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Tracial approximation in simple ${C}^{\ast }$-algebras

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  26 February 2021

Xuanlong Fu  and
Huaxin Lin
Xuanlong Fu*
Affiliation:
Shanghai Center for Mathematical Sciences, Fudan University, Shanghai200438, China
Huaxin Lin
Affiliation:
Department of Mathematics, East China Normal University, Shanghai, China and (Current) Department of Mathematics, University of Oregon, Eugene, OR97403, USA e-mail: hlin@uoregon.edu
*
e-mail: xlf@fudan.edu.cn
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Abstract

We revisit the notion of tracial approximation for unital simple $C^*$ -algebras. We show that a unital simple separable infinite dimensional $C^*$ -algebra A is asymptotically tracially in the class of $C^*$ -algebras with finite nuclear dimension if and only if A is asymptotically tracially in the class of nuclear $\mathcal {Z}$ -stable $C^*$ -algebras.

MSC classification

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 46L35: Classifications of $C^*$-algebras
Type
Article
Information
Canadian Journal of Mathematics , Volume 74 , Issue 4 , August 2022 , pp. 942 - 1004
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

Huaxin Lin is the corresponding author.

Xuanlong Fu was supported by China Postdoctoral Science Foundation, grant # 2020M670962, and partially supported by an NSFC grant (NSFC 11420101001). Huaxin Lin was partially supported by an NSF grant (DMS-1954600). Both authors acknowledge the support from the Research Center of Operator Algebras at East China Normal University which is partially supported by Shanghai Key Laboratory of PMMP, Science and Technology Commission of Shanghai Municipality (STCSM), grant #13dz2260400 and a NNSF grant (11531003).

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