The spatial Rokhlin property for actions of compact quantum groups

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.

We introduce the spatial Rokhlin property for actions of coexact compact quantum groups on 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 C * -algebraic properties relevant to the classification program pass from the underlying 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

Introduction
The Rokhlin property for finite group actions on unital C * -algebras was introduced and studied by Izumi in [15,16], building on earlier work of Herman-Jones [12] and Herman-Ocneanu [13]. Since the very beginning it has proven to be a useful tool in the theory of finite group actions. 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. This was in part addressed by Izumi in [15], and studied more in depth by Osaka-Phillips [27] and Santiago [30]. The second feature is rigidity with respect to approximate unitary equivalence; a result of Izumi [15] asserts that two Rokhlin actions of a finite group on a separable, unital C * -algebra are conjugate via an approximately inner automorphism if and only if the two actions are pointwise approximately unitarily equivalent; see [11] for the nonunital case and [10] for the case of compact groups. As demonstrated by Izumi in [15], this rigidity property is useful for classifying Rokhlin actions on classifiable C * -algebras via K-theoretic invariants. The third feature is duality theory; a result of Izumi [15] shows that an action of a finite abelian group on a separable, unital C * -algebra has the Rokhlin property if and only if the dual action is approximately representable, and vice versa. This has been generalized to the non-unital case by Nawata [25], and to actions of compact abelian groups by the first two authors [4]; see also [8].
In the present paper, we introduce and study the spatial Rokhlin property for actions of coexact compact quantum groups, generalizing and unifying the work mentioned above. In particular, we carry over various structure results from the classical to the general case. Firstly, this allows us to remove all commutativity assumptions in the study of duality properties for Rokhlin actions. This is relevant even for classical group actions. Indeed, the Pontrjagin dual of a nonabelian group is no longer a group, but can be viewed as a quantum group. Accordingly, a natural way to fully incorporate nonabelian groups into the picture is to work in the setting of quantum groups from the very beginning. Secondly, it turns out that some results can be given quite short and transparent proofs in this more abstract setup, simpler than in previous accounts. Finally, our results are also of interest from the point of view of quantum group theory. Indeed, they provide examples of quantum group actions that either allow for classification, or the systematical analysis of structural properties of crossed product C * -algebras, in particular whether they fall within the scope of the Elliott program.
Let us highlight two comparably new ingredients in our approach. The first is the notion of (equivariantly) sequentially split * -homomorphisms introduced by the first two authors in [4]. It has already been demonstrated in [4] that many structural results related to the Rokhlin property can be recast and conceptually proved in the language of sequentially split * -homomorphisms, and some new ones could be proved as well. The second ingredient is a purely quantum feature, namely the braided tensor product construction. This provides the correct substitute for tensor product actions in the classical theory, and it also gives a conceptual explanation of the fact that the central sequence algebra is no longer the right tool in the quantum setting. Being widely known in the algebraic theory of quantum groups, braided tensor products in the operator algebraic framework were first introduced and studied in [26].
As already indicated above, for most of the paper we will assume that our quantum groups satisfy exactness/coexactness assumptions. This may appear surprising at first sight; it is essentially due to the fact that we have chosen to work in a reduced setting, that is, with reduced crossed products and minimal (braided) tensor products. We shall indicate at several points in the main text where precisely exactness enters. Our setup yields the strongest versions of conceivable definitions of Rokhlin actions and approximately representable actions, however, at the same time our examples are restricted to the amenable/coamenable case. On the other hand, the reduced setting matches best with the existing literature on quantum group actions, see for instance [2,3,31,26] and references therein. The necessary modifications to set up a full version of our theory are mainly of technical nature; we have refrained from carrying this out here.
Let us now explain how the paper is organized. In Section 1, we gather some preliminaries and background on quantum groups, including a review of Takesaki-Takai duality and braided tensor products. Section 2 deals with induced actions of discrete and compact quantum groups on sequence algebras. Already for classical compact groups these actions typically fail to be continuous, and in the quantum setting this leads to a number of subtle issues. In Section 3, we define and study equivariantly sequentially split * -homomorphisms. We show that, as in the case of group actions, this notion behaves well with respect to crossed products and fixed point algebras. We also establish a general duality result for equivariantly sequentially split * -homomorphisms. In Section 4, we introduce the spatial Rokhlin property for actions of coexact compact quantum groups, and spatial approximate representability for actions of exact discrete quantum groups. We verify that various C * -algebraic properties pass to crossed products and fixed point algebras. Moreover, we show that the spatial Rokhlin property and spatial approximate representability are dual to each other. In Section 5, we present some steps towards a classification theory for actions with the spatial Rokhlin property. Among other things, we prove that two such G-actions on a C * -algebra A are conjugate via an approximately inner automorphism if and only if the actions are approximately unitarily equivalent as * -homomorphisms from A to C r (G) ⊗ A. This generalizes a number of previous such classification results, in particular those of Izumi [15], Gardella-Santiago [10] and Kodaka-Teruya [21]. In this section, we also generalize a K-theory formula for the fixedpoint algebra of a Rokhlin action, first proved for certain finite group cases by Izumi [15] and for compact group actions by the first two authors in [4]. Finally, in Section 6 we present some examples of Rokhlin actions. In particular, we show that any coamenable compact quantum group admits an essentially unique action with the spatial Rokhlin property on O 2 .
The work presented here was initiated while the first two authors participated in the conference CSTAR at the University of Glasgow in September 2014. Substantial parts of this work were carried out during research visits of the authors to Oberwolfach in June 2015, of the first two authors at the Mittag-Leffler Institute from January to March 2016, and of the third author to the University of Southern Denmark in April 2016. The authors are grateful to all these institutions for their hospitality and support.
The first author was supported by GIF Grant 1137-30.6/2011, ERC AdG 267079, SFB 878 'Groups, Geometry, and Actions' and the Villum Fonden project grant 'Local and global structures of groups and their algebras' (2014-2018), 7423. The second author was supported by SFB 878 'Groups, Geometry, and Actions' and the Engineering and Physical Sciences Research Council Grant EP/N00874X/1 'Regularity and dimension for C * -algebras'. The third author was supported by the Engineering and Physical Sciences Research Council Grant EP/L013916/1 and the Polish National Science Centre grant no. 2012/06/M/ST1/00169.
The authors would like to thank the referee for a thorough reading of the manuscript and for helpful suggestions.

Preliminaries
In this preliminary section we collect some definitions and results from the theory of quantum groups and fix our notation. We will mainly follow the conventions in [26] as far as general quantum group theory is concerned. For more detailed information and background we refer to [34,24,22,23].
Let us make some general remarks on the notation used throughout the paper. We write L(E) for the space of adjointable operators on a Hilbert A-module, and K(E) denotes the space of compact operators. The closed linear span of a subset X of a Banach space is denoted by [X]. If x, y are elements of a Banach space and ε > 0 we write Depending on the context, the symbol ⊗ denotes either the tensor product of Hilbert spaces, the minimal tensor product of C * -algebras, or the tensor product of von Neumann algebras. We write for algebraic tensor products.
If A and B are C * -algebras then the flip map If H is a Hilbert space we write Σ ∈ L(H ⊗ H) for the flip map Σ(ξ ⊗ η) = η ⊗ ξ. For operators on multiple tensor products we use the leg numbering notation. For instance, If B is a C * -algebra we write B for the smallest unitarization of B.

