A CLASSIFICATION OF ANOMALOUS ACTIONS THROUGH MODEL ACTION ABSORPTION

. We discuss a strategy for classifying anomalous actions through model action absorption. We use this to upgrade existing classiﬁcation results for Rokhlin actions of ﬁnite groups on C ∗ -algebras, with further assuming a UHF-absorption condition, to a classiﬁcation of anomalous actions on these C ∗ -algebras.


Introduction
Connes' classification of automorphisms on the hyperfinite II 1 factor R ( [8,7]) paved the way towards a classification of symmetries of simple operator algebras.Over the next decade, this was followed by Vaughan Jones' classification of finite group actions on R ( [29]) and Ocneanu's classification of actions of countable amenable groups on R ( [37]).To achieve these classification results, an important role is played by adaptations of Connes' non-commutative Rokhlin lemma, which yields that outer group actions on R satisfy a condition often called the Rokhlin property that is analogous to properties of ergodic measure preserving actions of amenable groups on probability spaces ( [41], [38]).In the C * -setting, the analogous property is not automatic.However, there has been substantial progress in the classification of those group actions on C * -algebras that satisfy the Rokhlin property ( [17,18,19,12,21,22,23]). Very recently, groundbreaking results towards a classification of group actions without the need for the Rokhlin property have appeared ( [15,25,26]).
Connes, Jones and Ocneanu also classify group homomorphisms G → Out(R) up to outer conjugacy ( [8,29,37]).Such a homomorphism is called a G-kernel on R. The classification of G-kernels on injective factors was completed by Katayama and Takesaki ([31]).These can be understood as the first classification results for quantum symmetries of R which do not arise as group actions.Quantum symmetry is a broad term that encapsulates generalised notions of symmetry that appear in topological and conformal field theories.These symmetries are often encoded through the action of a higher category equipped with a product operation such that the category weakly resembles a group.In the case of G-kernels, these can be understood as actions of 2-groups The author was supported by the Ioan and Rosemary James Scholarship awarded by St John's College and the Mathematical Institute, University of Oxford.
In comparison to the success in understanding the existence and classification of G-kernels on von Neumann algebras, the study of G-kernels on C * -algebras has up to recently been underdeveloped.In [27] Corey Jones studies the closely related notion of ω-anomalous action. 1 In his paper, Corey Jones provides a C * -adaptation of Vaughan Jones' work ( [28]), laying out a systematic way to construct anomalous actions on C * -crossed products.Corey Jones also establishes existence and no-go theorems for anomalous actions on abelian C * -algebras.In [13] Evington and the author lay out an algebraic K-theory obstruction to the existence of anomalous actions on tracial C * -algebras.Recently, Izumi has developed a cohomological invariant for G-kernels ( [24]).This invariant introduces new obstructions to the existence of G-kernels which also apply in the non-tracial setting.Further, Izumi uses this invariant to classify G-kernels of some poly-Z groups on strongly self-absorbing UCT Kirchberg algebras.
This paper provides a classification of anomalous actions with the Rokhlin property on C * -algebras where K-theoretic obstructions vanish.The Rokhlin property for finite group actions was first systematically studied by Izumi ([21,22]).In his work, Izumi uses the Rokhlin property to boost existing classification results of Kirchberg algebras in the UCT class ( [39,32]) and unital, simple, separable, nuclear, tracially approximate finite dimensional (TAF) algebras in the UCT class ( [34]) by their K-theory, to a classification of finite group actions with the Rokhlin property on these classes of C * -algebras by the induced module structure on K-theory ([22, Theorem 4.2, Theorem 4.3]). 2he strategy of this paper is to bootstrap Izumi's classification of G actions with the Rokhlin property, for finite groups G, to achieve analogous classification results for anomalous actions.To do this, we will assume that our C * -algebras satisfy a UHF absorbing condition.To be precise, that the C * -algebras are stable by the UHF algebra this sort of assumption is considered for example in [2]).Further assuming the Rokhlin property, we will establish a model action absorption result (Proposition 3.6).Second, we will use the model action absorption combined with a trick, that builds on ideas of Connes in the cyclic group case ([8, Section 6]).This trick lets us use the existence of anomalous action on the UHF-algebra M |G| ∞ to reduce the classification of anomalous actions to the classification of cocycle actions.This trick is not available through replacing M |G| ∞ by Z or O ∞ due to the obstruction results of [13, Theorem A] and [24,Theorem 3.6].This argument allows us to prove the following.
Theorem A. (cf.Theorem 4.2 and Theorem 4.3) Let G be a finite group and A ∼ = A⊗M |G| ∞ be either a Kirchberg algebra in the UCT class or a unital, simple, separable, nuclear TAF algebra in the UCT class.If (α, u), (β, v) are anomalous G actions on A with the Rokhlin property, then (α, u) is cocycle conjugate to (β, v) through an automorphism that is trivial on K-theory if and only if K i (α g ) = K i (β g ) for all g ∈ G and the anomalies of (α, u) and (β, v) coincide.
