On the unbounded picture of KK-theory

In the founding paper on unbounded KK-theory it was established by Baaj and Julg that the bounded transform, which associates a class in KK-theory to any unbounded Kasparov module, is a surjective homomorphism (under a separability assumption). In this paper, we provide an equivalence relation on unbounded Kasparov modules which describes the kernel of the bounded transform, and we thus complete the unbounded picture of KK-theory from the topological point of view. The equivalence relation consists of homotopies together with a new kind of degeneracy condition on unbounded Kasparov modules, which we term spectral decomposability since these unbounded Kasparov modules admit a decomposition into a part with positive spectrum and a part with negative spectrum.


Introduction
KK-theory, as introduced by Kasparov in [Kas75,Kas80b], has its roots in the Brown-Douglas-Fillmore extension theory of commutative C * -algebras, [BDF77], and in Atiyah's axiomatization of properties of elliptic operators on manifolds, [Ati70]. But KK-theory extends far beyond the context of commutative C *algebras and has become an important tool for accessing the algebraic topology of C * -algebras with applications ranging from Elliott's classification program to all aspects of index theory.
In practice, explicit classes in KK-theory often come from unbounded operators acting on Hilbert C * -modules and they constitute the main ingredient in the unbounded picture of KK-theory. The cycles there are called unbounded Kasparov modules and are mostly of a differential geometric origin with prototypical examples being Dirac operators (in the case of K-homology) or multiplication operators by symbols of Dirac operator (in the case of K-theory). In the unbounded picture, the relationship between KK-theory and the program of Connes on noncommutative geometry is in fact immediate: spectral triples are without any further modifications examples of unbounded Kasparov modules, [Con94,Con96].
The passage from the unbounded picture to the more commonly encountered bounded picture of KK-theory is furnished by the bounded transform which turns an unbounded Kasparov module into a class in KK-theory via the smooth approximation of the sign function given by t → t(1 + t 2 ) −1/2 . The richness of the unbounded picture is then witnessed by a fundamental theorem of Baaj and Julg stating that any class in KK-theory comes from an unbounded Kasparov module so that the bounded transform is in fact a surjection (under a mild separability condition), [BaJu83].
In this paper, we construct an equivalence relation on unbounded Kasparov modules which captures the kernel of the bounded transform and this equivalence relation is "geometric" in the sense that it can be formulated without any reference to the bounded picture of KK-theory. Our equivalence relation relies on a novel kind of degenerate elements together with a notion of homotopies using families of unbounded Kasparov modules parametrized by the unit interval. Our degeneracy condition on an unbounded Kasparov module is formulated in terms of a spectral decomposition of the unbounded operator in question building on the simple observation that the phase of an unbounded selfadjoint and regular operator with strictly positive spectrum is equal to the identity operator.
In summary, we complete the unbounded picture of KK-theory by defining a notion of topological unbounded KK-theory and showing that topological unbounded KK-theory is isomorphic to KK-theory via the bounded transform. We hereby give an affirmative answer to a question raised by Deeley, Goffeng, and Mesland in [DGM18]. It would thus be interesting to investigate the relationship between our topological unbounded KK-theory and the bordism group introduced in [DGM18], where the equivalence relation comes from Hilsum's notion of bordisms of unbounded Kasparov modules, [Hil10].
The word topological is a keyword in connection with our definition of unbounded KK-theory. In other approaches to unbounded KK-theory, the aim is to find an interesting equivalence relation which captures geometric content at the level of unbounded Kasparov modules while still admitting explicit formulae for the interior Kasparov product. The geometric content which could be valuable in this respect relates to the asymptotic behaviour of eigenvalues and the spectral metric aspects of noncommutative geometry. Certainly, this geometric content is invisible from a topological point of view and thus in particular from the point of view of topological unbounded KK-theory. The much more delicate questions on the geometric nature of unbounded KK-theory are part of ongoing research on the unbounded Kasparov product and the interested reader can consult the following (incomplete list of) references: [Mes14,KaLe13,BMvS16,MeRe16,Kaa16,KavS18].
1.1. Acknowledgements. The starting point for this paper was a couple of conversations with Bram Mesland during the thematic programme on "Bivariant K-theory in Geometry and Physics" at the Erwin Schrödinger Institute in Vienna in November 2018. I would like to thank the ESI for their hospitality and support and the organizers of the thematic programme for this great opportunity to meet and discuss the unbounded approach to KK-theory and its applications in mathematical physics.
