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

We define a notion of tracial Z-absorption for simple not necessarily unital C*-algebras. This extends the notion defined by Hirshberg and Orovitz for unital (simple) C*-algebras. We provide examples which show that tracially Z-absorbing C*-algebras need not be Z-absorbing. We show that tracial Z-absorption passes to hereditary C*-subalgebras, direct limits, matrix algebras, minimal tensor products with arbitrary simple C*-algebras. We find sufficient conditions for a simple, separable, tracially Z-absorbing C*-algebra to be Z-absorbing. We also study the Cuntz semigroup of a simple tracially Z-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.

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