Similarly, we can boost Nawata's classification of Rokhlin G actions on W (see [35]) to a classification of anomalous actions on W.
Theorem B. (cf.Theorem 4.4) Let G be a finite group and (α, u), (β, v) be anomalous G actions on W with the Rokhlin property, then (α, u) is cocycle conjugate to (β, v) if and only if the anomalies of (α, u) and (β, v) coincide.
The procedure utilised for the proof of Theorem A can be expected to work in more generality.The reason for restricting to unital, simple, nuclear TAF algebras in the tracial setting is due to the need to apply classification results for (cocycle) group actions.With more novel stably finite classification results in hand ( [5]), and using similar techniques to [21,22], a classification of finite group actions with the Rokhlin property on simple, separable, nuclear, Z-stable C * -algebra satisfying the UCT through the induced module structure on the Elliott invariant is plausible.A strategy to approach this classification problem has been proposed by Szabó in private communications.With such a result in hand, one could apply the abstract Lemma 4.1 to yield the equivalent to Theorem A in the generality of simple, separable, nuclear, M |G| ∞ -stable C * -algebras satisfying the UCT.
Recent advances in the classification of more general symmetries on C * -algebras pave the way towards a classification of quantum symmetries.Significant results in this direction are the classification of AF-actions of fusion categories on AF-algebras ( [6]), as well as Yuki Arano's announcement of an adaptation of Izumi's techniques in [21] to actions of fusion categories with the Rokhlin property.In the final section of this paper, we connect our results to the work in [6].We demonstrate the existence of an AF ω-anomalous G-action with the Rokhlin property on M |G| ∞ which we denote by θ ω G .This has structural implications for anomalous actions with the Rokhlin property on any AF-algebra A. Indeed, combined with Theorem A, the existence of θ ω G implies that every anomalous action on A with the Rokhlin property, that consists of automorphisms that act trivially on K-theory, is automatically AF (see Corollary 5.3).Under some assumptions on the anomaly, an application of the classification results of [6] establish the converse (see Corollary 5.3).This partial converse exhibits a difference in behavior between anomalous actions and group actions (see the discussion following Corollary 5.3).
The paper is organised as follows.In Section 1 we recall some necessary background on anomalous actions.Section 2 recalls the construction of model anomalous actions on UHF algebras.In Section 3 we prove a model action absorbing result for finite group anomalous actions.In Section 4 we set out an abstract lemma for the classification of anomalous actions (Lemma 4.1) which we use to prove our main results.Finally, in Section 5, we discuss an application of the classification result to AF-actions.
Acknowledgements.The author would like to thank Samuel Evington, Corey Jones and Stuart White for useful discussions related to the topic of this paper.This work forms part of the authors DPhil thesis [16].

Preliminaries
Throughout, A and B will be used to denote C * -algebras and G, Γ, K will be used to denote countable discrete groups.We let T ⊂ C be the circle group.We denote the multiplier algebra of A by M(A).Any automorphism α ∈ Aut(A) extends uniquely to an automorphism of M(A), we denote this extension also by α.For a unitary u ∈ M(A) we write Ad(u) for the automorphism a → uau * of A and the group of inner automorphisms on A by Inn(A).Recall that a G-kernel of A is a group homomorphism G → Aut(A)/Inn(A) = Out(A).We now recall the definition of an anomalous action from [27, Definition 1.1].In the case that A has trivial centre this notion coincides with a lift of a G-kernels into Aut(A).Definition 1.1.An anomalous action of a countable discrete group G on a C * -algebra A consists of a pair (α, u) where are a pair of maps such that Firstly, note that in (1.1) and (1.2) we have used the subscript notation α g and u g,h instead of α(g) and u(g, h) for g, h ∈ G.We will use this throughout when notationally convenient.
As shown in [10,Lemma 7.1] the formula in (1.2) defines a circle valued 3-cocycle i.e. an element of Z3 (G, T).We will call this the anomaly of the action and denote it by o(α, u).For ω ∈ Z 3 (G, T) we say (α, u) is a (G, ω) action on A to mean that (α, u) is an anomalous action of G on A with anomaly ω. 3 If ω = 1 then we call (α, u) a cocycle action.Note that any anomalous action (α, u) induces a G-kernel when passing to the quotient group Out(A), we denote its associated G-kernel by α.For any G-kernel α on A we denote by ob(α) ∈ H 3 (G, Z(U(M(A)))) its 3-cohomology invariant (see e.g.[13,Section 2.1]).