1.2. Standing assumptions. Throughout this text A = A 0 ⊕ A 1 and B = B 0 ⊕ B 1 will be Z/2Z-graded C * -algebras with A separable and B σ-unital (meaning that B has a countable approximate identity). We moreover fix a norm-dense Z/2Z-graded * -subalgebra A ⊆ A, which we require to be generated as a * -algebra by some countable subset {x j | j ∈ N} ⊆ A .

Kasparov modules and KK-theory
In this section we give a brief summary of the main definitions concerning Kasparov modules and KK-theory. For more details the reader can consult the following references: [Kas80b,Bla98,JeTh91]. The commutators appearing in this section are all graded commutators. For a Z/2Z-graded C * -correspondence X from A to B we usually suppress the even * -homomorphism π X : A → L(X), which determines the left action of A on X (where L(X) denotes the Z/2Z-graded C * -algebra of bounded adjointable operators on X).
Definition 2.2. Two Kasparov modules (X 0 , F 0 ) and (X 1 , F 1 ) from A to B are unitarily equivalent when there exists an even unitary isomorphism of Z/2Z-graded C * -correspondences U : X 0 → X 1 such that UF 0 U * = F 1 . In this case we write (X 0 , F 0 ) ∼ u (X 1 , F 1 ). Remark that U (by definition) has to intertwine the left actions as well so that Uπ X 0 (a)U * = π X 1 (a) for all a ∈ A.
When given a Z/2Z-graded C * -algebra C and an even * -homomorphism β : B → C we may "change the base" of a Z/2Z-graded C * -correspondence X from A to B. Indeed, we may consider C as a Z/2Z-graded C * -correspondence from B to C and form the interior tensor product X ⊗ β C which is a Z/2Z-graded C * -correspondence from A to C. Any bounded adjointable operator T : X → X then induces a bounded adjointable operator T ⊗ β 1 : X ⊗ β C → X ⊗ β C and this operation yields an even * -homomorphism L(X) → L(X ⊗ β C), see for example [Lan95,Chapter 4].
It is a non-trivial fact that the above homotopy relation is an equivalence relation and it can be difficult to find a record of this result in the standard litterature on KK-theory. We state the result as a proposition here and notice that the proof is very similar to the proof given in the unbounded setting later on, see Proposition 4.6 and in particular Lemma 4.5 which can be applied to prove the transitivity of the relation ∼ h . We may form the direct sum of two Kasparov modules (X, F ) and (X ′ , F ′ ) from A to B and this is the Kasparov module from A to B given by The zero module is the Kasparov module (0, 0) from A to B.
We quote the following two results from [Bla98, Chapter 17]: Proposition 2.6. Any degenerate Kasparov module from A to B is homotopic to the zero module.
Theorem 2.1. The direct sum operation and the zero module provide KK-theory from A to B with the structure of an abelian group.

Unbounded Kasparov modules
In this section we review the main results of the paper [BaJu83], which can be regarded as the founding paper on unbounded KK-theory. We recall that a symmetric unbounded operator D : Dom(D) → X acting on a Hilbert C * -module X over B is selfadjoint and regular when the operators D ± i : Dom(D) → X are surjective, see [Lan95, Lemma 9.7 and 9.8]. Unbounded selfadjoint and regular operators admit a continuous functional calculus as developed in [Wor91,WoNa92], see also [Lan95,Theorem 10.9]. Notice that in our convention all unbounded operators are densely defined. For an unbounded Kasparov module (X, D) it follows automatically that d(a) : X → X is adjointable for all a ∈ A and we have the formulae d(a) * = −d(a * ) for a even d(a * ) for a odd .
Definition 3.2. The direct sum of two unbounded Kasparov modules (X, D) and It was proved in [BaJu83] that every unbounded Kasparov module represents a class in KK-theory: It turns out that every class in KK-theory can be represented by an unbounded Kasparov module. This result is also due to Baaj and Julg, [BaJu83]. The standing hypothesis that A ⊆ A is countably generated as a * -algebra plays a crucial role in the proof.
