Sequentially split $*$-homomorphisms between $\mathrm{C}^*$-algebras

We define and examine sequentially split $*$-homomorphisms between $\mathrm{C}^*$-algebras and $\mathrm{C}^*$-dynamical systems. For a $*$-homomorphism, the property of being sequentially split can be regarded as an approximate weakening of being a split-injective inclusion of $\mathrm{C}^*$-algebras. We show for a sequentially split $*$-homomorphism that a multitude of $\mathrm{C}^*$-algebraic approximation properties pass from the target algebra to the domain algebra, including virtually all important approximation properties currently used in the classification theory of $\mathrm{C}^*$-algebras. We also discuss various settings in which sequentially split $*$-homomorphisms arise naturally from context. One particular class of examples arises from compact group actions with the Rokhlin property. This allows us to recover and extend the presently known permanence properties of Rokhlin actions with a unified conceptual approach and a simple proof. Moreover, this perspective allows us to obtain new results about such actions, such as a generalization of Izumi's original $K$-theory formula for the fixed point algebra, or duality between the Rokhlin property and approximate representability.


Introduction
In current C * -algebra theory, there is a substantial necessity to study interesting classes of examples of nuclear C * -algebras with the help of abstract regularity-type properties. The decisive motivation comes from the Elliott classification program for nuclear C * -algebras, and the fact that recent results in this area follow a rather abstract classification approach, in contrast to earlier results that presuppose some inductive limit decomposition of the C * -algebras under consideration. In this way, more satisfying and abstract classification results have emerged, largely depending on certain regularity properties.
On the one hand, these may include genuine (i.e. non-automatic) regularity properties such as finite decomposition rank [36], finite nuclear dimension [68], Z-stability [31,64] or regularity of the Cuntz semigroup [15,Section 3.2]. These properties play a key role in Toms-Winter's regularity conjecture, which asserts that these properties should be equivalent on a large scale [15,67,68]. On these properties alone, there exists a vast literature by now, and it would be impossible to do all the existing works justice within a mere paragraph of this introduction. Instead the reader is referred to [7] and the references therein, featuring an excellent overview of the current state-of-the-art in its introduction.
Another kind of regularity properties includes having finite tracial rank [38] or having generalized tracial rank at most one [24]. The latter class of C * -algebras is an important one because they have been recently shown to be both classifiable and to exhaust the Elliott invariant. Additionally, there are other kinds of more mysterious, yet important regularity-type properties for C * -algebras such as the universal coefficient theorem [53,54], where it is not known whether it is automatic for nuclear C * -algebras. In a very recent breakthrough by many hands [24,14,13,62], it has been shown that these (a priori) different regularity-type properties are linked. This has culminated in the classification of separable, unital, simple C * -algebras with finite nuclear dimension that satisfy the UCT, see [62,Section 6] for an overview.
If one wishes to show that a sample C * -algebra has one of the aforementioned C * -algebraic properties, then a frequent recipe for doing so is to find another C * -algebra, which is both easily seen to satisfy said property and has a certain connection to the original C * -algebra strong enough to carry it over. However, most of the proofs of this nature in the current literature follow a seemingly ad-hoc approach, and in particular it is often not clear what the crucial underlying common patterns really are. For instance, in order to study transformation group C * -algebras of the form C(X) ⋊ Z, one considers so-called orbit breaking subalgebras in the sense of Putnam [48]. Since these were shown by Lin-Phillips [40] to be ASH algebras whose inductive limit decomposition incorporates subsets of X, these subalgebras are easier to understand than the crossed product C * -algebra itself. Nevertheless, the subalgebra turns out to be large enough so that the crossed product inherits many of its properties. While this way of approach has been somewhat restricted only to a special crossed product setup for a long time, Phillips [47] has started to flesh out the general concept of a large subalgebra. This does not only have the obvious advantage of opening up a lot of potential further possible ways of employing such an approach in other contexts, but can overall improve the understanding of the previous special cases of application.
This paper aims to serve a similar purpose by introducing sequentially split * -homomorphisms between C * -algebras. This property of a * -homomorphism can be understood as an approximate weakening of being a splitinjective inclusion. In contrast to the large subalgebra setup, the existence of a sequentially split * -homomorphism from A to B guarantees that several approximation and perturbation properties pass from the larger algebra B to the smaller algebra A. As it turns out, there are many places in the literature, not least in lengthy and technical calculations, where such a setup has implicitly occurred before. Our motivation is to unify such arguments with a conceptual approach. Throughout the entire paper, we will thus have a great focus on drawing a parallel between some of our (partial) results and the literature, for example by recasting known concepts in the language of sequentially split * -homomorphisms.
Let us now describe how this paper is organized. In the first preliminary section, we will recall some notions and techniques related to sequence algebras and central sequence algebras. The techniques developed in Kirchberg's pioneering work [34] on this subject are quintessential for the entire paper to enable proper treatment of non-unital C * -algebras. For the reader's convencience and because we need some of these results in an equivariant context later, we will also reprove or prove slight variants of some of the results from [34], such as stability of the central sequence algebra.
In the second section, we introduce sequentially split * -homomorphisms between C * -algebras. We show that this notion behaves well with respect to standard constructions on separable C * -algebras, such as compositions, inductive limits, tensor products, or passing to hereditary subalgebras or quotients in a suitable manner. On the one hand, we will see that sequentially split * -homomorphisms are very rigid in the sense that they impose severe restrictions on the induced maps on C * -algebraic invariants, such as K-theory, traces or the Cuntz semigroup (cf. Theorem 2.8). On the other hand, the main result of the second section (cf. Theorem 2.9) asserts that if there exists any sequentially split * -homomorphism from a C * -algebra A to another C * -algebra B, than a host of C * -algebraic properties of interest pass from B to A. This includes many approximation properties such as regularity properties in the Toms-Winter conjecture, (generalized) tracial rank at most one, or being expressible as an inductive limit of certain weakly semiprojective C * -algebras, such as 1-NCCW complexes. Somewhat surprisingly, the property of being nuclear and satisfying the UCT also turns out to pass from B to A (cf. Theorem 2.10). This is a further manifestation of an observation by Dadarlat [10] that nuclearity together with the UCT can be understood as an approximation or perturbation property of some kind. To summarize, the abridged version of our combined main result in the second section reads as follows: Theorem. Let A and B be two C * -algebras. Suppose that there exists a sequentially split * -homomorphism from A to B. If B is classifiable in the sense of the Elliott program, then so is A.
In the third section, we consider sequentially split * -homomorphisms in the equivariant context, i.e. as a property of an equivariant * -homomorphism between C * -dynamical systems. This notion behaves well with respect to taking crossed products: if one has an equivariantly sequentially split *homomorphism between two C * -dynamical systems (A, α, G) and (B, β, G), then the induced * -homomorphism between A ⋊ α G and B ⋊ β G is also sequentially split. In the case that the acting group G is abelian, the induced map between the crossed products isĜ-equivariant (with respect to the dual actions), and is then also equivariantly sequentially split. Using Takaiduality, we will see that the converse holds as well (cf. Theorem 3.17). That is, a * -homomorphism ϕ : (A, α, G) → (B, β, G) is equivariantly sequentially split if and only if the dual morphismφ : (A ⋊ α G,α,Ĝ) → (B ⋊ β G,β,Ĝ) is equivariantly sequentially split.
In the fourth section, we will discuss several applications that we shall now summarize: First, we will see that a Rokhlin action α : G A of a compact group always gives rise to sequentially split * -homomorphisms A α → A and A ⋊ α G → A ⊗ K(L 2 (G)) in a natural way (cf. Theorem 4.6). We note that a similar observation was made by Gardella in [20], and also by the second author in [58] in the context of the continuous Rokhlin property. Therefore, the concept of sequentially split * -homomorphisms fleshes out many of the arguments related to permanence properties appearing in the literature of Rokhlin actions of either finite groups or compact groups [44,55,27,20]. Exploiting this viewpoint for Rokhlin actions some more, we prove a Ktheory formula for the fixed point algebra of a Rokhlin action (cf. Theorem 4.9), generalizing such a K-theory formula for finite groups established by Izumi in [29]. Another one of Izumi's observations from [29] is that for finite abelian groups, Rokhlin actions are in a natural way dual to approximately representable actions. Motivated by this, we extend Izumi-Matui's definition of approximately representable actions in [30] to the setting of discrete group actions on not necessarily unital C * -algebras. It turns out that, similar to the Rokhlin property, approximate representability can be characterized in terms of (equivariantly) sequentially split * -homomorphisms. Armed with this perspective, we then generalize Izumi's duality result and prove that Rokhlin actions of compact abelian groups are in a natural way dual to approximately representable actions of discrete abelian groups (cf. Theorem 4.27). Note that a similar observation was made by Gardella [18] for circle actions on unital C * -algebras; see also [19] for a further generalization in the unital case.
We then consider Osaka-Kodaka-Teruya's notion of an inclusion of unital C * -algebras A ⊂ B with the Rokhlin property. This was introduced in [43] and studied further in [45,46], motivated by the fact that in this setup many interesting C * -algebraic properties pass from B to A. We show that if an inclusion A ⊂ B has the Rokhlin property in the sense of [43], then the inclusion map is sequentially split (cf. Theorem 4.16). In particular, the main results of this paper recover and extend the previously known permanence properties for inclusions of unital C * -algebras with the Rokhlin property.
We continue in the fourth section by showing that for separable C *algebras, sequentially split * -homomorphisms can also be understood as a generalization of an existential embedding (cf. Theorem 4.19), as considered by Goldbring-Sinclair in [22,Section 2]. We note that since the initial preprint version of this paper was published, the strong connection between sequantially split * -homomorphisms and the model theory of C *algebras and C * -dynamical systems has been pinned down in [23,21]. In particular, it has since been discovered that the notion of sequentially split * -homomorphisms, restricted to separable C * -algebras, agrees with the notion of so-called positively existential embeddings.
We end the fourth section by considering C * -dynamical systems absorbing a given strongly self-absorbing action tensorially in the sense of [59]. As it turns out, this can also be expressed in the language of sequentially split * -homomorphisms. Exploiting some of our general observations for equivariantly sequentially split * -homomorphisms, we prove a few permanence properties for C * -dynamical systems absorbing a given strongly selfabsorbing action tensorially: this property turns out to be invariant under equivariant Morita equivalence, and moreover passes to a system that comes from an invariant, hereditary subalgebra.
We are confident that the concept of sequentially split * -homomorphisms has enough worth to be fleshed out, and is certain to find some possible new applications in the future. Since the initial preprint version of this paper was available, the recent papers [23,21] have emerged as evidence toward this claim. In work of the first author joint with Omland-Stammeier [3], the main results of this paper are applied for the purpose of K-theoretic computations. Moreover, building on the techniques presented in this paper, the authors have developed a theory for Rokhlin actions of compact quantum groups in collaboration with Christian Voigt; see [5]. We would like to thank him for some fruitful discussion on some of the topics of this paper. For the same reason, the authors are grateful to Ilijas Farah and Isaac Goldbring. We also thank Martino Lupini for pointing out an error in the first preprint version of this paper. We would like to express our gratitude to the referee for some useful suggestions.