Quantum groups
Although we will only be interested in compact and discrete quantum groups, let us first recall a few definitions and facts regarding general locally compact quantum groups.
Let ϕ be a normal, semifinite and faithful weight on a von Neumann algebra M . We use the standard notation and write M + * for the space of positive normal linear functionals on M . Assume that (1) for all x ∈ M + ϕ and ω ∈ M + * . Similarly one defines the notion of a right invariant weight.
Definition 1.1. A locally compact quantum group G is given by a von Neumann algebra L ∞ (G) together with a normal unital * -homomorphism Δ : and normal semifinite faithful weights ϕ and ψ on L ∞ (G) which are left and right invariant, respectively. The weights ϕ and ψ will also be referred to as Haar weights of G.
Remark 1.2. Our notation should help to make clear how ordinary locally compact groups can be viewed as quantum groups. Indeed, if G is a locally compact group, then the algebra L ∞ (G) of essentially bounded measurable functions on G together with the comultiplication Δ : defines a locally compact quantum group. The weights ϕ and ψ are given in this case by left and right Haar measures, respectively. Of course, for a general locally compact quantum group G the notation L ∞ (G) is purely formal. Similar remarks apply to the C * -algebras C * f (G), C * r (G) and C f 0 (G), C r 0 (G) associated to G that we discuss below. It is convenient to view all of them as different appearances of the quantum group G. [22]). Let G be a locally compact quantum group and let Λ : N ϕ → L 2 (G) be the GNS-construction for the Haar weight ϕ. Throughout the paper we will only consider second countable quantum groups, that is, quantum groups for which L 2 (G) is a separable Hilbert space.
One obtains a unitary W G = W on L 2 (G) ⊗ L 2 (G) by for all x, y ∈ N ϕ . This unitary is multiplicative, which means that W satisfies the pentagonal equation From W one can recover the von Neumann algebra L ∞ (G) as the strong closure of the algebra (id ⊗L(L 2 (G)) * )(W ), where L(L 2 (G)) * denotes the space of normal linear functionals on L(L 2 (G)). Moreover one has for all x ∈ M . The algebra L ∞ (G) has an antipode, which is an unbounded, σ-strong * closed linear map S given by S(id ⊗ω)(W ) = (id ⊗ω)(W * ) for ω ∈ L(L 2 (G)) * . Moreover, there is a polar decomposition S = Rτ −i/2 where R is an antiautomorphism of L ∞ (G) called the unitary antipode and (τ t ) is a strongly continuous one-parameter group of automorphisms of L ∞ (G) called the scaling group. The unitary antipode satisfies σ • The group-von Neumann algebra L(G) of the quantum group G is the strong closure of the algebra (L(L 2 (G)) * ⊗ id)(W ) with the comultiplication Δ : where Ŵ = ΣW * Σ and Σ ∈ L(L 2 (G) ⊗ L 2 (G)) is the flip map. It defines a locally compact quantum group Ĝ which is called the dual of G. The left invariant weight φ for the dual quantum group has a GNS-construction Λ : Nφ → L 2 (G), and according to our conventions we have L(G) = L ∞ (Ĝ). Remark 1.4. Since we are following the conventions of Kustermans and Vaes [22], there is a flip map built into the definition of Δ . As we will see below, this is a natural choice when working with Yetter-Drinfeld actions; however, it is slightly inconvenient when it comes to Takesaki-Takai duality. We will write Ǧ for the quantum group corresponding to L(G) cop . That is, L ∞ (Ǧ) is the von Neumann algebra L(G) equipped with the opposite comultiplication Δ =Δ cop = σ •Δ, where σ denotes the flip map. By slight abuse of terminology, we shall refer to both Ǧ and Ĝ as the dual of G. According to Pontrjagin duality, the double dual of G in either of these conventions is canonically isomorphic to G. Remark 1.5 (cf. [22]). The modular conjugations of the left Haar weights ϕ and φ are denoted by J and Ĵ , respectively. These operators implement the unitary antipodes in the sense that R(x) =Ĵx * Ĵ ,R(y) = Jy * J for x ∈ L ∞ (G) and y ∈ L(G). Note that L ∞ (G) = JL ∞ (G)J and L(G) =ĴL(G)Ĵ for the commutants of L ∞ (G) and L(G). Using J and Ĵ one obtains multiplicative unitaries for f ∈ L ∞ (G) and x ∈ L(G). We also record the formula We will mainly work with the C * -algebras associated to the locally compact quantum group G. The algebra [(id ⊗L(L 2 (G)) * )(W )] is a strongly dense C * -subalgebra of L ∞ (G) which we denote by C r 0 (G). Dually, the algebra [(L(L 2 (G)) * ⊗id)(W )] is a strongly dense C * -subalgebra of L(G) which we denote by C * r (G). These algebras are the reduced algebra of continuous functions vanishing at infinity on G and the reduced group C * -algebra of G, respectively. One has W ∈ M (C r 0 (G) ⊗ C * r (G)). Restriction of the comultiplications on L ∞ (G) and L(G) turns C r 0 (G) and C * r (G) into Hopf C * -algebras in the following sense. Definition 1.6. A Hopf C * -algebra is a C * -algebra H together with an injective nondegenerate * -homomorphism Δ : Definition 1.7. A compact quantum group is a locally compact quantum group G such that C r 0 (G) is unital. Similarly, a discrete quantum group is a locally compact quantum group G such that C * r (G) is unital.
We will write C r (G) instead of C r 0 (G) if G is a compact quantum group. A finite quantum group is a compact quantum group G such that C r (G) is finite dimensional. This is the case if and only if C * r (G) is finite dimensional.
A unitary representation of a locally compact quantum group G is a unitary corepresentation of C r 0 (G). Remark 1.11. Let G be a locally compact quantum group. The full group C * -algebra of G is a Hopf C * -algebra C * f (G) together with a unitary representation W ∈ M (C r 0 (G) ⊗ C * f (G)) satisfying the following universal property: for every unitary representation Similarly, one obtains the full C * -algebra C f 0 (G) of functions on G.
Definition 1.12. Let G be a locally compact quantum group. We say that G is amenable if the canonical quotient map If G is coamenable we will simply write C 0 (G) for C r 0 (G). By slight abuse of notation, we will also write C 0 (G) if a statement holds for both C f 0 (G) and C r 0 (G). In particular, if G is compact and coamenable we will simply write C(G) instead C r (G). Similarly, if G is amenable we will write C * (G) for C * r (G). We remark that every compact quantum group is amenable, and equivalently every discrete quantum group is coamenable. Remark 1.13 (cf. [34,24]). Let G be a compact quantum group. In analogy with the theory for compact groups, every unitary representation of G is completely reducible, and all irreducible representations are finite dimensional. We write Irr(G) for the set of equivalence classes of irreducible unitary representations of G. Our general second countability assumption amounts to saying that the set Irr(G) is countable.
A unitary representation of G on a finite dimensional Hilbert space H λ is given by a unitary u λ ∈ C r (G) ⊗ K(H λ ), so it can be viewed as an element u λ = (u λ ij ) ∈ M n (C r (G)) if dim(H λ ) = n. Moreover, the corepresentation identity translates into the formula for the comultiplication of the matrix coefficients u λ ij . By Peter-Weyl theory, the linear span of all matrix coefficients u λ ij for λ ∈ Irr(G) forms a dense * -subalgebra O(G) ⊂ C r (G). In fact, together with the counit : O(G) → C given by We will use the Sweedler notation Δ(f ) = f (1) ⊗ f (2) for the comultiplication of general elements of O(G). This is useful for bookkeeping of coproducts, let us emphasize however that this notation is not meant to say that Δ(f ) is a simple tensor. For higher coproducts one introduces further indices, for instance f (1) ⊗f (2) Again by Peter-Weyl theory, one obtains a vector space basis of O(G) consisting of the matrix coefficients u λ ij where λ ranges over Irr(G) and 1 ≤ i, j ≤ dim(λ). Here we abbreviate dim(λ) = dim(H λ ). If we write O(G) λ for the linear span of the elements u λ ij for 1 ≤ i, j ≤ dim(λ), we have a direct sum decomposition Note that the coproduct of O(G) takes a particularly simple form in this picture; from the above formula for Δ(u λ ij ) we see that it looks like the transpose of matrix multiplication. Let u λ be an irreducible unitary representation of G, and let u λ ij be the corresponding matrix elements in some fixed basis. The contragredient representation u λ c is given by the In general u λ c is not unitary, but as any finite dimensional representation of G, it is unitarizable. The representations u λ and u λ cc are equivalent, and there exists a unique positive invertible intertwiner F λ : H λ → H λ cc satisfying tr(F λ ) = tr(F −1 λ ). The trace of F λ is called the quantum dimension of H λ and denoted by dim q (λ).
With this notation, the Schur orthogonality relations are where λ, η ∈ Irr(G) and ϕ : C r (G) → C is the Haar state of G. In the sequel we shall fix bases such that F λ is a diagonal operator for all λ ∈ Irr(G).
Remark 1.14. Let again G be a compact quantum group. While the comultiplication for the C * -algebra C r (G) looks particularly simple in terms of matrix coefficients, dually the multiplication in the C * -algebra C * (G) is easy to describe. More precisely, we have where the right hand side denotes the c 0 -direct sum of the matrix algebras K(H λ ). If u λ ij are the matrix coefficients in O(G), then the dual basis vectors ω λ ij , that is, the linear functionals on O(G) given by form naturally a vector space basis of matrix units for the algebraic direct sum inside C * r (G). Let us note that D(G) should not be confused with a quantum double, but there will be no conflicting notation in this regard appearing in this paper.
Let us also note that according to the Schur orthogonality relations, see Remark 1.13, the functionals ω λ ij extend continuously to bounded linear functionals on C r (G).