The reader should be warned that there is a slight variation in Definition 1.1 to the definitions of anomalous actions in [27] and [13].Given our conventions in Definition 1.1, a (G, ω) action induces an ω anomalous action as in [27,Definition 1.1], this is seen by taking m g,h = u * g,h .Throughout this paper, we will denote the algebra of bounded sequences of A quotiented by those sequences going to zero in norm by A ∞ .For a * -closed subset S of A ∞ we may consider the commutant We may then denote Kirchberg's sequence algebra by In the case that S is the C * -algebra of constant sequences in A ∞ we denote this simply by F (A) = F (A, A ∞ ) and F (A) the central sequence algebra of A. Note that F (A) is a unital C * -algebra whenever A is σ-unital.Indeed, the unit is given by h = (h n ) for any sequential approximate unit h n for A.
Any automorphism θ ∈ Aut(A) induces an automorphism θ of A ∞ through (a n ) → (θ(a n )) for any (a n ) ∈ A ∞ . 4If a subset S of A ∞ is invariant under both θ and θ −1 , then so are A ∞ ∩ S ′ and A ∞ ∩ S ⊥ and θ induces an automorphism of F (S, A ∞ ).
Remark 1.2.When A is equipped with a (G, ω) action (α, u), it induces a (G, ω) action on A ∞ .In fact, α induces a group action on Similarly, if S = S * is an α invariant subset of A ∞ containing A and S is also invariant by u g,h for all g, h ∈ G (i.e.u g,h S + Su g,h ∈ S for all g, h ∈ G), then α induces a group action on F (S, A ∞ ) (see [43,Remark 1.8]).A subset which is invariant under both α and u will be called (α, u)-invariant.
We will be interested in anomalous actions with the Rokhlin property.This notion was introduced in [21, Definition 3.10] for actions of finite groups on unital C * -algebras and later generalised by Nawata to the generality of σ-unital C * -algebras in [35].Its definition in the setting of anomalous actions is ad verbatim.Definition 1.3.An anomalous action (α, u) of a finite group G on a σ-unital C * -algebra A is said to have the Rokhlin property, if there exist projections p g ∈ F (A) for g ∈ G such that: (1) g∈G p g = 1, (2) α g (p h ) = p gh .
Remark 1.4.The Rokhlin property also makes sense for G-kernels.In this case, a G-kernel α of a finite group G on a σ-unital C * -algebra A satisfies the Rokhlin property if for any/some lift (α, u) of α there exists a partition of unity of projections p g ∈ F (A) for g ∈ G such that α g (p h ) = p gh for all g, h ∈ G.
Our main goal is to classify anomalous actions with the Rokhlin property.To make sense of this question, we first need to introduce equivalence relations for anomalous actions.Before we do so, we start by introducing some notation that will allow us to streamline future definitions.
Definition 1.5.Let (α, u) be an anomalous action of a group G on a gh , g, h ∈ G is an anomalous action.We say that (α Ú , u Ú ) is a unitary perturbation of (α, u).
Definition 1.6.Let A, B be C * -algebras, (α, u) be an anomalous G action on A and (β, v) be an anomalous action on B. Then we say that We denote this by (α, u) ≃ (β, v).(iii) If A and B are equal and (α, u) ≃ (β, v) with the conjugacy holding through an automorphism θ such that K i (θ) = id K i (A) for i = 1, 2, we say (α, u) and (β, v) are K-trivially cocycle conjugate.We denote this by (α, u) Finally, recall the definition of a unitary one cocycle.
Definition 1.7.Let α be a (G, ω) action on a C * -algebra A. We call a map v :

Model actions
Given a finite group G and ω ∈ Z 3 (G, T) a 3-cocycle, [13, Theorem C] constructs a (G, ω) action on M |G| ∞ .This result is based on a construction of Corey Jones in [27] which in turn is based on a construction of Vaughan Jones in the setting of von Neumann algebras ( [28]).
