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SIMPLE TRACIALLY $\mathcal {Z}$-ABSORBING C*-ALGEBRAS

Part of: Selfadjoint operator algebras

Published online by Cambridge University Press:  04 June 2025

Massoud Amini ,
Nasser Golestani Open the ORCID record for Nasser Golestani [Opens in a new window] ,
Saeid Jamali  and
N. Christopher Phillips
Massoud Amini
Affiliation:
Faculty of Mathematical Sciences, https://ror.org/03mwgfy56Tarbiat Modares University, Tehran 14115-134
Nasser Golestani*
Affiliation:
Faculty of Mathematical Sciences, https://ror.org/03mwgfy56Tarbiat Modares University, Tehran 14115-134
Saeid Jamali
Affiliation:
Faculty of Mathematical Sciences, https://ror.org/03mwgfy56Tarbiat Modares University, Tehran 14115-134
N. Christopher Phillips
Affiliation:
Department of Mathematics, University of Oregon, Eugene OR 97403-1222
*
(n.golestani@modares.ac.ir)
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Abstract

We define a notion of tracial $\mathcal {Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital C*-algebras. The Razak-Jacelon algebra, simple nonelementary C*-algebras with tracial rank zero, and simple purely infinite C*-algebras are tracially $\mathcal {Z}$-absorbing. We obtain the first purely infinite examples of tracially $\mathcal {Z}$-absorbing C*-algebras which are not $\mathcal {Z}$-absorbing. We use techniques from reduced free products of von Neumann algebras to construct these examples. A stably finite example was given by Z. Niu and Q. Wang in 2021. We study the Cuntz semigroup of a simple tracially $\mathcal {Z}$-absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.

Keywords

tracial 𝒵-absorption𝒵-absorptionCuntz semigrouppurely infinite C*-algebraCuntz subequivalence

MSC classification

Primary: 46L55: Noncommutative dynamical systems 46L80: $K$-theory and operator algebras (including cyclic theory)

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Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 45
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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