Actions, crossed products and Takesaki-Takai duality
Let us now consider actions on C * -algebras. An action of a locally compact quantum group G on a C * -algebra A is a coaction α : We will also say that (A, α) is a G-C * -algebra in this case. A * -homomorphism ϕ : If H is a Hopf C * -algebra and α : A → M (H ⊗ A) a coaction, then the density condition [α(A)(H ⊗ 1)] = H ⊗ A implies in particular that the image of α is contained in the H-relative multiplier algebra of H ⊗ A, defined by In contrast to the situation for ordinary multiplier algebras, the tensor product map id ⊗ϕ : H ⊗ A → H ⊗ B of a (possibly degenerate) * -homomorphism ϕ : A → B admits a unique extension M H (H ⊗ A) → M H (H ⊗ B) to the relative multiplier algebra, which will again be denoted by id ⊗ϕ. If ϕ is injective, then this also holds for id ⊗ϕ : . We refer to [6, Appendix A] for further details on relative multiplier algebras. Remark 1. 16. For the construction of examples of Rokhlin actions we shall consider inductive limit actions of Hopf C * -algebras. Assume H is a Hopf C * -algebra and that A 1 → A 2 → · · · is an inductive system of H-C * -algebras with coactions α j : A j → M H (H ⊗ A j ) and injective equivariant connecting maps. Then the direct limit A = lim − − → A j becomes an H-C * -algebra in a canonical way. Firstly, we have lim Remark 1.17 (cf. [28]). Let G be a compact quantum group. Further below we will use the fact that any G-C * -algebra A admits a spectral decomposition. In order to discuss this we review some further definitions and results.
Let G be a compact quantum group and let (A, α) be a G-C * -algebra. Since G is compact, the coaction is an injective * -homomorphism α : be the λ-spectral subspace of A. Here we recall that O(G) λ ⊂ O(G) denotes the span of the matrix coefficients for H λ , see Remark 1.13.
The subspace A λ is closed in A, and there is a projection operator p λ : By definition, the spectral subalgebra S(A) ⊂ A is the * -subalgebra From the Schur orthogonality relations and For ω ∈ D(G) and a ∈ A let us define Then the Schur orthogonality relations imply that A ·D(G), the linear span of all elements a · ω as above, is equal to S(A). Moreover, from the coaction property of α it follows that A becomes a right D(G)-module in this way.
Definition 1.18. Let G be a locally compact quantum group and let A be a G-C * algebra with coaction α : Recall here that C r 0 (Ǧ) = C * r (G) as a C * -algebra, but equipped with the opposite comultiplication.
The reduced crossed product is equipped with a canonical dual action of Ǧ , which turns it into a Ǧ -C * -algebra. More precisely, the dual action is given by comultiplication on the copy of C r 0 (Ǧ) and the trivial action on the copy of A inside M (G α,r A). By our second countability assumption, the crossed product G α,r A is separable provided A is separable.
For the purpose of reduced duality, we have to restrict ourselves to regular locally compact quantum groups [2]. All compact and discrete quantum groups are regular, so this is not an obstacle for the constructions we are interested in further below. Remark 1.19. If G is a regular locally compact quantum group then the regular representation of L 2 (G) induces an action of G on the algebra K(L 2 (G)) of compact operators by conjugation. More generally, if A is any G-C * -algebra we can turn the tensor product K(L 2 (G)) ⊗ A into a G-C * -algebra by equipping K(L 2 (G)) ⊗ A ∼ = K(L 2 (G) ⊗ A) with the conjugation action arising from the tensor product representation of G on the Hilbert A-module E = L 2 (G) ⊗ A. Explicitly, following the notation in [26], we consider where X = ΣV Σ and V is as in Remark 1.5. This determines a coaction λ : , which in turn corresponds to a unitary corepresentation The conjugation action Ad λ : and K(E) becomes a G-C * -algebra in this way. Under the isomorphism K( If G is a classical group then the resulting action is nothing but the diagonal action on the tensor product K(L 2 (G)) ⊗ A. For further details we refer to [1,26].
Let us now state the Takesaki-Takai duality theorem for regular locally compact quantum groups [2], see [31, chapter 9] for a detailed exposition. Theorem 1.20. Let G be a regular locally compact quantum group and let A be a G-C * -algebra. Then there is a natural isomorphism Let us give a brief sketch of the proof of Theorem 1.20 for the sake of convenience. Recall from Remark 1.5 that J is the modular conjugation of the left Haar weight of L ∞ (G), and similarly Ĵ the modular conjugation of the left Haar weight of L(G). We shall write U = JĴ.
Conjugating by W * 12 and using that C r 0 (G) and JC r 0 (G)J commute, this is isomorphic to since α is injective. Moreover, [UC r 0 (G)UC * r (G)] identifies with K(L 2 (G)). Hence the right hand side is isomorphic to K(L 2 (G)) ⊗ A, taking into account that Let us also identify the bidual coaction. By construction, the bidual coaction α maps and leaves C * r (G) ⊗ 1 and α(A) fixed. Using the relations from Remark 1.5 one can show At some points we will also need the full crossed product G α,f A of a G-C * -algebra (A, α); we refer to [26] for a review of its definition in terms of its universal property for covariant representations.

Braided tensor products
Finally, let us discuss Yetter-Drinfeld actions and braided tensor products. We refer to [26] for more information. Here The braided tensor product, which we review next, generalizes at the same time the minimal tensor product of C * -algebras and reduced crossed products. Roughly speaking, it allows one to construct a new C * -algebra out of two constituent C * -algebras, together with some prescribed commutation relations between the two factors.
The braided tensor product A B becomes a G-C * -algebra with coaction α β : Example 1.23. A basic example of a G-YD-action is given by the C * -algebra A = C r 0 (G) for a regular locally compact quantum group G, equipped with α = Δ and the adjoint action We will mainly be interested in the special case of the braided tensor product construction where the first factor is equal to C r 0 (G) with the Yetter-Drinfeld structure from Example 1.23.

Lemma 1.24. Let G be a regular locally compact quantum group. For any
Proof. The map T α is obtained from the identifications where we use Ŵ = ΣW * Σ in the third step and the fact that α is injective in the penultimate step.
In particular, the canonical action of G on C r 0 (G) A corresponds to the translation action on the first tensor factor in C r 0 (G) ⊗ A under this isomorphism. 2

Induced actions on sequence algebras
The theory of Rokhlin actions for compact quantum groups relies on the possibility of obtaining induced actions on the level of sequence algebras. In this section we shall recall a few facts on sequence algebras, and then discuss the construction of induced actions, separately for the case of discrete and compact quantum groups.