Preliminaries
Let us first fix some notations: Notation. • If F is a finite subset inside some larger set M , we write F ⊂ ⊂M . • Let ε > 0 and a, b some elements in a normed space. We write a = ε b as a shortcut for a − b ≤ ε. • Let ε > 0 and let M 1 , M 2 be subsets of some normed space. If the distance from M 1 to M 2 is at most ε, then we write Let us now recall some notions related to sequence algebras and central sequence algebras. The techniques developed in Kirchberg's pioneering work [34] on this subject are quintessential for what follows: . Let A be a C * -algebra and B ⊂ A a C *subalgebra. We denote by Consider also the normalizer of B inside A There is a chain of inclusions where Ann(B, A) is an ideal in both of these C * -algebras. This allows one to define

Notation 1.2. Let
A be a C * -algebra and ω a free filter on N. Recall the (ω-)sequence algebra A ω of A given by Given a bounded sequence (a n ) n ∈ ℓ ∞ (N, A), the norm of the corresponding element in A ω is given by [(a n ) n ] = lim sup n→ω a n = inf J∈ω sup n∈J a n .
Then A embeds canonically into A ω as (representatives of) constant sequences. We will frequently use this identification of A inside A ω without mention. ). If B ⊂ A ω is a C * -subalgebra, then the constructions from 1.1 apply. In this context, we additionally set D B,Aω = B · A ω · B as a shortcut. If B is σ-unital, then the existence of a countable approximate unit in B, together with a reindexation argument, allows one to find a positive contraction e ∈ A ω with eb = be = b for all b ∈ B. Then e ∈ A ω ∩ B ′ and in fact, its class e + Ann(B, In the special case that we consider the standard inclusion A ⊂ A ω , we write D ω,A = D A,Aω and F ω (A) = F (A, A ω ). In particular, we see that if A is a σ-unital C * -algebra, then its central sequence algebra F ω (A) is unital. However, if A is in fact unital, then we see that Ann(A, A ω ) = 0 and so F ω (A) simply recovers the ordinary central sequence algebra A ω ∩ A ′ .
is a well-defined, natural * -homomorphism. This proves to be important in applications.
The following was proved by Kirchberg in [34, 1.9] for free ultrafilters on N. We give a more direct proof for arbitrary free filters on N, not making use of the full power of the ε-test as in [34].
(1) The canonical * -homomorphism N (B, A ω ) → M(B) given by the universal property of the multiplier algebra is surjective and its kernel coincides with Ann(B, A ω ).
Proof. (1): Since B ⊂ N (B, A ω ) is an ideal, there is a unique * -homomorphism π : N (B, A ω ) → M(B) extending the identity map on B. More explicitly, for m ∈ N (B, A ω ), π(m) is given by Moreover, Let h ∈ B be a strictly positive element. Let ρ ω : A ∞ → A ω denote the natural surjection. Take a positive lift k ∈ A ∞ for h and define Then C is a σ-unital C * -subalgebra of A ∞ satisfying ρ ω (C) = B. The surjection ρ ω : C → B extends to a strictly continuous * -epimorphism M(C) → M(B). Observe that ρ ω maps N (C, A ∞ ) into N (B, A ω ). Consider also the canonical * -homomorphism π ′ : N (C, A ∞ ) → M(C). We obtain a commutative diagram So in order to prove the assertion, we may without loss of generality restrict to the case ω = ∞. Let M ∈ M(B). For each n ∈ N, we find some k ∈ N such that a n = Let us now represent all these elements by bounded sequences, i.e. write , respectively. We find a strictly increasing sequence of natural numbers (n k ) k with the property that Define m ∈ A ∞ as the equivalence class of the bounded sequence given by Analogously, we also obtain that lim sup k→∞ h k m k − c k = 0. Hence, mh = M h and hm = hM . As h is strictly positive for B and mh, hm ∈ B, one concludes that m ∈ N (B, A ∞ ). As the multipliers M and π(m) coincide on the strictly positive element h ∈ B, we get that π(m) = M . This shows that π is surjective.
(2): First note that Ann(B, A ω ) = Ann(D B,Aω , A ω ), since the inclusion B ⊂ D B,Aω is non-degenerate. Therefore, (1) indeed induces an iso- Since y lies in the relative commutant of B, we know that [b, x] ∈ Ann(B, A ω ) for all b ∈ B by construction of π. But then xb = xb 1/2 b 1/2 = b 1/2 xb 1/2 = bx for all b ∈ B + , using this fact twice for the square root b 1/2 . Thus x ∈ A ω ∩ B ′ , which finishes the proof. Proposition 1.6. Let A be a C * -algebra, K the compact operators on some separable Hilbert space and ω a free filter on N. Then the canonical embed- Proof. Note that in the case of a finite-dimensional Hilbert space, this is obvious. In that case, for each n ∈ N, it is well-known that A ω ⊗ M n ∼ = (A ⊗ M n ) ω via the canonical embedding, and thus Now let K be the compacts on the separable and infinite-dimensional Hilbert space. If we view M n embedded into M n+1 as the upper left corner, then it follows that This finishes the proof.
The following is a well-known fact among C * -algebraists. However, we will include a proof for the reader's convenience, as the authors had trouble finding a reference where this is made explicit. Proposition 1.7. Let A be a C * -algebra and let K denote the C * -algebra of compact operators on some Hilbert space. Then Proof. Fix a generating set of matrix units {e kl | k, l ∈ I} for K, where I is an index set whose cardinality matches the dimension of the underlying Hilbert space. Let m ∈ M(A ⊗ K) ∩ (1 ⊗ K) ′ and (u λ ) λ an approximate unit for A. Using m(1 ⊗ e ij ) = (1 ⊗ e ij )m, we get m(u λ ⊗ e ii ) = m(u λ ⊗ e ij e jj e ji ) = (1 ⊗ e ij )m(u λ ⊗ e jj )(1 ⊗ e ji ) for i, j ∈ I. If i = j, this shows that m(u λ ⊗ e ii ) = a λ,i ⊗ e ii for some a λ,i ∈ A. The computation now shows that for all i, j ∈ I, This implies a λ,i = a λ,j = a λ for all i, j ∈ I and all λ. For L⊂ ⊂I and each λ, we get As the net i∈L u λ ⊗ e ii (λ,L) , with the obvious underlying directed set, is an approximate unit for A ⊗ K, we see that the left hand side of the equation converges strictly to m ∈ M(A ⊗ K). Hence, the net a λ ⊗ i∈L e ii (λ,L) also converges strictly to m ∈ M(A⊗ K). Let F ⊂ ⊂A be a finite subset and ε > 0. Fix some k ∈ I. Then for sufficiently large (λ 1 , L 1 ), (λ 2 , L 2 ) with k ∈ L 1 ∩L 2 and every a ′ ∈ F , we obtain and analogously a ′ (a λ 1 −a λ 2 ) ≤ ε. This shows that (a λ ) λ converges strictly to some c ∈ M(A). It follows that a λ ⊗ n i=1 e ii (λ,L) converges strictly to c ⊗ 1. This shows that m = c ⊗ 1 ∈ M(A) ⊗ 1. As the other inclusion is trivial, the proof is complete.

