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

We define a notion of tracial $mathcal{Z}$-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 $mathcal{Z}$-absorbing C*-algebras need not be $mathcal{Z}$-absorbing. We show that tracial $mathcal{Z}$-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 $mathcal{Z}$-absorbing C*-algebra to be $mathcal{Z}$-absorbing. We also study the Cuntz semigroup of a simple tracially $mathcal{Z}$-absorbing C*-algebra and prove that it is almost unperforated and weakly almost divisible.