Sequence algebras
Let us first recall some notions related to sequence algebras, see [4] and [17]. If A is a C * -algebra we write ∞ (A) = {(a n ) n | a n ∈ A and sup n∈N a n < ∞} for the C * -algebra of bounded sequences with coefficients in A. Moreover denote by the closed two-sided ideal of sequences converging to zero.
Given a bounded sequence (a n ) n∈N ∈ ∞ (A), the norm of the corresponding element in A ∞ is given by [(a n ) n∈N ] = lim sup n→∞ a n .
Note moreover that A embeds canonically into A ∞ as (representatives of) constant sequences. We will frequently use this identification of A inside A ∞ in the sequel.
We remark that Ann ∞ (A) sits inside N (D ∞,A , A ∞ ) as a closed two-sided ideal.
The multiplier algebra of D ∞,A admits the following alternative description. [4,Proposition 1.5(1)] and [17, 1.9(4) given by the universal property of the multiplier algebra is surjective, and its kernel coincides with Ann ∞ (A).
Let us also note that the construction of D ∞,A is compatible with tensoring by the compacts.
Lemma 2.4 (cf. [4, 1.6]). Let A be a C * -algebra and let K(H) be the algebra of compact operators on a separable Hilbert space H. The canonical embedding Given a C * -algebra equipped with an action of a quantum group G, we shall now discuss how to obtain induced actions on the sequence algebras introduced above.

Induced actions -discrete case
In the case of discrete quantum groups the situation is relatively simple. In fact, if G is a discrete quantum group then the C * -algebra C 0 (G) of functions on G is a C * -direct sum of matrix algebras. Explicitly, it is of the form where Λ = Irr(Ǧ) is the set of equivalence classes of irreducible representations of the dual compact quantum group, see Remark 1.13.
If (A, α) is a G-C * -algebra, then one of the defining conditions for the coaction α : With the notation as above, we have that is, we can identify the relative multiplier algebra with the ∞ -product of the algebras K(H λ ) ⊗ A. In other words, we have It follows that applying α componentwise induces a * -homomorphism α ∞ : Proof. Injectivity and coassociativity of α ∞ follow immediately from the corresponding properties of α. For the density condition for all finite rank (central) projections p ∈ C 0 (G). This in turn follows from the density condition for α, combined with fact that tensoring with finite dimensional algebras commutes with taking direct products. 2 The map α ∞ constructed above induces an injective * -homomorphism α ∞ : Coassociativity and the density conditions for α ∞ are inherited from α ∞ . We therefore obtain the following result.

Induced actions -compact case
Whereas for discrete quantum groups the extension of actions to sequence algebras always yields genuine actions, the situation for compact quantum groups is more subtle. Already classically, a strongly continuous action of a compact group G on a C * -algebra A induces an action on ∞ (A) and A ∞ , but these induced actions typically fail to be strongly continuous, compare [4].
We shall address the corresponding problems in the quantum case by using an adhoc notion of equivariant * -homomorphisms into sequence algebras. Our discussion also requires the technical assumption of coexactness.
Definition 2.7. A locally compact quantum group G is exact if the functor of taking reduced crossed products by G is exact. We say that G is coexact if the dual Ĝ is exact.
It is well-known that a discrete quantum group G is exact if and only if C * r (G) is an exact C * -algebra, see [33, 1.28]. In other words, a compact quantum group G is coexact if and only if C r (G) is an exact C * -algebra.
Remark 2.8. Let A and B be C * -algebras. If B is exact, then there exists a canonical injective * -homomorphism B⊗A ∞ −→ (B⊗A) ∞ coming from the following commutative diagram with exact rows obtained by applying α componentwise, and also a * -homomorphism The maps α ∞ and α ∞ , despite not being coactions in the sense of Definition 1.15 in general, turn out to be good enough to obtain a tractable notion of equivariance and suitable crossed products, at least when G is coexact. The reason for this is Remark 2.8, which is used in the definition below. Definition 2.9. Let G be a coexact compact quantum group and let (A, α) and (B, β) be Remark 2.10. As indicated above, if G is a compact group and α : G A is a strongly continuous action on a C * -algebra, there always exists a (not necessarily strongly continuous) induced action of G on A ∞ . If (B, β) is another G-algebra, then it is easy to see that any * -homomorphism ϕ : A → B ∞ is G-equivariant in the sense 2.9 if and only if it is G-equivariant in the usual sense.
Indeed, the equality (id ⊗ϕ) • α = β ∞ • ϕ clearly implies equivariance the usual sense. For the converse implication, one notes that if ϕ is equivariant in the usual sense, it maps A automatically into the continuous part of the action on B ∞ . Therefore, for a ∈ A the equality of (id ⊗ϕ) • α(a) = β ∞ • ϕ(a) in C(G) ⊗ B ∞ = C(G, B ∞ ) can be checked by evaluating both sides at the points of G.
In the quantum setting, we need a substitute of the continuous part of an action in order to define crossed products. We shall rely on the structure of compact quantum groups to obtain a construction suitable for the situation at hand.
Recall from Remark 1.13 that the dense Hopf * -algebra O(G) ⊂ C r (G) has a linear basis of elements of the form u λ ij where λ ∈ Irr(G) and 1 ≤ i, j, ≤ dim(λ). As explained in Remark 1.14, the linear functionals ω λ ij ∈ C r (G) * given by , which can be viewed as the dense * -subalgebra of C 0 (Ĝ) given by the algebraic direct sum of matrix algebras K(H λ ) for λ ∈ Irr(G). Moreover, in this picture the elements ω λ ij are matrix units in K(H λ ), that is, It is crucial for our purposes that such module structures also exist on ∞ (A) and A ∞ . Indeed, note that applying ω ⊗ id in each component we obtain slice which, by slight abuse of notation, will again be denoted by ω ⊗ id in the sequel. We will also continue to use the notation a · ω for the module structures obtained in this way.
In analogy with the constructions in Remark 1.17 we shall now define spectral subspaces of ∞ (A) and A ∞ , and use this to define corresponding continuous parts. Although the settings differ somewhat, this is similar to Kishimoto's definition of equicontinuous sequences for flows, cf. [20,19].
Definition 2.11. Let G be a coexact compact quantum group and let (A, α) be a G-C * -algebra. The spectral subspaces of ∞ (A) and A ∞ with respect to α are defined by respectively. The continuous parts of ∞ (A) and A ∞ with respect to α are defined by respectively.
At this point it is not immediately obvious that the subspaces in Definition 2.11 are closed under multiplication. We will show this further below.
Proof. Let x ∈ S(A ∞ ). Then we can write x = x · p = (p ⊗ id)α ∞ (x) for some finite rank idempotent p ∈ D(G). If x ∈ ∞ (A) is any lift of x, then x · p is a lift of x as well, which in addition is contained in S( ∞ (A)). 2 It follows from Lemma 2.12 that the canonical map ∞,α (A) → A ∞,α is surjective.
Proposition 2.13. Let G be a coexact compact quantum group and let (A, α) be a G-C * -algebra. Then we have Moreover, both ∞,α (A) and A ∞,α are G-C * -algebras in a canonical way.
Proof. Let us consider first the assertions for S( ∞ (A)). By construction, for an element for some finite set F ⊂ Irr(G) and elements x λ ij ∈ ∞ (A). By the definition of the counit : O(G) → C, we see that applying ω = λ∈F ii ∈ D(G) ⊂ C r (G) * in the first tensor factor gives Conversely, write x ∈ S( ∞ (A)) as a finite sum x = i y i · ω i of some elements ω i ∈ D(G) and y i ∈ ∞ (A). We may assume without loss of generality that each ω i is contained in K(H λ i ) for some λ i ∈ Irr(G). Let F ⊂ Irr(G) be the finite subset consisting of all λ j . Writing x = (x n ) n∈N and y = (y n ) n∈N this means that each Here we use that the construction of ∞ -products is compatible with tensoring by finite dimensional spaces. We conclude S( The corresponding assertion for S(A ∞ ) is obtained in a similar way. According to Lemma 2.12, we know that x ∈ S(A ∞ ) is represented by an element x ∈ S( ∞ (A)), so the above argument shows As a consequence of these considerations we obtain in particular that S( ∞ (A)) and S(A ∞ ) are * -algebras, and hence ∞,α (A) ⊂ ∞ (A) and A ∞,α ⊂ A ∞ are C * -subalgebras.
It remains to show that these C * -algebras are G-C * -algebras in a canonical way, with coactions induced by α ∞ and α ∞ , respectively.
Let us again first consider the case ∞,α (A). From coassociativity of α we obtain that α ∞ maps S( ∞ (A)) to O(G) S( ∞ (A)). Therefore it induces a * -homomorphism , which we will again denote by α ∞ .
Injectivity of the latter map is clear. Similarly, the coaction identity (id ⊗α ∞ )α ∞ = (Δ ⊗id)α ∞ follows immediately from the coaction identity for α. For the density condition note that we can write A)), using the Hopf algebra structure of O(G), and the Sweedler notation The case A ∞,α is analogous. The considerations for S( ∞ (A)) above and Lemma 2.12 which we will again denote by α ∞ . Coassociativity and density conditions are inherited from the corresponding properties of the coaction Using Proposition 2.13 we obtain an alternative way to describe the notion of equivariance introduced in Definition 2.9 Proposition 2.14. Let G be a coexact compact quantum group and let (A, α) and (B, β) be G-C * -algebras. For a * -homomorphism ϕ : A → B ∞ , the following are equivalent: b) ⇒ c) : As ϕ(a · ω) = ϕ(a) · ω for all ω ∈ D(G) and a ∈ A, it follows that ϕ(S(A)) ⊂ S(B ∞ ). Hence, β ∞ • ϕ maps S(A) into O(G) S(B ∞ ). For a ∈ S(A) and ω ∈ D(G) we therefore compute It is now straightforward to check that β ∞ • ϕ(a) = (id ⊗ϕ) • α(a) for all a ∈ S(A). As This implication follows immediately from the definitions. 2 We shall now define crossed products of induced actions on sequence algebras.
Definition 2.15. Let G be a coexact compact quantum group and let (A, α) be a G-C * -algebra. We define In a similar way we define Remark 2.16. The notation introduced in Definition 2.15 will allow us to unify our exposition of several results in subsequent sections. Remark that the crossed products G α ∞ ,r A ∞ and G α ∞ ,r ∞ (A) carry honest Ǧ -C * -algebra structures given by the dual actions.
At a few points we will need a notion of equivariance for * -homomorphisms with target D ∞,B or M (D ∞,B ).
Let G be a coexact compact quantum group and (B, β) be a G-C * -algebra. Note that nondegeneracy of the * -homomorphism β : is a nondegenerate C * -subalgebra. Hence β ∞ induces a * -homomorphism which we will again denote by β ∞ .

