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SIMPLE TRACIALLY
$\mathcal {Z}$-ABSORBING C*-ALGEBRAS
- Part of
-
- Journal:
- Journal of the Institute of Mathematics of Jussieu , First View
- Published online by Cambridge University Press:
- 04 June 2025, pp. 1-45
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- Article
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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.