Corollary 1.8. Let
A be a C * -algebra, K the compact operators on some separable Hilbert space and ω a free filter on N. Then Proof. By 1.6, M(D ω,A⊗K ) ∼ = M(D ω,A ⊗ K) naturally. In particular, the canonical subalgebras 1 ⊗ K on both sides get identified under this isomorphism. The claim now follows directly from 1.7.
Next, we use this observation to prove that F ω (A) is a stable invariant for σ-unital C * -algebras. For free ultrafilters this is already known due to Kirchberg's pioneering work on central sequences of C * -algebras [34]. . Let A be a σ-unital C * -algebra, K the compact operators on some separable Hilbert space and ω a free filter on N. Then the canonical * -monomorphism given by the identifications of 1.5 (2) is an isomorphism.
Proof. By 1.5, it is enough to show that the inclusion which is induced by the first-factor embedding, is surjective. We have and hence this embedding is indeed onto.
We will also make use of the following standard fact:  [61, 3.9]). Let C be a class of separable, weakly semiprojective C * -algebras. Let A be a separable C * -algebra that can be locally approximated by C * -algebras in C. Then A is an inductive limit of C * -algebras in C.

2.
Sequentially split homomorphisms: The non-equivariant case where the horizontal map is the canonical inclusion. If ψ : B → A ∞ is a * -homomorphism fitting into the above diagram, then we say that ψ is an approximate left-inverse for ϕ. We say that A is sequentially dominated by B or that B sequentially dominates A, if there exists a sequentially split * -homomorphism ϕ : A → B.
Proof. Evidently, we only have to prove the "if"-part. Find * -linear maps ψ (m,n) : B → A, m, n ∈ N, such that (ψ (m,n) ) (m,n) : B → ℓ ∞ (N 2 , A) is well-defined and lifts ψ. As ψ is a * -homomorphism, we have for all a ∈ A by choice of ψ.
By the above, there is a sequence of natural numbers (m k ) k∈N such that It now follows directly from the construction that the Q[i]- * -linear map This shows that ϕ is sequentially split.

Proposition 2.3.
Restricted to separable C * -algebras, the composition of any two sequentially split * -homomorphisms is sequentially split.
Proof. Let A, B and C be separable C * -algebras and assume that ϕ : A → B and ψ : B → C are sequentially split * -homomorphisms. We obtain a commutative diagram where the horizontal maps are the respective standard embeddings. It now follows from 2.2 that ψ • ϕ : A → C is sequentially split.
Proof. The "only if" part is clear because for every such C, the * -homomorphism from 1.4 provides an approximate left-inverse for the first-factor embedding. So let us show the "if" part. Assuming that id A ⊗1 : Since the first-factor embedding is non-degenerate, the image of ψ must be contained in D ∞,A , and the resulting * -homomorphism ψ : A ⊗ max C → D ∞,A is also non-degenerate. Let us consider the unique strictly continuous extension ψ : M(A ⊗ max C) → M(D ∞,A ). Since ψ(a ⊗ 1) = a for every a ∈ A, it follows that ψ(1 ⊗ C) ⊂ M(D ∞,A ) ∩ A ′ . By 1.5(2), the right-hand side is naturally isomorphic to F ∞ (A), so this yields the existence of a unital * -homomorphism from C to F ∞ (A).

Proposition 2.5. Let
A be a separable C * -algebra and B a σ-unital C *algebra. Assume that ϕ : A → B is a sequentially split * -homomorphism. Let C be a unital, separable C * -algebra and assume that there exists a unital * -homomorphism from C to F ∞ (B). Then there exists a unital * - Proof. By 2.4, the first-factor embedding id B ⊗1 : B → B ⊗ max C is sequentially split. By the same argument as in the proof of 2.3, the composition Because A and C are separable, it follows from 2.2 that id A ⊗1 is sequentially split. The proof is completed with an application of 2.4.
Although not stated directly in these terms, an important result of Toms and Winter on tensorial absorption of strongly self-absorbing C * -algebras fits nicely into the picture of sequentially split * -homomorphisms: Theorem 2.6 (cf. [64, 2.3]). Let A be a separable C * -algebra and D a strongly self-absorbing C * -algebra.
if and only if the first factor embedding from A into A ⊗ D is sequentially split.
In this way, 2.4 gives a conceptual reason why Kirchberg's variant [34, 4.11] is essentially the same result. We note that Toms-Winter's theorem can be viewed as a stronger version of a result found in Rørdam's book [52, 7.2.2].
The following shows that the property of being sequentially split is compatible with inductive limits. Proposition 2.7. Let {A n , κ n } n∈N and {B n , θ n } n∈N be two inductive systems of separable C * -algebras with inductive limits A and B, respectively. Let ϕ n : A n → B n be a sequence of * -homomorphisms compatible with the two inductive systems, i.e. θ n •ϕ n = ϕ n+1 •κ n for all n. Denote by ϕ : A → B the induced * -homomorphism given by the universal property of the inductive limits. If each of the * -homomorphisms ϕ n is sequentially split, then ϕ is sequentially split.
Proof. For n ∈ N, let ψ n : B n → (A n ) ∞ be an approximate left-inverse for ϕ n . Let η n : A n → A and ε n : B n → B denote the canonical *homomorphisms. The ψ n give rise to a * -homomorphism Consider the embedding ι : B ֒−→ n∈N B n / n∈N B n given by the standard construction of the inductive limit, that is, for every n ∈ N and b ∈ B n . Observe that this notation makes sense, as only the tail of a representing sequence is of interest. Let ψ : B → (A ∞ ) ∞ be the * -homomorphism given as the composition ofψ with ι. For n ∈ N and a ∈ A n , we have that This shows that ψ • ϕ coincides with the standard embedding of A into (A ∞ ) ∞ . The claim now follows from 2.2.
As it turns out, the property of being a sequentially split * -homomorphism forces the induced maps on C * -algebraic invariants to be very tractable. Moreover, numerous C * -algebraic approximation properties pass from the target algebra to the domain algebra. The next three theorems make this explicit and form the main result of this section. Theorem 2.8. Let A and B be two C * -algebras. Assume that ϕ : A → B is a sequentially split * -homomorphism. Then: The same is true for K-theory with coefficients in Z n for all n ≥ 2.