Equivariantly sequentially split * -homomorphisms
In this section we discuss the notion of sequentially split * -homomorphisms between G-C * -algebras, which was studied in [4] in the case of actions by groups.  1 (cf. [4, 2.1, 3.3]). Let G be a quantum group which is either discrete or compact and coexact. Moreover let (A, α), (B, β) be G-C * -algebras. We say that an equivariant * -homomorphism ϕ : (A, α) → (B, β) is equivariantly sequentially split if there exists a commutative diagram of G-equivariant * -homomorphisms of the form where the horizontal map is the standard embedding. If ψ : (B, β) → (A ∞ , α ∞ ) is an equivariant * -homomorphism fitting into the above diagram, then we say that ψ is an equivariant approximate left-inverse for ϕ.
An important feature of the theory of sequentially split * -homomorphisms is that it is compatible with forming crossed product C * -algebras. The proof makes use of the following fact.

compatible with the natural inclusions of G α,r A on both sides.
Proof. Assume first that G is discrete and exact. Since taking reduced crossed products with G is exact, the canonical map G α ∞ ,r Here

It is clear from the construction that the * -homomorphism G α ∞ ,r A ∞ → (G α,r A) ∞ is Ǧ -equivariant and compatible with the canonical inclusions of G α,r A.
Assume now that G is compact and coexact. Let us abbreviate K = K(L 2 (G)). Then the canonical map K ⊗ ∞ (A) → ∞ (K ⊗ A) induces the following commutative diagram with exact rows The middle vertical arrow restricts to a * -homomorphism This map is clearly compatible with the canonical embeddings of G α,r A. Moreover, This finishes the proof. 2 Then the induced * -homomorphism G r ϕ : G α,r A → G β,r B between the crossed products is Ǧ -equivariantly sequentially split.
Proof. Let a ∈ (A ∞,α ) α ∞ be represented by (a n ) n∈N ∈ ∞ (A). Then lim n→∞ α(a n ) − 1 ⊗ a n = 0 by the fixed point condition. Applying the Haar state ϕ : C r (G) → C in the first tensor factor gives lim n→∞ (ϕ ⊗ id) • α(a n ) − a n = 0 As we show next, a naturality property as in Proposition 3.3 also holds for fixed point algebras of actions of compact quantum groups. Then the induced * -homomorphism ϕ : A α → B β is a sequentially split.
Proof. Let ψ : (B, β) → (A ∞ , α ∞ ) be an equivariant approximate left-inverse for ϕ. By equivariance of ψ, we have According to Lemma 3.5 the latter identifies with (A α ) ∞ , and therefore ψ : The following stability result is an important feature for the theory of sequentially split * -homomorphisms.
Proof. Let us first consider the case that G is compact and coexact.
We shall write Ψ B for the restriction of Ψ to is a * -homomorphism whose image is contained in the relative commutant of K(L 2 (G)) ⊗ 1. According to [4, 1.8], its image im(Ψ B ) is therefore contained in 1 ⊗ M (D ∞,A ). Using again nondegeneracy of ϕ we see that im(Ψ B ) is in fact contained in 1 ⊗ D ∞,A . From these observations and the sequential split property we conclude that Ψ can be written in the form Ψ = id K(L 2 (G)) ⊗ψ for a non-degenerate * -homomorphism ψ : B → D ∞,A . It is easy to check that ψ is an approximate left-inverse for ϕ.
We claim that ψ : B → D ∞,A is G-equivariant. For this consider a simple tensor T ⊗ b ∈ K(L 2 (G)) ⊗ B and compute Here all expressions are viewed as elements of (C r (G) ⊗ K(L 2 (G)) ⊗ A) ∞ . Equivariance of Ψ means that the above expressions are equal. We conclude (id ⊗ψ)β(b) = α ∞ (ψ(b)) for all b ∈ B as desired.
In the case that G is discrete and exact we can follow the above arguments almost word by word, in this case the situation is even slightly easier since all algebras involved are honest G-C * -algebras. 2 Proposition 3.8. Let G be a quantum group which is either compact and coexact or discrete and exact. Moreover assume that (A, α), (B, β) are separable G-C * -algebras, and let ϕ : (A, α) → (B, β) be a non-degenerate equivariant * -homomorphism. ,r B,β) is Ǧ -equivariantly sequentially split.

Then ϕ is G-equivariantly sequentially split if and only if
Proof. If ϕ is G-equivariantly sequentially split, then Proposition 3.3 shows that φ is Ǧ -equivariantly sequentially split. On the other hand, if φ is Ǧ -equivariantly sequentially split, then Proposition 3.3 implies that φ is G-equivariantly sequentially split. Under the G-equivariant isomorphism given by Takesaki-Takai duality, see Theorem 1.20, the map φ corresponds to id K ⊗ϕ : (K ⊗ A, α K ) → (K ⊗ B, β K ). Here we abbreviate The claim now follows from Proposition 3.7. 2

The Rokhlin property and approximate representability
In this section we introduce the key notions of this paper, namely the spatial Rokhlin property and spatial approximate representability for actions of quantum groups. Moreover we prove that these notions are dual to each other.