In this section, we recall this construction as we will need its specific form to deduce properties of the action.In [27], Corey Jones shows that if ω is a normalised 3-cocycle and one has the following data: one can induce a (G, ω) action on the twisted reduced crossed product B ⋊ r π,c K, with K = ker(ρ) (see [4] for a reference on twisted crossed products). 5The automorphic data of this (G, ω) action is given by for a k ∈ B, v k the canonical unitaries in M(B ⋊ r π,c K), g ∈ G and g → ĝ a choice of set theoretic section to ρ : Γ → G. 6 In fact, given an arbitrary finite group G, Corey Jones constructs a finite group Γ, a surjection ρ and a 2 cochain c with the conditions needed above and additionally c| ker(ρ) = 1.Additionally to Γ and c, the extra data considered in [13, Theorem C] is: with λ Γ the left regular representation and Ad(λ Γ ) γ (T ) = λ Γ (γ)T λ Γ (γ) * for all T ∈ B(l 2 (Γ)) and γ ∈ Γ.In this case, the crossed product B⋊ r π K is shown to be isomorphic to the UHF algebra M |G| ∞ .Corey Jones' construction then yields a (G, ω) action on M |G| ∞ through (2.1) for any has the Rokhlin property.Proof.We use the notation set up in the previous paragraphs.Furthermore, denote by r i : B(l 2 (Γ)) → B the unital embedding into the i-th tensor factor.As for any complex scalars µ γ .Let p n = r n (e K ) for n ∈ N. Note that the projection p = (p n ) ∈ B ∞ commutes with any constant sequence of elements in B.Moreover, p commutes with the subalgebra We claim that the projections n∈N form a set of Rokhlin projections.We start by showing that the sum The maps r n are unital so it suffices to show that g∈G Ad(λ Γ ) ĝ(e K ) = 1 B(l 2 (Γ)) .To see this, let γ ∈ Γ, g ∈ G and δ γ ∈ l 2 (Γ) the point mass at γ, then The left K cosets are pairwise disjoint and cover the whole group Γ.Therefore, it follows that g∈G Ad(λ Γ ) ĝ(e K )(δ γ ) = δ γ for every γ ∈ Γ.As the operators g∈G Ad(λ Γ ) ĝ(e K ) and id B(l 2 (Γ)) coincide on a spanning set of l 2 (Γ), these operators are equal.
It remains to show that for g, h ∈ G the projections s ω G (g)p h = p gh .This follows as )p gh = p gh where the last equality in the chain holds as p gh commutes with A.

Absorption of model actions
In this section we show that any Rokhlin anomalous action of a finite group G, on an M |G| ∞ -stable C * -algebra, absorbs the action up to cocycle conjugacy. 7The methods utilised in this chapter are an adaptation of Vaughan Jones' work ( [29]) to the C * -setting.
In his work [43,44,42], Szabó establishes the theory of strongly selfabsorbing C * -dynamical systems as an equivariant version of strongly self-absorbing C * -algebras that were introduced in [45].We recall the main definition below.Definition 3.1.Let G be a locally compact group.A group action γ on a unital, separable C * -algebra D is called strongly self-absorbing if there exists an equivariant isomorphism ϕ : The relevant example of a strongly self-absorbing action for this paper is s G .That s G is strongly self-absorbing follows as a consequence of [43,Example 5.1].
In [43,Theorem 3.7] Szabó shows equivalent conditions for a cocycle action to tensorially absorb a strongly self-absorbing action.Although Szabó's theory only treats the case of cocycle actions absorbing a given strongly self-absorbing group action, many of the arguments follow in exactly the same way when replacing cocycle actions by anomalous actions that may have non-trivial anomaly.The proofs of [ We still require a few more results before we can achieve the model action absorption.These are based on known results in the setting of finite group actions on unital C * -algebras.These generalise line by line to anomalous actions of finite groups on unital C * -algebras, we adapt the arguments also for non-unital C * -algebras.Lemma 3.3 (cf.[20,Theorem 3.3]).Let A be a C * -algebra, G be a finite group and (α, u) be an anomalous action of G on A with the Rokhlin property.If B = B * is a separable (α, u)-invariant subset of A ∞ and there exists a unital * -homomorphism M → F (B, A ∞ ) for some separable, unital C * -algebra M, then there exists a unital Let S = B ∪ g∈G α g (ψ 0 (M)) ∪ g∈G α g (ψ 0 (M)) * so S = S * .By the Rokhlin property followed by a standard reindexing argument, there exist positive contractions

Now consider the linear mapping
Firstly, for m, m ′ ∈ M and b ∈ B it follows from (i) and (iiiv) Where in the last line we have used that B is u invariant and so the observation in Remark 1.2 applies.Therefore, the map This homomorphism is unital through combining (iii) and (iiv) and * -preserving by (ii).
In the next lemma, recall that if α is a action of a group G on a C * -algebra A, an α-cocycle is a family of unitaries v g ∈ U(M(A)) for g ∈ G such that v g α g (v h ) = v gh .Lemma 3.4 (cf.[18, Lemma III.1]).Let A be a separable C * -algebra and G a finite group.Let (α, u) be an anomalous action of G on A with the Rokhlin property.Let B = B * be a separable (α, u)-invariant subset of A ∞ .For any α-cocycle v g for the action induced by α on As in the previous lemma, one may apply the Rokhlin property combined with a reindexing argument to get a family of positive elements Then for any b ∈ B by (ii),(iiv) and (iiiv) it follows that Similarly uu * b = b for any b ∈ B.Moreover, (iii),(i),(iv) and (iiv) imply that for b ∈ B and g ∈ G Therefore, by passing to the quotient, u defines a unitary in F (B, A ∞ ) such that uα g (u * ) = v g for all g ∈ G.