(VII) The induced map between the simplices of tracial states
Then E = EAE, and in particular, e 1 ae 2 ∈ E for any e 1 , e 2 ∈ E and a ∈ A. We conclude that for e 1 , e 2 ∈ E and b ∈ B, So indeed, the restriction ϕ| E : E → ϕ(E)Bϕ(E) is sequentially split.
(II): Let J ⊂ A be an ideal and consider the induced * -homomorphism Hence, ϕ| J : J → Bϕ(J)B is sequentially split. This also shows that ψ induces a * -homomorphism It is clear from the construction thatψ • ϕ mod J recovers the standard embedding of A/J into its sequence algebra. This proves (II).
(III): Let I and J be two different ideals in A. We may assume that I J. By (II), ψ(Bϕ(J)B) ⊂ J ∞ and I ⊂ ψ(Bϕ(I)B) ⊂ I ∞ . As I J ∞ , the ideals Bϕ(I)B and Bϕ(J)B have to be different. This concludes the proof of (III).
(IV): Consider the canonical * -homomorphism There exists a commutative diagram showing that Cu(ϕ) is an order embedding. (VI): , it suffices to prove the claim for K 0 . Moreover, we may assume that A, B and ϕ are unital. Indeed, as ( we get that K 0 (ϕ) is injective. Let A, B and ϕ be unital and let p, By the definition of the K 0 -group, we find k, l ∈ N such that Since the relation of being a partial isometry is weakly stable, this implies that Injectivity of the induced map in K-theory with coefficients in Z n follows from the fact that [56, 6.4].
(VII): Let τ ∈ T (A) be a tracial state on A and ω a free ultrafilter on N. Consider the induced tracial state Then τ ′ = τ ω • ψ is a tracial state on B satisfying τ ′ • ϕ = τ . We conclude that T (ϕ) is surjective. Theorem 2.9. Let A and B be two separable C * -algebras. Assume that ϕ : A → B is a sequentially split * -homomorphism. The following properties pass from B to A: (4) decomposition rank at most r ∈ N. In fact, dr(A) ≤ dr(ϕ). (5) absorbing a given strongly self-absorbing C * -algebra D. We note that separability is only necessary in order to prove (5), (13), (14) and (15). For (16), σ-unitality is sufficient.
Proof. For what follows, ψ : B → A ∞ is an approximate left-inverse for ϕ.
(2): It is a well-known consequence of the Choi-Effros lifting theorem [9] that a C * -algebra is nuclear if and only if its standard embedding from A into A ∞ is nuclear. Now if ϕ : A → B is both nuclear and sequentially split, then this implies that the standard embedding from A into A ∞ is also necessarily nuclear.
(3) and (4): It is well-known that the nuclear dimension of a C * -algebra is identical to the nuclear dimension of its standard embedding into its sequence algebra, see [63, 2.5]. The same is true for decomposition rank. Thus the same argument as in (2) holds as the standard embedding of A factorizes through ϕ, and thus the claim follows.
(5): Assume that B ∼ = B ⊗ D. As pointed out in 2.6, this is equivalent to the first factor embedding B ֒−→ B ⊗ D being sequentially split. We obtain (6): Suppose that A, B and ϕ are unital. Assume that B is strongly selfabsorbing. In particular, B has approximately inner half-flip by [64, 1.5].
After lifting each v n to a sequence of unitaries, a simple diagonal sequence argument now shows that A also has approximately inner half-flip.
As B is strongly self-absorbing and admits a unital embedding into its own central sequence algebra, we can also embed A unitally into the central sequence algebra of B. So [52, 7.2.2] implies that B absorbs A tensorially. On the other hand, we can use (5) to see that A absorbs B tensorially. We conclude that A and B are isomorphic. (7): Suppose that A, B and ϕ are unital. If B is approximately divisible, then by definition, there exists a unital * -homomorphism Assume that B is purely infinite (see [33, 1.3] for a definition). Let a ∈ A be a positive element and let a ′ ∈ A be another positive element contained in the ideal generated by a.
A simple diagonal sequence argument yields elements c n ∈ A, n ∈ N, such that lim n→∞ c n ac * n = a ′ . Assume that A admits a character χ : A → C. Let ω be a free ultrafilter on N. Then This implies that χ ω • ψ : B → C is a character, which contradicts the fact that B is purely infinite. We have proven (8). (9): Let x, y ∈ Cu(A), n ∈ N with (n+1)x ≤ ny. Then (n+1) Cu(ϕ)(x) ≤ n Cu(ϕ)(y). As Cu(B) is almost unperforated (see [51,3.1] for a definition), Cu(ϕ)(x) ≤ Cu(ϕ)(y). As Cu(ϕ) is an order embedding by 2.8(V), we conclude that x ≤ y. This shows that Cu(A) is almost unperforated.
(10): It follows from [51, 3.2] and [2, 5.7] that a C * -algebra C has strict comparison (see [1, 7.6.4] for a definition) if and only if the uncompleted Cuntz semigroup W (C) is almost unperforated. Moreover, for any C *algebra, the uncompleted Cuntz semigroup is almost unperforated if and only if the Cuntz semigroup is unperforated, see [1, 7.6.4]. The claim now follows from (9). (11): Let a ∈ A be a self-adjoint element. If B has real rank zero, then there are mutually orthogonal projections p 1 , . . . , p n and λ 1 , . . . , λ n ∈ R such that Given m ∈ N, a diagonal sequence argument yields positive elements Since C n is weakly semiprojective, we find mutually orthogonal projection q 1 , . . . , q n ∈ A satisfying a = 3ε n j=1 λ j q j .
Hence, A has real rank zero. (12): Assume that B has stable rank one. Since (A ∞ ) ∼ ⊂ (A ∼ ) ∞ canonically, the map ϕ ∼ : A ∼ → B ∼ is sequentially split. Given a ∈ A ∼ , we find some invertible element b ∈ B ∼ such that ϕ ∼ (a) = ε b. Then ψ ∼ (b) ∈ (A ∼ ) ∞ is an invertible element satisfying a = ε ψ ∼ (b). As any invertible element in any sequence algebra lifts to a sequence of invertible elements, we can represent ψ ∼ (b) by a bounded sequence of invertible elements in A ∼ . Picking a suitable member of this sequence yields some invertible c ∈ A ∼ with a = 2ε c. Since ε > 0 was arbitrary, this shows that A has stable rank one. Now assume that B has almost stable rank one in the sense of Robert, see [50, 3.1]. Then the identical argument as above yields that any a ∈ A can be approximated by invertibles in A ∼ . Moreover, by 2.8(I), the same holds if we replace A by some hereditary subalgebra. This shows that A has almost stable rank one. (13): Assume that B is locally approximated by C * -algebras in C. Let F ⊂ ⊂A be a finite subset and ε > 0. Find some C * -subalgebra C ⊂ B such that C ∈ C and ϕ(F ) ⊂ ε C. Then F ⊂ ε ψ(C) in A ∞ . As C is weakly semiprojective, ψ| C : C → A ∞ lifts to a * -homomorphism C → ℓ ∞ (N, A). Composing it with the canonical projection onto a suitable coordinate ℓ ∞ (N, A) → A, we obtain a * -homomorphism κ : C → A satisfying F ⊂ 2ε κ(C). As κ(C) can be locally approximated by C * -algebras in C, we find a C * -subalgebra D ⊂ κ(C) ⊂ A such that D ∈ C and F ⊂ 3ε D. Hence, A can be locally approximated by C * -algebras in C. The corresponding statement about expressing A as an inductive limit now follows directly from 1.10. (14): Recall that 1-NCCW complexes are semiprojective, see [12, 6.2.2] and also [16, 3.4]. The same is true for finite dimensional C * -algebras. Applying (13) to the class C of all matrix algebras, we therefore conclude that the property of being UHF passes from B to A. Similarly, if C is the class of all finite dimensional C * -algebras, we get that A is an AF-algebra if this is true for B.
Assume now that B is an AT-algebra. Let C be the class of all C * -algebras of the form F 1 ⊕ F 2 ⊗ C(T), where F 1 and F 2 are finite-dimensional. Every quotient of a circle algebra can be locally approximated by C * -algebras in C. Hence, A can be written as an inductive limit of C * -algebras in C by (13). As every C * -algebra in C is also a quotient of a circle algebra, it follows from [52, 3.2