The spatial Rokhlin property
Let us start by defining the spatial Rokhlin property. Definition 4.1. Let G be a coexact compact quantum group and (A, α) a separable G-C * -algebra. We say that α has the spatial Rokhlin property if the second-factor embedding  [4, 4.3]. Indeed, for a classical compact group G, the braided tensor product C(G) A agrees with the ordinary tensor product C(G) ⊗ A. The term spatial in Definition 4.1 refers to the fact that we have chosen to work with minimal (braided) tensor products; we will comment further on the implications of this choice in Remark 4.9 below.
In special cases the Rokhlin property can be recast in the following way. Recall that if G is a compact quantum group and (A, α) a G-C * -algebra we write S(A) for the spectral subalgebra of A. We shall use the Sweedler notation α(a) = a (−1) ⊗ a (0) for the coaction α : S (A) → O(G) S(A). G be a coexact compact quantum group and (A, α) a separable G-C * -algebra. a) If α has the spatial Rokhlin property, then there exists a unital and G-equivariant exists such that κ(S(a (−1) ))a (0) ≤ a for all a ∈ S(A), then α has the spatial Rokhlin property.
Proof. a) Assume first that α has the spatial Rokhlin property. Let be an equivariant approximate left-inverse for ι α as required by Definition 4.1. Since ι α is non-degenerate, the image of this * -homomorphism is contained in D ∞,α , and ψ : C r (G) A → D ∞,A is again nondegenerate. Let us also denote the unique strictly continuous extension of ψ to multipliers by the same letter, so that we have By equivariance of ψ, the unital * -homomorphism is also equivariant. According to the definition of the braided tensor product, we have for all f ∈ O(G) and a ∈ S(A). The desired twisted commutation relation for κ then follows by applying ψ. Moreover, the norm condition κ(S(a (−1) ))a (0) ≤ a is a consequence of the formula κ(S(a (−1) ))a (0) = ψ(T −1 α (1 ⊗ a)) for a ∈ S(A) and the fact that ψ and the isomorphism T −1 α from Lemma 1.24 are * -homomorphisms between C * -algebras. (−1) ))a (0) . Then the commutation relation for κ gives

b) Consider the map ι : S(A) → D ∞,A given by ι(a) = κ(S(a
for any f ∈ O(G), using the antipode relation for the Hopf algebra O(G). Using that S is antimultiplicative we therefore obtain for a ∈ S(A). It follows that ι is a * -homomorphism. By assumption ι is bounded, so that it extends to a * -homomorphism ι : A → D ∞,A .
Combining κ and ι we obtain a * -homomorphism ψ = κ ⊗ ι : C(G) ⊗ A ∼ = C(G) ⊗ max A → D ∞,A , using the universal property of the maximal tensor product and nuclearity of C(G). Since κ is equivariant one checks that ι maps into the fixed point algebra of A ∞,α , and together with equivariance of κ it follows that ψ : Using the isomorphism from Lemma 1.24 we see that it defines an approximate left-inverse for ι A : Remark 4.4. Let us point out that the norm condition in part b) of Proposition 4.3 is automatically satisfied if G is a finite quantum group. Indeed, in this case the spectral subalgebra S(A) is equal to A, and the claim follows from the fact that * -homomorphisms between C * -algebras are contractive. It can also be shown that the norm condition always holds if G is a classical compact group, but it seems unclear whether it is automatic in general.
Classically, a Rokhlin action of a compact group on an abelian C * -algebra C 0 (X) induces a free action of G on X. In the quantum case, an analogue of the notion of freeness has been formulated by Ellwood in [7]. Namely, an action α : It is shown in [7, Theorem 2.9] that this generalizes the classical concept of freeness.
Let us verify that the spatial Rokhlin property implies freeness also in the quantum case.
Proposition 4.5. Let G be a coexact compact quantum group and let (A, α) be a separable G-C * -algebra. If α has the spatial Rokhlin property, then it is free.
In fact, for any f ∈ O(G) ⊂ C r (G) and a ∈ A we find finitely many elements Applying id ⊗ψ to this equality and using equivariance, we obtain Let us point out that the Rokhlin property is strictly stronger than freeness; this is already the case classically. For instance, the antipodal action of G = Z 2 on S 1 does not have the Rokhlin property.
Here comes the first main result of this paper: Theorem 4.6. Let G be a coexact compact quantum group and let (A, α) be a separable G-C * -algebra. If α has the spatial Rokhlin property, then the two canonical embeddings are sequentially split. In particular, if A has any of the following properties, then so do the fixed-point algebra A α and the crossed product G α,r A: • being simple; • being nuclear and satisfying the UCT; • having finite nuclear dimension or decomposition rank; • absorbing a given strongly self-absorbing C * -algebra D.
Proof. Let ψ : C r (G) A → A ∞ be an equivariant approximate left-inverse for the embedding ι A : A → C r (G) A. The resulting commutative diagram of equivariant * -homomorphisms by Proposition 3.6. Notice here that Lemma 1.24 implies that (C r (G) A) Δ α ∼ = A in such a way that the canonical embeddings of A α on both sides are compatible.
For the statement about the crossed product G α,r A, observe that the spatial Rokhlin property for α means that α : A → C r (G) ⊗ A is sequentially split, taking into account the isomorphism from Lemma 1.24. According to Proposition 3.3, it follows that the map is sequentially split. Moreover, by the Takesaki-Takai duality Theorem 1.20, we have G Δ,r C r (G) ∼ = K(L 2 (G)), and the resulting map G α,r A → K(L 2 (G)) ⊗ A is the standard embedding.
The asserted permanence properties are then a consequence of [4, Theorem 2.9]. 2

Spatial approximate representability
Let us now define spatial approximate representability. Let G be a discrete quantum group and (A, α) a separable G-C * -algebra. Denote by r A) is the canonical embedding. The unitary W α implements the inner action of G on the crossed product, more precisely Ad(W * α ) turns M (G α,r A) into a G-C * -algebra such that for all a ∈ A ⊂ M (G α,r A).
Definition 4.7. Let G be a discrete quantum group and (A, α) a separable G-C * -algebra. We say that α is spatially approximately representable if the natural embedding is G-equivariantly sequentially split.
as in a), then α is spatially approximately representable.
Proof. a) Assume that α is spatially approximately representable and let ψ : G α,r A → A ∞ be a G-equivariant approximate left-inverse for the embedding A → G α,r A. Since this embedding is nondegenerate, the image of ψ is contained in D ∞,A , and ψ : G α,r A → D ∞,A is a nondegenerate * -homomorphism. Let us denote the unique strictly continuous extension of ψ by the same letter, so that We let be the unitary representation corresponding to the restriction of ψ to C * r (G). Equivariance of ψ : for all x ∈ G α,r A. In particular, for every a ∈ A we obtain Since this holds for all ω ∈ L(L 2 (G)) * we conclude (id unitary satisfying the conditions in a). Clearly, the canonical map ι : A → D ∞,A is equivariant and nondegenerate, and the formula ι(a))V for all a ∈ A means that ι and V define a covariant pair. Hence they combine to a nondegenerate * -homomorphism a for all a ∈ A. Since G is amenable, we can identify the full crossed product G α,f A with the reduced crossed product G α,r A. To verify that ψ : G α,r A → M (D ∞,A ) is G-equivariant, it suffices to check this separately on the copies of C * r (G) and A inside M (G α,r A). For a ∈ A ⊂ G α,r A, the equivariance condition follows immediately from the relation ψ • ι(a) = a. On C * r (G) it is obtained by slicing the equation in the first tensor factor and using C * r (G) = [(L(L 2 (G)) * ⊗ id)(W )]. We conclude that ψ determines a G-equivariant approximate left-inverse for the inclusion A → G α,r A. 2 Remark 4.9. Definition 4.7 generalizes approximate representability for actions of discrete amenable groups, see [4, 4.23]. In the same way as already indicated in Remark 4.2, the term spatial in our definition is included since we work with minimal (braided) tensor products and reduced crossed products. In fact, approximate representability for classical discrete groups is defined in terms of the full crossed product instead, see [4, 4.23]. Notice that the trivial action of the free group F 2 on C is clearly approximately representable, but it is easily seen not to be spatially approximately representable in the sense of Definition 4.7.
It would therefore be more natural to develop the theory with maximal tensor products and full crossed products instead. However, this would mean in particular that one would have to work with full coactions taking values in maximal tensor products, which is technically less convenient.
Let us point out that all the above mentioned issues disappear for coamenable compact quantum groups and amenable discrete quantum groups, respectively; in these cases, we may omit the term spatial, and speak of the Rokhlin property and approximate representability.