Notice that a bounded positive operator ∆ : X → X is strictly positive precisely when the image of ∆ : X → X is dense in X and in this case ∆ −1 : Im(∆) → X is an unbounded positive and regular operator, see [Lan95,Lemma 10.1]. In particular, Dom(∆ −1 ) := Im(∆).
Theorem 3.2. Suppose that A ⊆ A is a norm-dense and countably generated Z/2Zgraded * -subalgebra. Suppose moreover that (X, F ) is a Kasparov module from A to B with F = F * and F 2 = 1. Then there exists an even strictly positive compact operator ∆ : X → X such that (1) the operator F preserves the domain of ∆ −1 and [F, ∆ −1 ] = 0 on Dom(∆ −1 ); (2) each a ∈ A preserves the domain of ∆ −1 and [a, ∆ −1 ] : Dom(∆ −1 ) → X extends to a compact operator on X; in KK(A, B). In particular, it holds that the Baaj-Julg bounded transform is surjective.
In the context of the above theorem, it is worthwhile to notice that the graded commutator [∆ −1 F, a] : Dom(∆ −1 ) → X does in fact extend to a compact operator for all a ∈ A and that (i + ∆ −1 F ) −1 : X → X is compact even though the separable C * -algebra A need not be unital.

Equivalence relations on unbounded Kasparov modules
In this section we introduce an equivalence relation on unbounded Kasparov modules and use this equivalence relation to construct the topological unbounded KKtheory. A key ingredient in our approach is the following notion of a degenerate cycle: Definition 4.1. An unbounded Kasparov module (X, D) from A to B is spectrally decomposable when there exists an orthogonal projection P : X → X such that (1) P preserves the domain of D and DP − P D is trivial on Dom(D); (2) DP and D(P − 1) are unbounded positive and regular operators; (3) aP = P a for all even elements a ∈ A and aP = (1−P )a for all odd elements a ∈ A; For a spectrally decomposable unbounded Kasparov module (X, D) with spectral decomposition given by an orthogonal projection P : X → X, we apply the notation for the associated unbounded positive and regular operators. We remark that } so that we obtain the decomposition: For more information on products of unbounded selfadjoint and regular operators with bounded adjointable operators we refer to [Kaa17, Section 6]. Proof. By (3) and (4) in Definition 4.1, the pair (X, 2P − 1) is a degenerate Kasparov module from A to B and by Proposition 2.6 it therefore suffices to show that (X, D(1 + D 2 ) −1/2 ) is homotopic to (X, 2P − 1). We let a ∈ A and show that D(1 + D 2 ) −1/2 P a − P a and D(1 + D 2 ) −1/2 (1 − P )a − (P − 1)a are compact operators on X. By (1) in Definition 4.1, it holds that implying the identities We thus conclude that Using (2) in Definition 4.1, we see that D + − (1 + D 2 + ) 1/2 : Dom(D + ) → X and But this implies that Unitary equivalence of unbounded Kasparov modules is indeed an equivalence relation and we denote it by ∼ u .
Suppose now that C is an extra Z/2Z-graded σ-unital C * -algebra and that β : B → C is an even * -homomorphism. As in Section 2 we have the change of base operation given by the interior tensor product of Z/2Z-graded C * -correspondences: X ⊗ β C. Moreover, any unbounded selfadjoint and regular operator D : Dom(D) → X induces an unbounded selfadjoint and regular operator D ⊗ β 1 : Dom(D ⊗ β 1) → X ⊗ β C, which has resolvents given by In particular, if a(i + D) −1 : X → X is a compact operator for some a ∈ A, then Before proving that homotopies of unbounded Kasparov modules yields an equivalence relation it is worthwhile to spend a little time on a glueing construction for Z/2Z-graded C * -correspondences. Consider two countably generated Z/2Z-graded C * -correspondences X and X ′ both from A to C([0, 1], B) and suppose that is an even unitary isomorphism of C * -correspondences. This data gives rise to a Z/2Z-graded C * -correspondence X × U X ′ from A to C([0, 1], B) obtained by glueing X and X ′ using the unitary U to identify the fibres sitting at 1 and 0, respectively. Indeed, we may define and endow this set with the vector space structure, left action of A and Z/2Z-grading inherited from the direct sum X ⊕ X ′ . To construct the right action of C([0, 1], B) and the inner product on X × U X ′ , we introduce the even * -endomorphisms β 0 , β 1 : Proof. We start by constructing an adjointable isometry S : Since B is σ-unital by our standing assumptions this will imply that X × U X ′ is countably generated. Since X and X ′ are countably generated over C([0, 1], B) it follows by Kasparov's stabilization theorem, [Kas80a, Theorem 2], that we may find unitary isomorphisms of Hilbert C * -modules and We let V : ℓ 2 (N, B) → ℓ 2 (N, B) denote the unique unitary isomorphism of Hilbert C * -modules, which makes the diagram here below commute: .