The proof of the next lemma is based on the proof of [29, Proposition 3.4.1].Lemma 3.5.Let G be a finite group and A be a separable C * -algebra such that A ∼ = A ⊗ M |G| ∞ .Let (α, u) be an anomalous action with the Rokhlin property of G on A .Then there exists a G-equivariant unital embedding Proof.To prove this we inductively construct unital equivariant *homomorphisms φ n : (B(l 2 (G)), Ad(λ G )) → (F (A), α) for n ∈ N with commuting images.Then the map defined by Suppose φ 1 , φ 2 , . . ., φ n : (B(l 2 (G)), Ad(λ G )) → (F (A), α) are equivariant maps with commuting images and let Then S is separable, S = S * and S is (α, u) invariant.We check that u g,h S ⊂ S for all g, h ∈ G, the remaining conditions follow similarly.For a ∈ A, m ∈ M and 1 [45,Theorem 2.2].Moreover by reindexing one can also choose a homomorphism as stated.)It follows from Lemma 3.3 that there exists a unital embedding B(l 2 (G)) → F (S, A ∞ ) α .Let (e ′ g,h ) g,h∈G in F (S, A ∞ ) α be the images of e g,h under this unital embedding.The permutation unitary v g = h∈G e ′ gh,h gives a unitary representation of G on F (S, A ∞ ) α and as α g (v h ) = v h it follows that v g is an α-cocycle.Therefore, by Lemma 3.4 there exists a unitary u ∈ F (S, A ∞ ) such that uα g (u * ) = v g .Now f g,h = u * e ′ g,h u for g, h ∈ G is a set of matrix units such that Hence the * -homomorphism defines an Ad(λ G ) to α equivariant * -homomorphisms.Moreover, the image of φ n+1 commutes with φ i for all 1 ≤ i ≤ n.Considering φ n+1 as a unital equivariant homomorphism into A ∞ ∩A ′ /A ∞ ∩A ⊥ the induction argument is complete.
We have collected all the necessary ingredients to prove the model action absorption.Proposition 3.6.Let G be a finite group and A a separable C * -algebra such that A ∼ = A ⊗ M |G| ∞ .Let (α, u) be a (G, ω) action on A with the Rokhlin property.Then (α, u) and (α⊗s G , u⊗1 M |G| ∞ ) are cocycle conjugate through an isomorphism that is approximately unitarily equivalent to id A ⊗1 M |G| ∞ .

Classification
We now discuss the abstract approach to bootstrapping the classification of group actions on a given class of C * -algebras to a classification of anomalous actions.This method is a generalisation of that used by Connes in [8, Section 6], a similar strategy was recently used in [24] to classify G-kernels of poly-Z groups on O 2 .
Before proceeding with the result, we set up notation.For a group G, we say "(α, u) is an anomalous G-action on A" and "(A, α, u) is an anomalous G-C * -algebra" interchangebly.Let F be a functor whose domain category is the category of C * -algebras (denoted C*alg).We say F is invariant under approximate unitary equivalence if F (α) = F (θ) whenever α ≈ a.u θ.We also say that F restricted to a subcategory C ⊂ C*alg is full on isomorphisms , if whenever Φ ∈ Hom(F (A), F (B)) is an isomorphism for A, B ∈ C, then there exists an isomorphism ϕ : A → B in C with F (ϕ) = Φ.The sort of functors with these properties are those used in the classification of C * -algebras.For example, the functor consisting of pointed K 0 and K 1 is invariant under approximate unitary equivalence, it is also full on isomorphisms when restricted to the category of unital Kirchberg algebras satisfying the UCT (see [39]).Similarly, the functors KT u and KT u of [5] are invariant under approximate unitary equivalence and is full on isomorphisms when restricted to classifiable C * -algebras.
If F is invariant under unitary equivalence, an anomalous action (A, α, u) induces a G-action on F (A) through the automorphisms F (α g ).If (A, α, u) and (B, β, v) are anomalous actions, we say the induced actions F (α g ) and F (β g ) are conjugate if there exists an isomorphism Φ : F (A) → F (B) with ΦF (α g )Φ −1 = F (β g ) for all g ∈ G.We denote this by F (α) ∼ F (β).