Duality
We shall now show in several steps that the spatial Rokhlin property and spatial approximate representability are dual to each other.
Then there exists a Ǧ -equivariant * -isomorphism that makes the following diagram commutative: Proof. Using Lemma 1.24 and Theorem 1.20, we obtain Ψ α as the composition of the following identifications: Note that the copy of A inside M (G Δ α,r (C r (G) A)) identifies with α(A) ⊂ M (K(L 2 (G)) ⊗ A), and that the same holds for the copy of A inside M (Ǧ α,r G α,r A). Similarly, the copies of C * r (G) on both sides identify with C * r (G) ⊗1 ⊂ M (K(L 2 (G)) ⊗A). Moreover, the above identifications are compatible with the action of Ǧ on K(L 2 (G)) ⊗ A implemented by conjugation with ΣV Σ. More precisely, the coaction on K(L 2 (G)) ⊗ A corresponds to the dual coaction on G Δ α,r (C r (G) A) and to the conjugation coaction γ = Ad(W * α ) : For the latter observe ΣV Σ = (1 ⊗ U )Ŵ (1 ⊗ U ) and take into account the passage from C * r (G) to C * r (G) cop . 2 Proposition 4.11. Let G be a discrete quantum group and (A, α) a separable G-C * -algebra. Consider the G-equivariant inclusion Then there exists a Ǧ -equivariant * -isomorphism that makes the following diagram commutative: Proof. We obtain Φ α as the composition of the following identifications: Under these identifications, the copy of C * r (G) inside M (G α,r A) on the left hand side gets identified with 1 (C * r (G) ⊗ 1) inside C r (Ǧ) (G α,r A), and the copy of A in G α,r A is mapped to 1 α(A). In other words, we indeed obtain a commutative diagram as desired.
Moreover, it is not hard to check that the dual action on G Ad(W * α ),r (G α,r A) corresponds to the action of Δcop =Δ on the first tensor factor of C r (Ǧ) ⊗ (G α,r A). It follows that Φ α is Ǧ -equivariant. 2 As a consequence, we obtain the duality between the spatial Rokhlin property and spatial approximate representability.
is Ǧ -equivariantly sequentially split. By Proposition 4.10, there exists a commutative diagram of Ǧ -equivariant * -homomorphisms We conclude that ι α is G-equivariantly sequentially split if and only if jα is Ǧ -equivariantly sequentially split. This means that α has the spatial Rokhlin property if and only if α is spatially approximately representable. The claim in the discrete case is proved in an analogous fashion. Again by Proposition 3.8, the G-equivariant * -homomorphism is G-equivariantly sequentially split if and only if the induced * -homomorphism is Ǧ -equivariantly sequentially split. By Proposition 4.11, there exists a commutative diagram of Ǧ -equivariant * -homomorphisms We conclude that j α is G-equivariantly sequentially split if and only if ια is Ǧ -equivariantly sequentially split. Hence α is spatially approximately representable if and only if α has the spatial Rokhlin property. 2

Rigidity of Rokhlin actions
In this section we provide a classification of actions of coexact compact quantum groups with the spatial Rokhlin property on separable C * -algebras. This type of result was first obtained by Izumi in [15]. Our basic approach follows Gardella-Santiago [11,Section 3], who proved corresponding results for finite group actions. We note that Gardella-Santiago have also announced the results of this section for actions of classical compact groups, see [10].
Recall that if (B, β) is a G-C * -algebra for a compact quantum group G then B β ⊂ B denotes the fixed point subalgebra. We shall also write β for the induced coaction on the minimal unitarization B of B; note that β(1) = 1 ⊗ 1.
Definition 5.1. Let G be a compact quantum group. Let α : A → C r (G) ⊗ A and β : B → C r (G) ⊗ B be two G-actions on C * -algebras, and assume that A is separable. Let ϕ 1 , ϕ 2 : (A, α) → (B, β) be two equivariant * -homomorphisms. We say that ϕ 1 and ϕ 2 are approximately G-unitarily equivalent, written ϕ 1 ≈ u,G ϕ 2 , if there exists a sequence of unitaries v n ∈ U(B β ) such that Remark 5.2. For the trivial (quantum) group G, the above definition recovers the usual notion of approximate unitary equivalence between * -homomorphisms. We write simply ϕ 1 ≈ u ϕ 2 instead of ϕ 1 ≈ u,G ϕ 2 in this case. Let us now consider a series of partial results that will lead to the classification of Rokhlin actions.
Proof. For convenience, the term id will always denote the identity map on C r (G) in this proof. Identity maps on other sets are decorated with the corresponding set.
Using our assumptions on ϕ, we may choose unitaries u n ∈ (C r (G) ⊗ B) ∼ such that in point-norm. By Lemma 1.24, we have the equivariant isomorphism Let us also denote by T β the obvious extension to the unitarizations. Set v n = T −1 β (u n ). We calculate One should note that even though the existence of all these limits is a priori not clear at the beginning of this calculation, it follows a posteriori from the steps in this calculation. Moreover, we calculate for all x ∈ A that From these two calculations, it is clear that for given F ⊂ ⊂A and ε > 0, any of the unitaries v n satisfies the desired property for sufficiently large n. 2 Lemma 5.5. Let G be a coexact compact quantum group. Let α : and for all x ∈ F .
Proof. As β is assumed to have the spatial Rokhlin property, let be an equivariant * -homomorphism satisfying We also denote by ψ the canonical extensions to the smallest unitarizations on both sides. Now let F ⊂ ⊂A and ε > 0 be given. Apply Lemma 5.4 and choose a unitary w ∈ (C r (G) B) ∼ such that Combining the equivariance of ψ with (e5.4), (e5.5) and (e5.6), we obtain for all x ∈ F . Now represent v by some sequence of unitaries v n ∈B. Then these equations translate to the conditions lim sup for all x ∈ F . It follows that for sufficiently large n, any of the unitaries v n satisfies the desired inequalities with respect to ε in place of ε/2. This finishes the proof. 2 Proposition 5.6 (cf. [11, 3.2]). Let G be a coexact compact quantum group. Let α : A → C r (G) ⊗ A and β : B → C r (G) ⊗ B be two G-actions on separable C * -algebras. Assume that β has the spatial Rokhlin property.
Proof. Let be an increasing sequence of finite subsets with dense union. Let (ε n ) n∈N be a decreasing sequence of strictly positive numbers with ∞ n=1 ε n < ∞. Using Lemma 5.5 we find a unitary v 1 ∈B satisfying for all x ∈ F 1 . Applying Lemma 5.5 again (but now for Ad(v 1 ) • ϕ in place of ϕ), we find a unitary v 2 ∈B satisfying for all x ∈ F 2 . Applying Lemma 5.5 again (but now for Ad(v 2 v 1 ) • ϕ in place of ϕ), we find a unitary v 3 ∈B satisfying for all x ∈ F 3 . We inductively repeat this process and obtain a sequence of unitaries v n ∈B satisfying for all n ≥ 1 and for all x ∈ F n and n ≥ 2. For m > n ≥ k and x ∈ F k this implies As the ε n were chosen as a 1-summable sequence and the union of the F n is dense, this estimate implies that the sequence Ad(v n · · · v 1 ) • ϕ(x) is Cauchy for every x ∈ A.
In particular, the point-norm limit ψ = lim n→∞ Ad(v n · · · v 1 ) • ϕ exists and yields a well-defined * -homomorphism from A to B. By construction we have ψ ≈ u ϕ, and the equivariance condition follows from (e5.7). This finishes the proof. 2 Proof. Let u n ∈ U(B) be a sequence of unitaries satisfying Using Lemma 1.24, we consider the equivariant isomorphism that satisfies condition (e5.3). We shall also denote by T β the obvious extension to the unitarizations. Set As 1 ⊗ u n is in the fixed-point algebra of Δ ⊗ id B , it follows that v n is in the fixed-point algebra of Δ β. We have This shows our claim. 2 Proposition 5.8. Let G be a coexact compact quantum group. Let α : be equivariant * -homomorphisms. Assume that ψ is equivariantly sequentially split. Then Here comes the main result of this section, which generalizes analogous results for finite group actions due to Izumi [15, 3.5], Nawata [25, 3.5] and Gardella-Santiago [11, 3.4]. It also generalizes the corresponding results for finite quantum groups by Osaka-Teruya [21, 10.7] and for classical compact groups by Gardella-Santiago [10]. Proof. First assume that θ : (A, α) → (A, β) is an equivariant * -isomorphism which is approximately inner as a * -automorphism. Then Now assume that α and β are approximately unitarily equivalent. Then clearly Since both α and β have the spatial Rokhlin property, it follows from Proposition 5.6 that there exist equivariant * -homomorphisms ϕ 1 : (A, α) → (A, β) and ϕ 2 : (A, β) → (A, α) that are both approximately inner as * -homomorphisms. Hence Corollary 5.9 implies ϕ 1 • ϕ 2 ≈ u,G id A and ϕ 2 • ϕ 1 ≈ u,G id A . According to Proposition 5.3 we conclude that there exists an equivariant * -isomorphism θ : (A, α) → (A, β) with θ ≈ u,G ϕ 1 . In particular, θ is also approximately inner as a * -automorphism. 2 To conclude this section, we generalize the K-theory formula for fixed-point algebras of Rokhlin actions, which is originally due to Izumi and was recently extended by the first two authors.
Theorem 5.11 (cf. [15, 3.13] and [4, 4.9]). Let G be a coexact compact quantum group. Let α : A → C r (G) ⊗ A be an action on a separable C * -algebra with the spatial Rokhlin property. Then the inclusion A α −→ A is injective in K-theory, and its image coincides with the subgroup Proof. If x ∈ im(K * (A α ) → K * (A)), then clearly K * (α)(x) = K * (1 ⊗ id A )(x). For the converse, let x = [p] − [1 k ] be an element of K 0 (A), where p ∈ M n (Ã) and for the canonical extension of α to unitarizations and matrix amplifications.
Similarly, we write M n ((1 ⊗id A ) ∼ ) for the extension of By definition of K 0 , we therefore find natural numbers m, l such that in M n ((C r (G) ⊗ C r (G) ⊗ A) ∼ ). By equivariance of T and ψ, the same applies to that is, the latter element satisfies Now the invariant part of M n (Ã) ∞ equals M n ((A α ) ∼ ) ∞ by Lemma 3.5. Since the relation of being a partial isometry with a fixed range projection is well-known to be weakly stable, this shows that there exists a projection q ∈ M r ((A α ) ∼ ) such that is contained in im(K 0 (A α ) → K 0 (A)) as desired.
For the statement about the K 1 -group, one uses suspension to reduce matters to K 0 , see the proof of [4, 4.9]. 2

