229-284
Number of pages	66
Journal	Annals of Mathematics
Volume	185
Issue number	1
DOIs	https://doi.org/10.4007/annals.2017.185.1.4
Publication status	Published - 31 Jan 2017

Bibliographical note

Research partially supported by EPSRC (EP/N002377), NSERC (PDF, held by AT), by
an Alexander von Humboldt foundation fellowship (held by SW) and by the DFG (SFB 878).

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author = "Aaron Tikuisis and Stuart White and Wilhelm Winter",

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AU - White, Stuart

AU - Winter, Wilhelm

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N2 - Abstract. We prove that faithful traces on separable and nuclear C∗-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C∗-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms–Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C∗-algebras.

AB - Abstract. We prove that faithful traces on separable and nuclear C∗-algebras in the UCT class are quasidiagonal. This has a number of consequences. Firstly, by results of many hands, the classification of unital, separable, simple and nuclear C∗-algebras of finite nuclear dimension which satisfy the UCT is now complete. Secondly, our result links the finite to the general version of the Toms–Winter conjecture in the expected way and hence clarifies the relation between decomposition rank and nuclear dimension. Finally, we confirm the Rosenberg conjecture: discrete, amenable groups have quasidiagonal C∗-algebras.

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DO - 10.4007/annals.2017.185.1.4

M3 - Article

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VL - 185

SP - 229

EP - 284

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -

Quasidiagonality of nuclear C*-algebras

Abstract

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