.3] that A is AT.
If B is an AI-algebra then it is also an AT-algebra. Indeed, AI-algebras are exactly the AT-algebras with trivial K 1 -group, see [52, 3.2.17]. By 2.8(VI), we get that K 1 (A) = 0 and we conclude that A is an AI-algebra.
Lastly, assume that B is expressible as an inductive limit of 1-NCCW complexes. Basically the same proof as in [24, 3.20] shows that every quotient of a 1-NCCW complex can be locally approximated by C * -algebras of the form C 1 ⊕ C 2 , where C 1 is finite dimensional and C 2 is a 1-NCCW complex. Every C * -algebra of this form is the image of a split surjection starting from some 1-NCCW complex. We therefore conclude that A is expressible as an inductive limit of 1-NCCW complexes. This shows (14). (15): Let S be either one of the following classes of C * -algebras: • all finite-dimensional C * -algebras; • all C * -algebras isomorphic to F 1 ⊕ F 2 ⊗ C[0, 1], where F 1 and F 2 are finite-dimensional; • All unital 1-NCCW-complexes (also known as Elliott-Thomson building blocks [24,Section 3]) and all finite-dimensional C * -algebras.
Assume now that B is simple, exact and B ∈ TAS, see [24, 9.4]. Then A is simple by (1), and moreover exact because ϕ is injective. This implies that T (A) = QT (A) by Haagerup's theorem [26]. Let us show that also A ∈ TAS. We note (as in the proof of (14)) that the class S consists of weakly semiprojective C * -algebras, and that quotients of C * -algebras in S can be locally approximated by C * -algebras in S. By [24, 9.11], B has strict comparison of positive elements, and so has A by (10). In particular, it suffices to show (cf. [38, 6.15]) that for every ε > 0 and F ⊂ ⊂A, there exists for all x ∈ F and τ ∈ T (A). Since B ∈ TAS, it follows that we can find for all x ∈ F and τ ∈ T (B). Consider the restriction ψ| C 1 : C 1 → A ∞ , and use weak semiprojectivity to lift this to a sequence of * -homomorphisms κ n : for all x ∈ F 1 . In particular, we can pick a member of this sequence κ : for all x ∈ F and τ ∈ T (A). Using the previously mentioned fact that quotients of C * -algebras in S are locally approximated by C * -algebras in S, we can find C ∈ S with C ⊂ κ(C 1 ) unitally, satisfying for all x ∈ F and τ ∈ T (A). Since ε > 0 and F ⊂ ⊂A were arbitrary, this shows that indeed A ∈ TAS. (16): By [28, 2.1 and 2.2], a σ-unital C * -algebra C is stable if and only if for each positive c ∈ C and ε > 0, there exists some d ∈ C such that d * d = ε c and d * ddd * ≤ ε. Assume that B is stable and let a ∈ A be positive. Then there exists some x ∈ B such that x * x = ε ϕ(a) and x * xxx * ≤ ε. If y = ψ(x) ∈ A ∞ , we therefore get that y * y = ε a and y * yyy * ≤ ε. Picking a suitable member z ∈ A of a representing sequence for y, we can arrange that z * z = 2ε a and z * zzz * ≤ 2ε. We conclude that A is stable. Somewhat less obvious than most of the properties listed in 2.9, it turns out that the UCT together with nuclearity is inherited under sequential dominance as well. We note that the key arguments in the proof below are a combination of a nearly identical argument due to Kirchberg in [32], where he reduces the UCT problem to the purely infinite setting, and a nearly identical argument due to Dadarlat in [11], where he gives a simplified proof of a special case of his theorem [10] that the UCT is a local property. Proof. Assume that B is nuclear and satisfies the UCT. Let ϕ : A → B be a sequentially split * -homomorphism. By 2.9(2), we already know that A is nuclear. As ϕ ∼ : A ∼ → B ∼ is also sequentially split, we may assume that A, B and ϕ were unital to begin with. By passing to the sequentially split (see 2.
is a unital embedding, and we define a *endomorphism {B, ψ B } denote the corresponding stationary inductive limits. Clearly, A ♯ and B ♯ are again separable, unital, nuclear and O ∞ -absorbing C * -algebras. Since for all x = 0, the element It is immediate that KK(ψ A ) = 1+KK(ι•κ•ϕ) = 1, since ι•κ•ϕ factors through O 2 . In particular, the connecting maps of this inductive system induce KK-equivalences. Hence it follows that the canonical embedding ψ A,∞ : A → A ♯ induces a KK-equivalence. This is implied by [11, 2.4], which basically boils down to plugging in the Milnor sequence [6, 21.5.2] for the functor KK( _ , B) in this situation. By a similar argument, we also get that the canonical embedding ψ B,∞ : B → B ♯ is a KK-equivalence. We conclude that A ♯ and B ♯ are unital Kirchberg algebras KK-equivalent to A and B, respectively.
Hence it suffices to show that A ♯ satisfies the UCT. Note that by construction, the following diagram commutes and thus induces a * -homomorphism Since B ♯ is a UCT Kirchberg algebra, it is expressible as an inductive limit of UCT Kirchberg algebras with finitely generated K-theory, see [52, 8.4.13]. These C * -algebras are known to be weakly semiprojective, see [39,57]. Using 2.9(13), we therefore conclude that A ♯ is expressible as an inductive limit of UCT Kirchberg algebras with finitely generated K-theory. This implies that A ♯ indeed satisfies the UCT, which finishes the proof.

Sequentially split homomorphisms: The equivariant case
Notation 3.1. Let G be a locally compact group, A a C * -algebra and α : G A a point-norm continuous action. Componentwise application of {α g } g∈G on representative sequences yields a (discrete) G-action α ∞ on A ∞ . If B ⊂ A ∞ is a (globally) α ∞ -invariant C * -subalgebra, then we get induced actionsα ∞ on F (B, A ∞ ) and also M(D B,A∞ ). 2 These actions are in general not continuous. However, we may restrict to the continuous parts of these actions, for instance for α ∞ on A ∞ we consider In this way, we obtain C * -dynamical systems (A ∞,α , α ∞ ), (M α (D B,A∞ ),α ∞ ) and (F α (B, A ∞ ),α ∞ ). For brevity, we denote F α (A, A ∞ ) = F ∞,α (A).