Examples
In this final section we present some examples of Rokhlin actions. Example 6.1. Let G be a coamenable compact quantum group acting on A = C(G) by the regular coaction α = Δ. Then α has the spatial Rokhlin property.
Indeed, in this case the embedding ι A : A → C(G) A ∼ = C(G) ⊗A is given by ι A = Δ. Since G is coamenable, the counit : O(G) → C extends continuously to C(G) = C r (G), and id ⊗ is an equivariant left-inverse for ι A . Hence composition with the canonical embedding of C(G) = A into A ∞ yields an equivariant approximate left-inverse.
The * -homomorphism κ : C(G) → A ∞ corresponding to this Rokhlin action according to Proposition 4.3 is induced by the canonical embedding of C(G) into its sequence algebra.
Remark 6.2. 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]. 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. A substantial difference between our approach and [21] is that the duality of these two notions becomes a theorem rather than a definition. given by T → 1 T are G-equivariant. As explained in Remark 1.16, we may therefore form the inductive limit action α : A → C(G) ⊗ A of G on the corresponding inductive limit A.
We remark that A can be identified with the UHF-algebra M n ∞ . Indeed, the braided tensor product M n M n is easily seen to be isomorphic to the ordinary tensor product M n ⊗ M n as a C * -algebra, using that An analogous statement holds for iterated braided tensor products.
We obtain a G-equivariant * -homomorphism κ : C(G) → A ∞ by setting κ(f ) = [(ι k (f )) k∈N ], where ι k : M n → A is the embedding into the k-th braided tensor factor of A. Moreover, for a ∈ ι m (M n ) ⊂ A and f ∈ C(G) we have aι k (f ) = a (−2) ι k (f )S(a (−1) )a (0) provided k > m. It follows that κ satisfies the commutation relations required by Proposition 4.3, and the norm condition in Proposition 4.3 b) is automatic since G is a finite quantum group. Hence (A, α) has the spatial Rokhlin property. Proposition 6.4. Let G be a coexact compact quantum group and D a strongly selfabsorbing C * -algebra. Then there exists at most one conjugacy class of G-actions on D with the spatial Rokhlin property.
Proof. By [32, Corollary 1.12], any two unital * -homomorphisms from D to C r (G) ⊗D are approximately unitarily equivalent. Therefore the claim follows from Theorem 5.10. 2 Remark 6.5. As a consequence of Proposition 6.4 we see that the action of a finite quantum group G of order n = dim(C(G)) constructed in Example 6.3 is the unique Rokhlin action of G on M n ∞ up to conjugacy. In particular, it is conjugate to the action constructed by Kodaka-Teruya in [21,Section 7].
Finally, we shall construct a Rokhlin action of any coamenable compact quantum group on O 2 . As a preparation, recall that an element a in a C * -algebra A is called full if the closed two-sided ideal generated by a is equal to A. Proof. Let f ∈ C(G) = C r (G) be nonzero. To show that h = (id ⊗ι)(Δ(f )) is full it is enough to verify that (π ⊗ id)(h) is nonzero for all irreducible representations π of C(G). Indeed, if the ideal generated by h is proper, there must exist a primitive ideal of C(G) ⊗ O 2 containing h. Since O 2 is nuclear and simple these ideals are of the form I ⊗ O 2 for primitive ideals I ⊂ C(G), see [5,Theorem 3.3]. Now if π : C(G) → L(H π ) is any * -representation, then (π ⊗ id)(Δ(f )) = (π ⊗ id)(W ) * (1 ⊗ f )(π ⊗ id)(W ) is nonzero in L(H π ) ⊗ C(G), and hence (id ⊗ι)((π ⊗ id)(Δ(f ))) = (π ⊗ ι)(Δ(f )) = (π ⊗ id)(h) is nonzero as well since ι is injective. 2 Theorem 6.7. Let G be a coamenable compact quantum group. Then up to conjugacy, there exists a unique G-action on the Cuntz algebra O 2 with the spatial Rokhlin property.
Proof. According to Proposition 6.4 it suffices to construct some G-action on O 2 with the spatial Rokhlin property. Since Notice that Φ n = Φ 0 ⊗ id O ⊗n 2 for all n ≥ 1. We have This means that Φ 0 is an injective equivariant * -homomorphism We thus also have that each is injective and equivariant. Define the inductive limit where α denotes the inductive limit coaction, compare Remark 1.16. Notice that each building block in this inductive limit has the Rokhlin property, and moreover the first-factor embedding of C(G) into C(G) ⊗ O ⊗n Hence α has the Rokhlin property.
By Lemma 6.6, we know that Φ 0 , and thus also each Φ n is a full * -homomorphism. It follows that the inductive limit A is simple, and it is clearly separable, unital and nuclear. Moreover A is O 2 -absorbing by [32, 3.4]. This implies A ∼ = O 2 due to Kirchberg-Phillips [18]. 2