2021/09/11 by Massoud Amini, Nasser Golestani, Amini, Massoud +5
Mathematics · #Advanced Operator Algebra Research #Spectral Theory in Mathematical Physics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2109.05192
We define a notion of tracial 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 Z-absorbing. We obtain the first purely infinite examples of tracially Z-absorbing C*-algebras which are not 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 Z-absorbing C*-algebra and prove that it is almost unperforated and the algebra is weakly almost divisible.