Let (A, α, u) and (A, β, v) be two anomalous G-C * -algebras.We write (α, u) ≃ F (β, v) if (α, u) ≃ (β, v) through an automorphism θ with F (θ) = id F (A) .This notion recovers K-trivial cocycle conjugacy of Definition 1.6 when F is taken to be the functor consisting of K 0 ⊕K 1 .Finally, if R is a class of anomalous G-C * -algebras, we will say R is closed under conjugacy, if whenever (A, α, u) ∈ R and ϕ : A → B is an isomorphism in C*alg then (B, ϕαϕ −1 , ϕ(u)) ∈ R. Lemma 4.1.Let G be a group, D a strongly self-absorbing C * -algebra and R a class of anomalous G-C * -algebras that is closed under conjugation.Let F be a functor with domain category the category of C *algebras such that F is invariant under approximate unitary equivalence and is full on isomorphisms for C * -algebras in R. Suppose further that, (A1) there exists a G-action (D, s G , 1) such that if (A, α, u) ∈ R, then (A, α, u) ≃ (A⊗D, α ⊗s G , u ⊗1) through an automorphism that is approximately unitarily equivalent to id A ⊗1 D ; (A2) if there exists a (G, ω) action in R for some ω ∈ Z 3 (G, T), there exist a (G, ω) and (G, ω) action (D, s ω G , u ω ) and (D, With the same hypothesis but replacing (A3) with the condition that (A3') for cocycle actions (A, α, u) and Proof.First we show that if (A1)-(A3) hold and (A, α, u), (B, β, v) are anomalous actions in R, then (A, α, u) ≃ (B, β, v) if and only if ) and also that F (α) ∼ F (β) as F is trivial when evaluated at inner automorphisms.We now turn to the converse.Suppose F (α) ∼ F (β) and o(α, u) = o(β, v).First note that this implies that also F (α ⊗ id D ) ∼ F (β ⊗ id D ).Indeed, by (A1) let φ A : A → A ⊗ D and φ B : B → B ⊗ D be isomorphisms which are approximately unitarily equivalent to the first factor embeddings and Φ : ) (and similarly replacing A by B).Hence we compute that We now prove our classification theorems Theorem 4.2.Let G be a finite group and A be a unital Kirchberg algebra satisfying the UCT with are anomalous actions of G on A with the Rokhlin property then (α, u) ≃ K (β, v) if and only if o(α, u) = o(β, v) and K i (α g ) = K i (β g ) for all g ∈ G and i = 0, 1.
Proof.We check that the hypothesis of Lemma 4.1 is satisfied.Let D = M |G| ∞ , F be the functor consisting of the pointed K 0 group direct sum the K 1 group and R the class of Rokhlin anomalous G-actions on unital Kirchberg algebras satisfying the UCT.That F is full on isomorphisms follows from [39].Condition (A1) follows from Proposition 3.6.For any ω ∈ Z 3 (G, T), we have actions (D, follows from [18, Theorem III.6] combined with [21, Lemma 3.12] as the actions (D, s ω G , u ω ) have the Rokhlin property (and hence property R ∞ ) by Proposition 2.1.Therefore, (A2) is also satisfied.Finally (A3') is satisfied by Izumi's classification result [22,Theorem 4.2] and that every cocycle action with the Rokhlin property is a unitary perturbation of a group action [22,Lemma 3.12].Theorem 4.3.Let G be a finite group and A be a unital, simple, nuclear TAF-algebra in the UCT class such that A ∼ = A ⊗ M |G| ∞ and (α, u), (β, v) are anomalous actions on A with the Rokhlin property, then (α, u) ≃ K (β, v) if and only if o(α, u) = o(β, v) and K i (α g ) = K i (β g ) for all g ∈ G.
Proof.We apply Lemma 4.1 with D = M |G| ∞ , R the class of Rokhlin anomalous actions on M |G| ∞ -stable unital, simple, separable, nuclear TAF-algebras satisfying the UCT and F the functor consisting of ordered K 0 and K 1 .Firstly, F is full on isomorphisms by [34].(A1) holds by Proposition 3.6.(A2) holds for the same reason as in the proof of Theorem 4.2.Condition (A3') follows from a combination of [22,Theorem 4.3] and [21,Lemma 3.12].
We have shown a classification of anomalous actions on some classes of simple C * -algebras.Such a classification also implies a classification of G-kernels, we illustrate it by using Theorem 4.2, the same argument may also be used to rewrite the results of Theorem 4.4, Theorem 4.3 and Corollary 4.5.
Corollary 4.6.Let A be a unital Kirchberg algebra satisfying the UCT with A ∼ = A⊗M |G| ∞ and α, β be G-kernels with the Rokhlin property on A. Then α and β are K trivially conjugate if and only if ob(α) = ob(β) and K i (α g ) = K i (β g ) for all g ∈ G and i = 0, 1.

Applications
We start this section by giving an alternative construction, of a (G, ω) action on the UHF algebra M |G| ∞ which is visibly compatible with a Bratteli diagram of M |G| ∞ .This action is an AF-action in the sense of [11] and [6,Definition 4.8] (see also the discussion in [16,Section 6.1]).The existence of an AF ω-anomalous action on M |G| ∞ follows from an adaptation of the Ocneanu compactness argument to the C * -setting ( [36]).We build it explicitily below.Proposition 5.1.Let G be a finite group and ω ∈ Z 3 (G, T), then there exists an AF-(G, ω) action with the Rokhlin property on M |G| ∞ .We denote this action by θ ω G .
