The spatial Rokhlin property for actions of compact quantum groups

Abstract

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on $\mathrm{C}^*$-algebras, generalizing the Rokhlin property for both actions of classical compact groups and finite quantum groups. Two key ingredients in our approach are the concept of sequentially split $*$-homomorphisms, and the use of braided tensor products instead of ordinary tensor products. We show that various structure results carry over from the classical theory to this more general setting. In particular, we show that a number of $\mathrm{C}^*$-algebraic properties relevant to the classification program pass from the underlying $\mathrm{C}^*$-algebra of a Rokhlin action to both the crossed product and the fixed point algebra. Towards establishing a classification theory, we show that Rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence. Regarding duality theory, we introduce the notion of spatial approximate representability for actions of discrete quantum groups. The spatial Rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group, and vice versa.Comment: 47 pages; v2 minor corrections. This version is going to appear in J. Funct. Ana

Rokhlin dimension: absorption of model actions

“…, D n ) as the n-fold join D 1 ⋆ · · · ⋆ D n . We can also observe that 2 The reader should keep in mind that an element f in the domain is a continuous function on [0, 1] whose values are in turn (certain) continuous functions from ∆ (n) to the tensor product D1 ⊗max · · · ⊗max Dn+1. this isomorphism is natural in each C * -algebra, and therefore becomes equivariant as soon as we equip each C * -algebra D j with an action α (j) of some group G.…”

“…Further work pushed these techniques to actions of infinite higher-rank groups as well [67,44,60,61,62,40,39]. The case of finite groups was treated in work of Izumi [37,38], where it was shown that such actions with the Rokhlin property have a particularly rigid theory; see also [75,28,22,23,24,1,2]. In contrast to von Neumann algebras, however, the Rokhlin property for actions on C * -algebras has too many obstructions in general, ranging from obvious ones like lack of projections to more subtle ones of K-theoretic nature.…”

Model theory and Rokhlin dimension for compact quantum group actions

“…Building on this work, the notion of Rokhlin property and the above-mentioned preservation results have been generalized to the more general setting of actions of coexact compact quantum groups on C*-algebras in [2]. This work has significantly expanded the scope of the preservation results for Rokhlin actions, paving the way of finding several new examples of classifiable C*-algebras arising from compact quantum group actions.…”

“…This work has significantly expanded the scope of the preservation results for Rokhlin actions, paving the way of finding several new examples of classifiable C*-algebras arising from compact quantum group actions. Furthermore, approach from [2] has also contributed to a simplification and better understanding of the Rokhlin property, even in the classical setting.…”

An Intrinsic Order-Theoretic Characterization of the Weak Expectation Property

“…The model-theoretic notion of positively existentially closed structure, and the related notion of positively existential embedding, have recently found several applications to the study of C*-algebras and C*-dynamics, as in the work of Goldbring and Sinclair on the Kirchberg embedding problem [30] and the works of Barlak and Szabo [7] and Gardella and the author [27] providing a unified approach to several preservation results in C*-dynamics for actions of compact groups with finite Rokhlin dimension. This perspective has also been used to generalize these preservation results to the case of compact quantum groups [8,26].…”