We specify that the lower vertical isomorphisms are induced by f ⊗ ev i b → f (i) · b, for i ∈ {0, 1}, and the top vertical unitary isomorphisms come from the distributivity of the interior tensor product together with the lower vertical isomorphisms. The notation ι : X → X ⊕ ℓ 2 (N, C([0, 1], B)) and ι ′ : refers to the standard inclusions given on matrix form as 1 0 . We define our adjointable isometry S : X × U X ′ → ℓ 2 (N, C([0, 1], B)) by the formula for all (ξ, ξ ′ ) ∈ X × U X ′ and t ∈ [0, 1]. We leave it to the reader to verify that S is well-defined (in particular that V (W ι ′ xi)(1) = (W ′ ι ′ ξ ′ )(0)). The adjoint of S is given explicitly by where σ 0 (f )(t) = V * f (t/2) and β 1 (f ) = f ((t + 1)/2) for all f ∈ ℓ 2 (C ([0, 1], B)) and t ∈ [0, 1]. The compactness of K ⊕K ′ : X × U X ′ → X × U X ′ is equivalent to the compactness of S(K ⊕ K ′ )S * : ℓ 2 (N, C([0, 1], B)) → ℓ 2 (N, C([0, 1], B)). The compact operators on the standard module ℓ 2 (N, C([0, 1], B)) can be identified with the operator norm continuous maps from [0, 1] to the compact operators on ℓ 2 (N, B). Using this identification, we compute that where L := W ιKι * W * and L ′ : 1], B)). Since L and L ′ are compact by assumption this proves the com- Transitivity: Suppose that (X 0 , D 0 ), (X 1 , D 1 ) and (X 1 , D ′ 1 ) are unbounded Kasparov modules from A to B such that (X 0 , D 0 ) ∼ h (X 1 , D 1 ) and (X 1 , D 1 ) ∼ h (X ′ 1 , D ′ 1 ) via the unbounded Kasparov modules (X, D) and (X ′ , D ′ ), respectively. Let us choose an even unitary isomorphism of C * -correspondences such that U(D ⊗ ev 1 1)U * = D ′ ⊗ ev 0 1. We consider the associated Z/2Z-graded C *correspondence X × U X ′ from A to C([0, 1], B) and notice that X × U X ′ is countably generated by Lemma 4.5. We define the odd unbounded selfadjoint and regular operator D ′′ : Dom(D ′′ ) → X × U X ′ by To see that D ′′ is indeed selfadjoint and regular, we notice that D ′′ is symmetric and that the resolvents are given by We claim that (X × U X ′ , D ′′ ) is an unbounded Kasparov module from A to C ([0, 1], B). It is indeed clear that each a ∈ A preserves the domain of D ′′ and that the graded commutator [D ′′ , a] : Dom(D ′′ ) → X × U X ′ extends to a bounded adjointable operator on X × U X ′ . Moreover, it follows from Lemma 4.5 that a · (i + D ′′ ) −1 : X × U X ′ → X × U X ′ is a compact operator for all a ∈ A.
The unbounded Kasparov module (X × U X ′ , D ′′ ) from A to C([0, 1], B) implements the homotopy from (X 0 , D 0 ) to (X ′ 1 , D ′ 1 ) and this ends the proof of the proposition.
Let us shortly discuss the notion of bounded perturbations of unbounded Kasparov modules: We remark that the above notion of bounded perturbations yields an equivalence relation on unbounded Kasparov modules. We will not discuss this equivalence relation any further at this point but refer the reader to [Kaa16] for more details.
In this paper the relevant equivalence relation on unbounded Kasparov modules is a stabilized version of homotopies of unbounded Kasparov modules, where we are using spectrally decomposable unbounded Kasparov modules in the stabilization procedure: Definition 4.8. Two unbounded Kasparov modules (X 0 , D 0 ) and (X 1 , D 1 ) from A to B are stably homotopic when there exist two spectrally decomposable unbounded Kasparov modules . In this case we write (X 0 , D 0 ) ∼ sh (X 1 , D 1 ).
