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arXiv · 2109.05192

Simple tracially $\mathcal{Z}$-absorbing C*-algebras

Massoud Amini·Nasser Golestani·Saeid Jamali·N. Christopher Phillips
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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 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.

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Massoud Amini, Nasser Golestani, Saeid Jamali, N. Christopher Phillips. 2021-09-11. Simple tracially $\mathcal{Z}$-absorbing C*-algebras. https://arxiv.org/abs/2109.05192

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