Remark 3.2.
The natural * -homomorphism in 1.4 restricts to a * -homomorphism which clearly isα ∞ ⊗α-to-α ∞ -equivariant. Observe that this * -homomorphism indeed maps into A ∞,α , since the action on the left-hand side is point-norm continuous.
where the horizontal map is the canonical inclusion. If ψ : (B, β) → (A ∞,α , α ∞ ) is an equivariant * -homomorphism fitting into the above diagram, then we say that ψ is an equivariant approximate left-inverse for ϕ.
Similarly as for path algebras in [25, 1.8], it turns out that continuous elements in the sequence algebra have a particularly strong continuity property also on the level of their representatives. We note that the technical proof appearing in the initial preprint version of this paper turned out to be redundant upon discovering a much more general result due to Brown [8].  (N, A), i.e. the map g → α g (x n ) n ∈ ℓ ∞ (N, A) is continuous. In particular, if x ∈ A ∞,α , then the following holds: For any g 0 ∈ G and δ > 0, there exists a neighbourhood U of g 0 such that Lemma 3.5. Let G be a locally compact group, A a C * -algebra and α : G A a continuous action. Let x ∈ A ∞,α and (x n ) n ∈ ℓ ∞ (N, A) a representing sequence. Then the following statement holds: For every compact set K ⊂ G and δ > 0, there exists some n 0 ∈ N such that for all g ∈ K, Proof. Let K ⊂ G be compact and δ > 0. Given g 0 ∈ K, we apply 3.4 and find some open neighbourhood U 0 of g 0 such that By passing to a possibly smaller open neighbourhood of g 0 and using the point-norm continuity of α ∞ on A ∞,α , we may assume that By compactness of K, we find some N ∈ N, {g j } N j=1 ⊂ K, an open covering K ⊂ N j=1 U j such that g j ∈ U j and sup for all j = 1, . . . , N and g ∈ U j . Moreover, we find n 0 ∈ N such that for j = 1, . . . , N , We then compute for g ∈ U j sup k≥n 0 Since the sets U j formed an open cover of K, this concludes the proof.
Using this simple technical tool, we can generalize most of the basic properties of sequentially split * -homomorphisms, which were shown in the second section, to the equivariant context. Since doing this is very routine, we recommend the reader to skip the next few technical statements upon first reading, and jump right ahead to 3.11, where we start discussing properties that are exclusively interesting in the equivariant context.
Proof. Since the "only if"-part is trivial, we only show the "if"-part. For this, let ψ : (B, β) → ((A ∞,α ) ∞,α∞ , (α ∞ ) ∞ ) be an equivariant * -homomorphism satisfying ψ • ϕ(a) = a for all a ∈ A. Find * -linear maps ψ m : Let H 1 ⊂ H 2 ⊂ . . . ⊂ ⊂G be an increasing sequence of finite subsets H k such that G ′ = k∈N H k is a dense subgroup of G. Let K ⊂ G be a compact neighbourhood of 1 G . Let S 1 ⊂ S 2 ⊂ . . . ⊂ ⊂A be an increasing sequence of finite sets S k such that A ′ = k∈N S k is dense in A. Let F 1 ⊂ F 2 ⊂ . . . ⊂ ⊂B be an increasing sequence of finite sets F k such that B ′ = k∈N F k is a dense Q[i]- * -subalgebra of B. We may assume that ϕ(A ′ ) ⊂ B ′ and β g (B ′ ) = B ′ for all g ∈ G ′ .
As ψ : (B, β) → ((A ∞,α ) ∞,α∞ , (α ∞ ) ∞ ) is an equivariant * -homomorphism, we find a sequence of natural numbers (m k ) k such that for k ∈ N, h ∈ H k Using furthermore that ϕ • ψ coincides with the standard embedding A ֒−→ (A ∞,α ) ∞,α∞ , we may also assume that for all k ∈ N and a ∈ S k , Applying 3.5, we may assume that for all k ∈ N, b ∈ F k and g ∈ K, we have Similarly, we find a sequence of natural numbers (n k ) k such that for k ∈ N, Moreover, we may assume that for all k ∈ N and a ∈ S k , Using that for k ∈ N and b ∈ B, ψ m k (b) = [(ψ (m k ,n) (b)) n ] ∈ A ∞,α , we may by 3.5 also assume that for g ∈ K, that by construction is a contractive * -homomorphism satisfying ψ ′ • ϕ(a) = a for all a ∈ A ′ . As ψ ′ is contractive, it extends uniquely to a * -homomorphism ψ ′ : B → A ∞ . Using that A ′ ⊂ A is dense, we conclude that ψ ′ • ϕ coincides with the canonical embedding A ֒−→ A ∞ . It remains to show that ψ ′ is β-to-α ∞ -equivariant. It follows from the construction of ψ ′ that α ∞,g • ψ ′ (b) = ψ ′ • β g (b) for all b ∈ B ′ and g ∈ G ′ .
As B ′ ⊂ B is dense, we conclude that ψ ′ is equivariant with respect to the induced G ′ -actions. We claim that ψ ′ (B) ⊂ A ∞,α . For b ∈ B ′ and g ∈ K, As β is a continuous action, we find for given δ > 0 an open neighbourhood we conclude that ψ ′ indeed maps into A ∞,α . Now let b ∈ B, g ∈ G and find a sequence (g n ) n ⊂ G ′ that converges to g. Using that ψ ′ is equivariant with respect to the G ′ -actions and that ψ ′ (B) ⊂ A ∞,α , we get that This shows that ϕ : (A, α) → (B, β) is sequentially split.
As in the non-equivariant case, one can use 3.6 in order to conclude that compositions of equivariantly sequentially split * -homomorphisms are equivariantly sequentially split. We omit the proof as it is completely analogous.

Proposition 3.7.
Restricted to separable C * -algebras with point-norm continuous actions by second countable, locally compact groups, the composition of two equivariantly sequentially split * -homomorphisms is equivariantly sequentially split.
The following is completely analogous to 2.8(I). Next comes the equivariant analogue of 2.7. Since its proof is analogous, we omit it.  α) and (B, β), respectively. Let ϕ n : (A n , α n ) → (B n , β n ) be a sequence of equivariant * -homomorphisms compatible with the two inductive systems, i.e. θ n • ϕ n = ϕ n+1 • κ n for all n. Denote by ϕ : (A, α) → (B, β) the induced * -homomorphism given by the universal property of the inductive limits. If each of the * -homomorphisms ϕ n is equivariantly sequentially split, then ϕ is equivariantly sequentially split.
Remark 3.10. The notion of equivariantly sequentially split homomorphisms can be extended in a straightforward way to C * -algebras equipped with endomorphic actions by semigroups. More precisely, given C * -algebras A and B, a discrete semigroup P , and actions α : P A and β : P B by endomorphisms, an equivariant * -homomorphism ϕ : (A, α) → (B, β) is called (equivariantly) sequentially split, if there exists an equivariant *homomorphism ψ : (B, β) → (A ∞ , α ∞ ) making the following diagram commute At least when A, B and ϕ are unital, this notion behaves well with respect to the corresponding semigroup crossed products, that is, the C * -algebras that are universal for unital covariant pairs; see [37] for a precise definition. In this situation, many of the facts proved in this section hold as well. This is used in [3, Section 5] as an important tool related to the computation of the K-groups of C * -algebras arising from integral dynamics. However, we will not pursue this type of generalization in this paper.
From now on, we start discussing properties of sequentially split * -homomorphisms that are exclusively interesting in the equivariant context. Namely, they turn out to enjoy the following crucial functoriality property with respect to formation of crossed products. (i) The induced * -homomorphism ϕ ⋊ G : A ⋊ α G → B ⋊ β G between the crossed products is sequentially split. (ii) If G is abelian, then the dual morphismφ : is (Ĝ-)equivariantly sequentially split.
Proof. Let ψ : (B, β) → (A ∞,α , α ∞ ) be an equivariant approximate leftinverse for ϕ. We obtain a commutative diagram There is a canonical * -homomorphism A ∞,α ⋊ α∞ G → (A ⋊ α G) ∞ extending the canonical inclusion of A ⋊ α G into its sequence algebra. This shows that ϕ ⋊ G : A ⋊ α G → B ⋊ β G is sequentially split. If G is abelian, ψ ⋊ G and ϕ ⋊ G are equivariant with respect to the respective dual actions of G. Moreover, the canonical * -homomorphism If the acting group is compact, a similar functoriality also applies to the fixed point algebra. For the proof, we need the following fact. A be a C  *  -algebra, G a compact group and α : G A a continuous action. Then the canonical embedding from (A α ) ∞ into (A ∞ ) α∞ is an isomorphism.

