Rokhlin property and approximate representability for inclusions of index-finite type
Let $P\subset A$ be an inclusion of unital $C\sp*$-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 $\mc Z$ 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 $\mc{Z}$-absorption, $\mc{Z}$-stability, strict comparison, tracial $m$-comparison, tracial $m$-almost divisibility, and tracial nuclear dimension from $A$ to $P$.