Rokhlin property and approximate representability for inclusions of index-finite type

Let $P\subset A$ be an inclusion of unital $C\text{sp}\ast$-algebras of index-finite type with a fixed conditional expectation $E:A\to P$. We introduce the weak tracial Rokhlin property and its dual notion, weak tracial approximate representability, using positive contractions in central sequence algebras; in particular, the definitions apply to projectionless algebras. Under natural simplicity, finiteness, and outerness hypotheses, we prove that these properties are exchanged by the Jones–Watatani basic construction and its dual conditional expectation. We show that the associated Rokhlin contractions produce tracially large injective completely positive order-zero maps and that our inclusion-theoretic definition recovers the weak tracial Rokhlin property for finite-group actions. The duality also yields inclusions of index-finite type with the weak tracial Rokhlin property which are not isomorphic to fixed-point inclusions arising from actions of ordinary finite groups. Explicit actions on an infinite-type UHF algebra, a simple monotracial AF algebra, and $\text{mc}Z$ separate approximate, tracial approximate, and weak tracial approximate representability. Finally, by showing that the inclusion $P\hookrightarrow A$ is tracially sequentially-split by order zero, we obtain permanence of tracial $\text{mc}Z$-absorption, $\text{mc}Z$-stability, strict comparison, tracial $m$-comparison, tracial $m$-almost divisibility, and tracial nuclear dimension from $A$ to $P$.