Definition 4.9. The topological unbounded KK-theory from A to B consists of the unbounded Kasparov modules from A to B modulo stable homotopies, thus modulo the equivalence relation ∼ sh . The topological unbounded KK-theory from A to B is denoted by UK top (A , B).
We may equip the topological unbounded KK-theory from A to B with the structure of a commutative monoid, where the addition is induced by the direct sum operation from Definition 3.2 and the neutral element is the class of the zero module (0, 0). We shall see in Section 6 that UK top (A , B) is in fact an abelian group.
The next result is a combination of Theorem 3.2 and Lemma 4.2 together with the observation that the Baaj-Julg bounded transform is compatible with direct sums and homotopies of unbounded Kasparov modules. The main result of this paper is that the surjective homomorphism F : UK top (A , B) → KK(A, B) is in fact an isomorphism. In particular, it holds that UK top (A , B) is independent of the norm-dense Z/2Z-graded * -subalgebra A ⊆ A as long as A is countably generated. This will be proved in Section 7.

Lipschitz regularity and invertibility
We shall now see that given a class in topological unbounded KK-theory one may always choose a Lipschitz regular representative (X, D) with the extra property that the unbounded selfadjoint and regular operator D : Dom(D) → X is invertible (so that D −1 : X → X is a bounded adjointable operator with image equal to the domain of D).
We start with Lipschitz regularity: Proposition 5.1. Let r ∈ (0, 1/2) and suppose that (X, D) is an unbounded Kasparov module from A to B. Then (X, D(1+D 2 ) −r ) is a Lipschitz regular unbounded Kasparov module from A to B and it holds that Proof. For each t ∈ [0, 1], define the functions We remark that f ± t ∈ C 0 (R) and that the maps [0, 1] → C 0 (R) given by t → f ± t are continuous with respect to the supremum norm on C 0 (R). Notice in this respect that we have the estimates |f ± t (x)| ≤ 2 r |x| 2r−1 whenever |x| ≥ 1 and t ∈ [0, 1].
For p ∈ (0, 1/2], we are going to apply the integral formula where the integrand is continuous in operator norm and the integral converges absolutely in operator norm. Let a ∈ A be homogeneous. For each t ∈ [0, 1], the domain of D is a core for D(1 + D 2 ) −tr and for each ξ ∈ Dom(D), we compute the graded commutator The first term extends to the bounded adjointable operator (1+D 2 ) −tr d(a) : X → X and we remark that the map [0, 1] → L(X) defined by t → (1 + D 2 ) −tr d(a) is continuous with respect to the strict operator topology on L(X). The second term in Equation (5.2) is more complicated and, using the integral formula in Equation (5.1), we obtain the expression for all t ∈ [0, 1].
In particular, we have that the unbounded Kasparov modules (X, D) and (X, D(1 + D 2 ) −r ) are homotopic.
To finish the proof of the proposition we only need to argue that the unbounded Kasparov module (X, D(1+D 2 ) −r ) is Lipschitz regular. Let a ∈ A be homogeneous. Since the function x → |x| − x 2 (1 + x 2 ) −1/2 is bounded on R, we just have to prove that the graded commutator extends to a bounded operator on X. Notice in this respect that D(1+D 2 ) −1/2 D(1+ D 2 ) −r : Dom(|D| 1−2r ) → X agrees with |D(1 + D 2 ) −r | = |D|(1 + D 2 ) −r : Dom(|D| 1−2r ) → X up to a bounded selfadjoint operator and moreover that Dom(D) ⊆ X is a core for |D(1 + D 2 ) −r |. Since we already know that where the left hand side only makes sense on Dom(|D| 1−2r ), but the right hand side makes sense as a bounded operator on X. Indeed, both of the integrals in Equation (5.5) have operator norm continuous integrands and converge absolutely because of the operator norm estimates which are valid for all λ ∈ (0, ∞).
We continue with invertibility: Proposition 5.2. Suppose that (X, D) is an unbounded Kasparov module from A to B. Then (X, D) is stably homotopic to a Lipschitz regular unbounded Kasparov module (X ′ , D ′ ) with D ′ : Dom(D ′ ) → X ′ invertible.
