Suppose A is a C*-algebra, and \alpha is an automorphism. One would like to find conditions which guarantee that the crossed product absorbs the Jiang-Su algebra Z tensorially. Previous work on the topic (by Winter and myself, Winter, Zacharias and myself and by Matui and Sato) focused on permanence results: if A is already assumed to be Z-absorbing, and \alpha satisfies an appropriate generalization of the Rokhlin property, then the crossed product is Z-absorbing as well.

I'll discuss some work in progress which addresses the case in which A is not assumed to be Z-absorbing. Here the finite Rokhlin dimension (or just the Rokhlin property) in its own is not sufficient to imply that the crossed product is Z-absorbing, however, I will show that it is if we furthermore require that the action satisfies a weak form of approximate innerness.