ON DUALITIES OF ACTIONS

The paper proves that for a finite abelian group action α on a simple, separable, unital, infinite-dimensional C*-algebra A with A ⋊α G simple, α has the weak tracial Rokhlin property iff its dual α̂ has weak tracial approximate representability, and conversely (Theorem 3.21). The proof factors both weak tracial properties through “tracially sequentially split by order zero” characterizations and then transfers them across crossed products using a general duality theorem (Theorem 3.18), together with Takai-type identifications (Lemmas 3.19–3.20) and the ultrapower order-zero formulations (Corollaries 3.7 and 3.14). The candidate solution follows the same blueprint: it (i) reduces wTRP and wTAR to tracially sequentially split-by-order-zero embeddings, and (ii) uses the duality of such embeddings to shuttle the property across α and α̂. One minor imprecision is the model’s claim that wTAR is equivalent to a first-factor embedding into B ⊗ C*(H); the paper instead characterizes wTAR via the natural embedding ιB: B → B ⋊β H (Corollary 3.14). Aside from this phrasing, the logical steps and dependencies align with the paper’s argument. Key steps are explicitly present in the paper: the wTRP characterization via first-factor embedding (Corollary 3.7), the wTAR characterization via ι (Corollary 3.14), the duality for tracially sequentially split maps (Theorem 3.18), and the canonical identifications used in the diagrams (Lemmas 3.19, 3.20) leading to Theorem 3.21.