Lemma 3.12. Let
Proof. Let x ∈ (A ∞ ) α∞ be represented by a bounded sequence (x n ) n in A. Since in particular x ∈ A ∞,α , it follows from 3.5 that Let µ be the normalized Haar measure on G. In particular, we have showing that x can be represented by a sequence in A α .

Remark 3.15. Let
A be a C * -algebra, G a locally compact, abelian group and α : G A a continuous action. By the Takai duality theorem [60], it is well-known that (A⋊ α G⋊αĜ,α) is conjugate to (A⊗K(L 2 (G)), α⊗ρ), where ρ is the G-action on K(L 2 (G)) induced by the right-regular representation. Moreover, this isomorphism is natural in (A, α). In particular, this means that for any equivariant * -homomorphism ϕ : (A, α) → (B, β), there are equivariant isomorphisms such that the following diagram is commutative: Combining this with 3.14, we obtain the following: is equivariantly sequentially split.
This, in turn, immediately implies the following duality: Proof. This follows from 3.11 and 3.16.

Rokhlin actions of compact groups.
Definition 4.1 (cf. [27]). Let A be a separable C * -algebra and G a second countable, compact group. Let σ : G C(G) denote the canonical G-shift, that is, σ g (f ) = f (g −1 · _) for all f ∈ C(G) and g ∈ G. A continuous action α : G A is said to have the Rokhlin property if there exists a unital and equivariant * -homomorphism The Rokhlin property turns out to fit formidably into the concept of sequentially split * -homomorphisms. This will be a consequence of the following equivariant generalization of 2.4. The proof is a straightforward generalization from the non-equivariant case.  For the rest of this subsection, we will use this observation to provide a conceptual proof of the fact that crossed product C * -algebras by Rokhlin actions of compact groups inherit many properties from the coefficient C *algebra. But first, we need some preparation.

Notation 4.4. Let
A be a C * -algebra, G a compact group and α : G → Aut(A) a continuous action. We will denote byᾱ ∈ Aut(C(G) ⊗ A) the induced automorphism given bȳ α(f )(g) = α g (f (g)) for all g ∈ G and f ∈ C(G, A).
Moreover, we will denote by α co : A → C(G) ⊗ A the corresponding coaction of C(G), viewed as a Hopf-C * -algebra, on A. That is, α co (a)(g) = α g (a) for all g ∈ G and a ∈ A.
The following is a well-known fact:

Proposition 4.5. Let
A be a C * -algebra, G a compact group and α : G A continuous action. Then the C * We note that variants of the following statement have been observed by Gardella in [20, Section 2] and by the second author in the proofs of [58, 2.5, 2.6]. Proof. (i): Since α has the Rokhlin property, 4.3 implies that is sequentially split. As G is compact, we can apply 3.13 and conclude that is also sequentially split. One easily checks that a function f ∈ C(G, A) is fixed under σ ⊗ α if and only if there is some a ∈ A such that f (g) = α g (a) for all g ∈ G. In particular, α co : A → (C(G) ⊗ A) σ⊗α is an isomorphism. Moreover the following diagram commutes: This shows that the canonical embedding A α ֒−→ A is sequentially split.
The second statement (ii) follows directly from 3.11, 4.3 and 4.5.
The following result arises as an immediate consequence, and generalizes many permanence property results of [44,55,27,20]. The statement about the UCT is a significant improvement of the main result of [58]. Corollary 4.7. Let A be a separable C * -algebra, G a second-countable, compact group and α : G A a continuous action with the Rokhlin property. Then all the properties listed in 2.9 pass from A to A α and from A ⊗ K(L 2 (G)) to A ⋊ α G. Moreover, if A is nuclear and satisfies the UCT, then so do A α and A ⋊ α G. Remark 4.8. Another consequence of 4.6 (together with 2.8(VI)) is that the canonical inclusion A α ֒−→ A is injective in K-theory, whenever α has the Rokhlin property. In the case that G is finite and the C * -algebra A is unital and simple, this was shown by Izumi [29, 3.13]. Izumi moreover proved that for Rokhlin actions of finite groups, the image K * (A α ) → K * (A) coincides with the subgroup of fixed points of the induced action on the Ktheory group K * (A). This striking result allows one, in contrast to many situations where the Rokhlin property is absent, to determine the K-theory of the crossed product in a very straightforward manner. Using the language of sequentially split * -homomorphisms, we will now see that this generalizes to the case of compact group actions with the Rokhlin property on separable C * -algebras. Theorem 4.9. Let A be a separable C * -algebra, G a second countable, compact group and α : G A a continuous action. If α has the Rokhlin property, then The analogous statement is true for K-theory with coefficients.
Find some n ∈ N and a projection p ∈ M n (Ã) such that Here, ε :Ã → C is the canonical character, which we also view as extended to ε : (C(G) ⊗ A) ∼ → C. Observe that we denote the matrix amplification of a * -homomorphisms again by the same symbol. We have By definition of the K 0 -group, we find k, l ≥ 0 satisfying Write r = n + k + l. As α has the Rokhlin property and the property of being equivariantly sequentially split passes to unitazations and matrix amplifications (endowed with the respective G-actions), we conclude from 4.3 that be an equivariant approximate left-inverse for (1 ⊗ id A ) ∼ . We have that By equivariance of ψ and 3.12, we get that Since the relation of being a partial isometry with a fixed range projection is weakly stable, this shows that there exists some projection q ∈ M r ((A α ) ∼ ) with the property that By definition of K 0 (A), we get that and we conclude that x ∈ im(K 0 (A α ) ֒−→ K 0 (A)).
For the assertion for K 1 , observe first that the continuous action Sα : G SA on the suspension has the Rokhlin property. The fixed point algebra (SA) Sα equals SA α . Therefore, the injective homomorphism One also has (Sα) co = Sα co . Hence, the assertion for K 0 ((SA) Sα ) ֒−→ K 0 (SA) translates to This concludes the proof for K * . The assertion for K-theory with coefficients in Z n , n ≥ 2, follows from the K 0 -formula with the completely analogous argument, by tensoring with the trivial action on O n+1 instead of taking the suspension.