Proof.In this proof we will use the symbols g, h, k, x, y, x i , y i , s i for i ∈ N to denote elements of the group G.
) be the multiplication operator by f .Consider the *homomorphisms ϕ n : ).The inductive system (A n , ϕ n ) has an inductive limit (we write the limit by A) which is known to be isomorphic to M |G| ∞ .Indeed, the Bratelli diagram of this AF-algebra is easily seen to be the complete bipartite graph on |G|-vertices, it is common knowledge that this coincides with the UHF-algebra of type |G| ∞ (see [9,Example III.2.4] for the case |G| = 2) We construct a (G, ω) action on each finite dimensional algebra A n such that the actions commute with the inclusion maps ϕ n .This will induce an AF ω-anomalous G action on M |G| ∞ by the universal property of the inductive limit (see [16,Section 6.1]).
Let e g,h ∈ B(l 2 (G)) be defined by with d n (g) defined inductively and for all n > 2 with the convention that x 0 = y 0 = k.As we have defined d n (g) on a spanning set of A n , d n (g) extend to linear maps from A n to itself.In fact each d n (g) is an endomorphism of A n .First, it is clear that they preserve the * -operation.To show the multiplicativity, it is sufficient to check on a spanning set.We show this by induction.For the case n = 2 it is only non-trivial to check that The left hand side is given by y 2 ) which coincides with the right hand side.To show that d n (g) is multiplicative for n > 2 it suffices to show that This follows immediately from the induction hypothesis and a direct computation of the left hand side (as in the case for n = 2).Notice that each d n (g) fixes elements of the form To construct a (G, ω) action on the first stage A 1 , we let u 1 (g, h)(k) = ω k −1 ,g,h .That (θ 1 , u 1 ) defines a (G, ω) action on C(G) is a straightforward computation (this is computed in [3,Section 4]).We proceed to extend this action on A 1 to all of M |G| ∞ through the inductive limit.Let u n (g, h) = ϕ 1,n (u 1 (g, h)) and θ n (g) = d n (g)θ ′ n (g).For the remaining part of the proof we check that (θ n , u n ) satisfy (1)-( 4) for all n ∈ N. We will repeatedly use the 3-cocycle formula during the calculations, instead of commenting on this every time, we will instead colour code the parts of our equations to which we apply the 3-cocycle formula.
We start by showing (1).Firstly, This holds trivially for n = 1.For n = 2 it follows from the 3-cocycle formula that We now proceed with an inductive argument for arbitrary n.We assume that (1) holds for n − 2, preforming a similar computation to the case n = 2; For (4) it suffices to show that ϕ n d n (g) = d n+1 (g)ϕ n .For n = 1 as d 1 is the identity map.The case n = 2 follows too Assuming that the case n − 2 holds, we now argue by induction, Condition ( 3) is immediate.It remains to show that (2) holds for arbitrary n.This follows from (2) for the case n = 1 and from (4).For To show that θ ω G has the Rokhlin property we construct a family of Rokhlin projections.The projections δ g ⊗ id B(l 2 (G)) ⊗n−1 ∈ Z(A n ) satisfy θ n (g)(δ h ⊗ id B(l 2 (G)) ⊗n−1 ) = δ gh ⊗ id B(l 2 (G)) ⊗n−1 and also g∈G δ g ⊗ id B(l 2 (G)) ⊗n−1 = id An .Therefore, the projections p g ∈ A ∞ with n-th coordinate given by ϕ n,∞ (δ g ⊗ id B(l 2 (G)) ⊗n−1 ) for g ∈ G satisfy the conditions of Definition 1.3.Remark 5.2.In the case that ω = 1 the construction in Proposition 5.1 greatly simplifies.Indeed, d n (g) is the identity automorphism and u n (g, h) is the unit for all g, h ∈ G and n ∈ N. Therefore, θ 1 G restricts to the group action θ n = λ G n−1 i=0 Ad(λ G ) on each A n with λ G the left regular representation.This action coincides with the infinite tensor product action s G (see Section 3).To see this, consider the inductive system (B n , φ n ) with B 2n−1 = A n , B 2n = n i=0 B(l 2 (G)) and φ 2n−1 (f ⊗ T ) = M f ⊗ T , φ 2n (S) = 1 ⊗ S for all n ∈ N, f ⊗ T ∈ A n and S ∈ B 2n .The even terms of the inductive system (B 2n , φ 2n+1 • φ 2n ) coincide with the inductive limit ( n i=1 B(l 2 (G)), M → id B(l 2 (G)) ⊗M).The odd terms (B 2n−1 , φ 2n • φ 2n−1 ) coincide with the inductive system (A n , ϕ n ) from the proof of Proposition 5.1.This allows to interpolate between ( n i=1 B(l 2 (G)), M → id B(l 2 (G)) ⊗M) and (A n , ϕ n ).It is immediate that θ G and s G are conjugate.Moreover, it follows from Theorem 4.3 that θ ω G is cocycle conjugate to s ω G for any ω ∈ Z 3 (G, T).We end this paper by studying to what extent Rokhlin anomalous actions on AF-algebras are AF-actions and vice versa.To do this, we will require results of [6].In [6], the authors associate an invariant to any AF-action F , of a fusion category C, on a AF-algebra A. Vaguely, this invariant consists of the K 0 -groups of all Q-system extensions of A by F and all natural maps between these extensions.The authors also show that any two AF-actions on AF-algebras A and B are equivalent if and only if their invariants are isomorphic.As observed in [6, Section 5.1], if the acting category C is torsion-free (see [1,Definition 3.7]), the invariant of [6] simplifies to just the module structure of K 0 (A) under the action of the fusion ring of C. We apply this when the acting category is Hilb(G, ω) and the action is induced by an anomalous action (α, u) as explained in [13,Proposition 5.6].The fusion ring of Hilb(G, ω) is Z[G] and the module structure of K 0 (A) is given by K 0 (α g ).
