  We introduce and study a notion of Rokhlin property for an inclusion of
unital $C^*$-algebras which could have no projections like the Jiang-Su
algebra. We also introduce a notion of approximate representability and show a
duality between them. We demonstrate the importance of these notions by showing
the permanence of the tracial $\mathcal{Z}$-absorbingness and the strict
comparison property.