Inclusions of C * -algebras with the Rokhlin property.
In [43], Osaka, Kodaka and Teruya defined the Rokhlin property for an inclusion A ⊂ B of unital C * -algebras. Definition 4.10 (cf. [65]). Let B be a unital C * -algebra and A ⊂ B a unital C * -subalgebra. Moreover, let E : B → A be a conditional expectation. Then E is said to have a quasi-basis, if there exist elements u 1 , v 1 , . . . , u n , v n ∈ B such that In this case, one defines the Watatani index of E as If A ⊂ B is some inclusion of unital C * -algebras such that there exists a conditional expectation E : B → A with a quasi-basis, one also says that this inclusion has finite Watatani Index.   Remark 4.14. It turns out that this notion is indeed a generalization of finite group actions with the Rokhlin property. It is a simple exercise to show the following, see also [43, 3.2]: Let α : G A be a finite group action on a separable, unital, simple C * -algebra. Consider the conditional expectation E : A → A α onto the fixed point algebra given by E(a) = |G| −1 g∈G α g (a), which is well-known to have finite index |G| −1 1 A . Then the inclusion A α ⊂ A has the Rokhlin property if and only if the action α has the Rokhlin property.
Then this is obviously a u.c.p. map. Since p is in the central sequence algebra of B, we apply 4.15 and obtain for all x, y ∈ B. By uniqueness, we obtain ψ(xy) = ψ(x)ψ(y) for all x, y ∈ B, and thus ψ is a unital * -homomorphism. Lastly, observe that for all a ∈ A. This finishes the proof.
Combining this observation with the permanence properties established in the second section, we can recover and extend the main results of [43,45,46]: Corollary 4.17 (cf. [43,45,46]). Let A ⊂ B be an inclusion of separable, unital C * -algebras with the Rokhlin property. If B satisfies any of the properties listed in 2.9, then so does A. Moreover, if B is nuclear and satisfies the UCT, then so does A.

Existential embeddings.
Let us briefly recall the notion of an existential embedding, which originally stems from model theory of metric structures and was introduced for C *algebras in [22]. See also [17]. As it turns out, an existential embedding into a separable C * -algebra is a special case of a sequentially split * -homomorphism. We thank Ilijas Farah for pointing this out to us. Also compare with [22, 2.14]. Proof. Assume that ι is existential. Without loss of generality, assume that A ⊂ B and ι is the inclusion map. As A and B are separable, we can choose countable, dense, Q[i]- * -subalgebras A ′ ⊂ A and B ′ ⊂ B. Moreover, we may choose this in such a way that A ′ = A ∩ B ′ . Let us consider the countable set of indeterminants {X b } b∈B ′ indexed over B ′ . Then the expressions and a ∈ A ′ , define a countable set of quantifierfree formulas with parameters in Q[i] and A ′ ⊂ A. Evaluating X b = b for all b ∈ B ′ yields that in B, the norms of the above formulas evaluate at zero simultaneously. Note that the minimum of finitely many quantifier-free formulas is a quantifier-free formula. As ι is an existential embedding, we have for every H⊂ ⊂Q[i], r > 0 and F ⊂ ⊂B ′ ≤r that inf Now pick increasing finite sets F n ⊂ ⊂B ′ with B ′ = n∈N F n . By applying the above condition, it follows that there exist sequences x and a ∈ A ′ . These relations imply that the map ψ : ) n ] is a well-defined, contractive * -homomorphism with ψ(ι(a)) = ψ(a) = a for all a ∈ A ′ . Since A ′ ⊂ A and B ′ ⊂ B are dense, it follows that there is a unique continuous extension ψ : B → A ∞ , which is a * -homomorphism with ψ(ι(a)) = a for all a ∈ A. Thus ψ is an approximate left-inverse for ι.

Remark 4.20.
Let ω ∈ βN \ N be a free ultrafilter. Using some more continuous logic from [17], one can improve 4.19 and show that in fact, a * -homomorphism ι : A → B between separable C * -algebras is an existential embedding if and only if ι has a faithful, approximate left-inverse into A ω . By separability, this is equivalent to having it into A ∞ , by virtue of a reindexation argument. This was pointed out to the authors by Ilijas Farah in personal communication. As this would require recalling more machinery from [17,22], we omit the proof for the sake of brevity. Since the initial preprint version of this paper was available, more connections to model theory were discovered in [23,21]. In particular, it turns out that a * -homomorphism between separable C * -algebras is sequentially split if and only if it is positively existential.    Notation. Let G be a locally compact group. The canonical unitary representation G → U(M(C * (G))) will be denoted by g → λ G g . If α : G A is a continuous action on a C * -algebra, we denote the canonical unitary representation G → U(M(A ⋊ α G)) by g → λ α g . Like the Rokhlin property, approximate representability can be characterized in terms of sequentially split * -homomorphisms: x h e 2 = x h ex * n,h x n,h e = α h (e)x n,h e ∈ D ∞,A . A similar computation shows that e 2 x h = ex h α h −1 (e) ∈ D ∞,A . Since e 2 is also strictly positive as an element in D ∞,A , we conclude that (1), so the elements x * h x h and x h x * h act like a unit on D ∞,A . Condition (2) means x g x h − x gh ∈ Ann(A, A ∞ ) and condition (4) implies . We may assume that each x n,h is a contraction. Using that this * -homomorphism is equivariant with respect to the induced H-actions, one checks that the x n,h , n ∈ N and h ∈ H, must satisfy the conditions (1)-(4) from 4.21.
We continue with a duality result for actions of second-countable, compact, abelian groups with the Rokhlin property and approximately representable actions of countable, discrete, abelian groups on separable C *algebras. This generalizes the well-known duality result by Izumi [29, 3.8] in the case of finite abelian group actions on separable, unital C * -algebras. Note that Gardella [18] has observed a similar phenomonon for circle actions on unital C * -algebras; see also [19] for a further generalization in the unital case. The essential ingredients of the proof will be the characterizations 4.3 and 4.23 of the Rokhlin property and approximate representability in terms of sequentially split * -homomorphisms. The duality result will turn out to be an application of the general duality principle 3.17. Before turning to the proof, some further preparation is needed. The following result is well-known.
The following is taken from the proof of the Takai duality theorem [60] presented in [66, 7.1], which is a variant of Raeburn's proof [49].
Let α : G A be a continuous action. By 3.17, we know that is G-equivariantly sequentially split if and only if the induced * -homomorphism between the crossed products isĜ-equivariantly sequentially split. By applying 4.25, we obtain a commutative diagram We conclude that ι A is H-equivariantly sequentially split if and only if 1 ⊗ id A⋊ β H isĤ-equivariantly sequentially split. It now follows from 4.3 and 4.23 that β is approximately representable if and only ifβ has the Rokhlin property. This shows (ii) and and finishes the proof.

Strongly self-absorbing actions.
Let us briefly recall from [59] the definition of a strongly self-absorbing action: Definition 4.28. Let D be a separable, unital C * -algebra and G a secondcountable, locally compact group. Let γ : G D be a continuous action. We say that (D, γ) is a strongly self-absorbing C * -dynamical system, or that γ is a strongly self-absorbing action, if the equivariant first-factor embedding id D ⊗1 D : (D, γ) → (D ⊗ D, γ ⊗ γ) is approximately G-unitarily equivalent to an isomorphism µ, that is, there exists a sequence u n ∈ U(D ⊗ D) such that µ = lim n→∞ Ad(u n ) • (id D ⊗1 D ) and lim n→∞ (γ ⊗ γ) g (u n ) − u n = 0 uniformly on compact subsets of G.
Combining the main result of [59] with 4.2, we see that equivariant tensorial absorption of a strongly self-absorbing action can also be expressed in the language of sequentially split * -homomorphisms: Theorem 4.29 (cf. [59, 3.7]). Let G be a second-countable, locally compact group. Let A be a separable C * -algebra and α : G A a continuous action. Let D be a separable, unital C * -algebra and γ : G D a continuous action such that (D, γ) is strongly self-absorbing. The following are equivalent: (i) (A, α) is strongly cocycle conjugate to (A ⊗ D, α ⊗ γ).
With the help of some observations from the third section, we deduce the following interesting consequences with the help of the above perspective: Theorem 4.30. Let G be a second-countable, locally compact group. Let A be a separable C * -algebra and α : G A a continuous action. Let D be a separable, unital C * -algebra and γ : G D a strongly self-absorbing action. Assume that (A, α) is (strongly) cocycle conjugate to (A ⊗ D, α ⊗ γ).
Remark. One can also obtain the two theorems above for semi-strongly self-absorbing actions γ (see [59,Section 4]) with the identical argument.