The Rohlin property for coactions of finite dimensional $C^*$-Hopf algebras on unital $C^*$-algebras

Abstract

We shall introduce the approximate representability and the Rohlin property for coactions of a finite dimensional C * -Hopf algebra on a unital C * -algebra and discuss some basic properties of approximately representable coactions and coactions with the Rohlin property of a finite dimensional C * -Hopf algebra on a unital C * -algebra. Also, we shall give an example of an approximately representable coaction of a finite dimensional C * -Hopf algebra on a simple unital C * -algebra which has also the Rohlin property and we shall give the 1-cohomology vanishing theorem for coactions of a finite dimensional C * -Hopf algebra on a unital C * -algebra and the 2-cohomology vanishing theorem for twisted coactions of a finite dimensional C * -Hopf algebra on a unital C * -algebra. Furthermore, we shall introduce the notion of the approximately unitary equivalence of coactions of a finite dimensional C * -Hopf algebra H on a unital C * -algebra A and show that if ρ and σ, coactions of H on a separable unital C * -algebra A, which have the Rohlin property, are approximately unitarily equivalent, then there is an approximately inner automorphism α on A such thatLet ρ be a coaction of H on A and A ρ the fixed point C * -subalgebra of A for ρ, that is,Let E ρ be the canonical conditional expectation from A onto A ρ defined by E ρ (a) = τ · ρ a = (id ⊗ τ )(ρ(a)) for any a ∈ A. We note that E ρ is faithful by [10, Proposition 2.12].Definition 2.3. We say that ρ is saturated if the action of H 0 on A induced by ρ is saturated in the sense of [10].In Sections 4, 5 and 6 of [6], we suppose that the action of H on A is saturated. But, without saturation, all the statements in Sections 4 and 5 and Theorem 6.4 of [6] hold. Hence we obtain the following proposition.Proof. Since the dual coaction of a twisted coaction is saturated, this is immediate by Proposition 2.3. DualityLemma 3.1. With the above notations, V I V * J = 1 ⋊ ρ τ if I = J 0 if I = J. Proof. Let I = (i, j, k) and J = (s, t, r) be any elements in Λ. Then 1 0 )V L = Ψ([a IJ ][b IJ ]) by Lemma 3.1. For any [a IJ ] ∈ M N (A), Ψ([a IJ ])

Model theory and Rokhlin dimension for compact quantum group actions

“…A rigidity result for Rokhlin actions of coexact second countable compact quantum groups, generalizing results for finite and compact from [20,22,29,42] and for finite quantum groups from [33], has been obtained in [2,Theorem 5.10]. In the rest of this section, we observe here that such a result holds for arbitrary (not necessarily coexact) second countable compact quantum groups.…”

The spatial Rokhlin property for actions of compact quantum groups

“…It follows from Proposition 4.8 that α is spatially approximately representable in the sense of Definition 4.7 if and only if it is approximately representable in the sense of Kodaka-Teruya [21,Section 4]. As a consequence, Theorem 4.12 shows that α has the spatial Rokhlin property in the sense of Definition 4.1, if and only if it has the Rokhlin property in the sense of Kodaka-Teruya [21,Section 5]. In particular, our definitions recover Kodaka-Teruya's notions of the Rokhlin property and approximate representability and extend them to the non-unital setting.…”

“…Let G be a finite quantum group and α : A → C(G) ⊗ A an action on a separable, unital C * -algebra. In [21], Kodaka-Teruya introduce and study the Rokhlin property and approximate representability in this setting; in fact they also allow for twisted actions in their paper. It follows from Proposition 4.8 that α is spatially approximately representable in the sense of Definition 4.7 if and only if it is approximately representable in the sense of Kodaka-Teruya [21,Section 4].…”

“…In [21], Kodaka-Teruya introduce and study the Rokhlin property and approximate representability in this setting; in fact they also allow for twisted actions in their paper. It follows from Proposition 4.8 that α is spatially approximately representable in the sense of Definition 4.7 if and only if it is approximately representable in the sense of Kodaka-Teruya [21,Section 4]. As a consequence, Theorem 4.12 shows that α has the spatial Rokhlin property in the sense of Definition 4.1, if and only if it has the Rokhlin property in the sense of Kodaka-Teruya [21,Section 5].…”

“…The Rokhlin property was subsequently generalized by Hirshberg-Winter [14] to the case of compact groups, and studied further by Gardella [9]; see also [11,10]. For finite quantum group actions, Kodaka-Teruya introduced and studied the Rokhlin property and approximate representability in [21]. The established theory, which we shall now briefly summarize in the initial setting of finite group actions, has three particularly remarkable features: The first is a multitude of permanence properties; it is known that many C * -algebraic properties pass from the coefficient C * -algebra to the crossed product and fixed point algebra.…”