Corollary 5.3.Let G be a finite group and A a simple, unital AFalgebra such that A ∼ = A ⊗ M |G| ∞ .Let (α, u) be a (G, ω)-action on A such that K 0 (α g ) = id A for all g ∈ G.If (α, u) has the Rokhlin property, then (α, u) is an AF-action.Moreover, if [ω| H ] = 0 for any subgroup H < G then the converse holds.
Proof.If (α, u) is a (G, ω)-action with the Rokhlin property on an AFalgebra A, then by Theorem 4.3 it is cocycle conjugate to the AF ω-anomalous G-action id A ⊗ θ ω G on A. Therefore (α, u) is AF as (by definition) being AF is preserved under cocycle conjugacy (see [16,Remark 6.1.7]).
We now consider the converse statement.An AF ω-anomalous G action (α, u) induces an AF-action of the fusion category Hilb(G, ω) in the sense of [6] (to see how a (G, ω)-action induces a Hilb(G, ω) action see [13,Proposition 5.6], that this is AF is discussed [16,Remark 6.1.7]).By the hypothesis on ω, the fusion category Hilb(G, ω) is torsion free, so as K 0 (α g ) = id A and K 0 (id A ⊗ θ ω G ) = id A , then [6, Theorem A] yields that the AF ω-anomalous G actions induced by (α, u) and id A ⊗ θ ω G are cocycle conjugate.So (α, u) has the Rokhlin property.
Remark 5.4.One may drop the hypothesis that A ∼ = A ⊗ M |G| ∞ in Corollary 5.3 if one instead assumes that the anomaly ω of (α, u) is such that [ω] has order |G|.Indeed, it follows from [16,Corollary 5.4.4] that in this case A will automatically absorb M |G| ∞ .Also, note that under this assumption on [ω] it is automatic that [ω| H ] = 0 for any subgroup H < G.
The behavior observed in the converse of Corollary 5.3 is quite different from the behaviour of group actions.It was already observed in [14] that there exist AF-actions of Z 2 on M 2 ∞ which do not have the Rokhlin property.

Theorem 4 . 4 .
Let G be a finite group and (α, u), (β, v) be anomalous G actions with the Rokhlin property on W.Then (α, u) ≃ (β, v) if and only if o(α, u) = o(β, v).Proof.We check the conditions of Lemma 4.1 with D = M |G| ∞ , R the class of Rokhlin anomalous actions on W and F the trivial functor.Firstly, (A1) holds by Proposition 3.6.Moreover, (A2) holds as in the proof of Theorem 4.2.Finally, (A3) follows from[35, Corollary 3.7]  as every cocycle action of a finite group on W is cocycle conjugate to a group action (this follows as W ∼ = W ⊗ M |G| and hence [15, Remark 1.5] applies).In light of[5, Theorem B], it follows from[21, Theorem 3.5] that all Rokhlin anomalous actions of G on classifiable M |G| ∞ -stable C *algebras are classified up to cocycle conjugacy by their induced action on the total invariant KT u (see[5, Section 3]) and their anomaly.

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43, Lemma 2.1, Theorem 2.6] and [43, Theorem 3.7, Corollary 3.8] for example, make no use of the anomaly associated to (α, u) and (β, w) being trivial.Under this observation, we can state a specific case of [43, Corollary 3.8].Theorem 3.2 (cf.[43, Theorem 2.8]).Let A and D be separable C *algebras and G a finite group.Assume (α, u) : G A is an anomalous action.Let γ : G D be a group action such that (D, γ) is strongly self-absorbing.If there exists an equivariant and unital *