Proof. By Proposition 5.1 we may assume that (X, D) is already Lipschitz regular. Let us denote the Z/2Z-grading operator on X by γ : X → X. Define the Z/2Zgraded C * -correspondence X from A to B which agrees with X as a Hilbert C *module over B, but X has grading operator −γ : X → X and the left action of A is trivial. Then the unbounded Kasparov module ( X, −D) is homotopic to the zero module (0, 0). Indeed, we may consider the Z/2Z-graded C * -correspondence C 0 ((0, 1], X) from A to C ([0, 1] which is indeed an invertible operator.

Group structure
We show in this section that the commutative monoid UK top (A , B) is in fact an abelian group. This result relies on a more general proposition stating (at least roughly speaking) that two unbounded Kasparov modules (X, D) and (X, D ′ ) from A to B are stably homotopic when the odd unbounded selfadjoint and regular operators D and D ′ have the same phase. This proposition will also be of key importance later on when we prove the injectivity of the Baaj-Julg bounded transform. where X op agrees with X as a Hilbert C * -module over B, but X op is equipped with the opposite Z/2Z-grading and with left action π X op : A → L(X op ) defined by π X op (a) = π X (a) for a even −π X (a) for a odd . Proof. We are going to show that is homotopic to a spectrally decomposable unbounded Kasparov module. We denote the grading operator on X by γ : X → X so that the grading operator on X op is given by −γ : X → X.
We claim that the pair (Y, where it is understood that the unbounded selfadjoint and regular operator in question acts as ( where Y t is the countably generated Z/2Z-graded C * -correspondence from A to B which agrees with X ⊕ X as a Hilbert C * -module but with grading operator σ t = 0 −γ −γ 0 and with left action given by the even * -homomorphism π t : A → L(X ⊕ X).
We let t ∈ [0, 1] be given and compute for each even a ∈ A that π t (a) − π 0 (a) = 1 2 1 − cos(tπ/2) − sin(tπ/2) sin(tπ/2) 1 − cos(tπ/2) (6.2) and for each odd a ∈ A that (6.3) Since the graded commutator [F, a] : X → X is compact, this computation implies that π t (a)−π 0 (a) : X ⊕X → X ⊕X is a compact operator for all a ∈ A and t ∈ [0, 1] and moreover that the associated map [0, 1] → K(X ⊕ X) is continuous in operator norm. Using assumption (3), the above computation also implies that π t (a) − π 0 (a) preserves the domain of (D + − D ′ − ) ⊕ (D − − D ′ + ) for all a ∈ A and t ∈ [0, 1] (the image of π t (a) − π 0 (a) is in fact contained in this domain) and moreover that the graded commutator , π t (a) − π 0 (a)] : Dom(DP + ∩ D ′ P − ) ⊕ Dom(DP − ⊕ D ′ P + ) → X ⊕ X extends to a bounded operator on X ⊕ X (in fact each of the terms have this property). The associated map [0, 1] → L(X ⊕ X) given by The fibre at t = 1 is given by the pair (Y 1 , where Y 1 is unitarily isomorphic to X ⊕ X op as a Z/2Z-graded C * -correspondence via the unitary operator U 1 : X ⊕ X → X ⊕ X defined in Equation (6.1). Moreover, we have that By assumption (1) we know that the off-diagonal entries extend to bounded operators on X and it follows that the fibre at t = 1 agrees with the unbounded By what has been achieved so far, we have reduced the proof of the proposition to showing that the fibre of (Y, (D + − D ′ − ) ⊕ (D − − D ′ + )) at t = 0 is homotopic to a spectrally decomposable unbounded Kasparov module from A to B.
The unbounded Kasparov module sitting as the fibre at t = 0 is given by the pair where the upper diagonal entry and minus the lower diagonal entry are both unbounded positive and regular operators. But this shows that the fibre at t = 0 is a bounded perturbation of a spectrally decomposable unbounded Kasparov module. Indeed, the unbounded Kasparov module (Y 0 , (P + ∆P + + P − ∆ ′ P − ) ⊕ (−P − ∆P − − P + ∆ ′ P + )) from A to B is spectrally decomposable (using the orthogonal projection 1 0 0 0 : X ⊕ X → X ⊕ X when verifying the conditions in Definition 4.1).