Tracially Z-absorbing C⁎-algebras

We study a tracial notion of Z-absorption for simple, unital C⁎-algebras. We show that if A is a C⁎-algebra for which this property holds then A has almost unperforated Cuntz semigroup, and if in addition A is nuclear and separable we show this property is equivalent to having A≅A⊗Z. We furthermore show that this property is preserved under forming certain crossed products by actions satisfying a tracial Rokhlin type property.