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Conformal covariance and the split property

We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, M\"obius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.


Introduction
More than half a century passed away since the first formulation of an axiomatic quantum field theory. There are several existing different settings (differing e.g. on the chosen spacetime, or whether their fundamental notion is that of a quantum field or a local observable) with many "additional" properties that are sometimes included among the defining axioms. For an introduction and overview of the topic we refer to the book of Haag [24].
Whereas properties like locality are unquestionably among the basic axioms, some other properties are less motivated and accepted. Haag-duality has an appealing mathematical elegance, but there seems to be no clear physical motivation for that assumption. Technicalities, like the separability of the underlying Hilbert space are sometimes required with no evident physical reason.
The split property is the statistical independence of local algebras associated to regions with a positive (spacelike) separation. It might be viewed as a stronger version of locality, and contrary to the previous two examples, it was formulated on direct physical grounds. However, traditionally it is not included among the defining axioms, as in the beginning it was unclear how much one can believe in it. Indeed, many years passed till this stronger version of locality was first established at least for the massive free field by Buchholz [2]. Only after the introduction of the nuclearity condition (which was originally motivated by the need of a particle interpretation [25]) it became more of a routine to verify the split property in various models, when its connection to nuclearity was discovered [6]. Another important step was the general mathematical understanding of split inclusions brought by the work of Doplicher and Longo [15].
In the meantime, interest rose in conformal quantum field theories, especially in the low dimensional case; i.e. conformal models given on the 2-dimensional Minkowski space and their chiral components that can be naturally extended onto the compactified lightray, the circle. The theory of conformal net of local algebras on S 1 is rich in examples and it provides an essential "playground" to people studying operator algebras as it turned out to have incredibly deep connections to the modular theory of von Neumann algebras as well as to subfactor theory; see e.g. [19,41,29]. In particular, the modular group associated to a local algebra and the vacuum vector always acts in a certain geometric manner: the so-called Bisognano-Wichmann property is automatic. In turn, this was used to conclude that several further important structural properties -e.g. Haag-duality and Additivityare also automatic in this setting. We refer to the original works [19,18,1,4] for more details on this topic.
The case of the split property seemed to be different -but there is an important detail to mention here. Initially, when studying chiral conformal nets, in the so-far cited works only Möbius covariance was exploited. There were several reasons behind this choice. First, because it is a spacetime symmetry implemented by a unitary representation for which the vacuum is an invariant vector. This is exactly how things go in higher dimension, but this is not how diffeomorphism covariance is implemented (no invariant vectors and one is forced to consider projective representations rather than true ones). Second, because the mentioned connection to modular theory of von Neumann algebras relies on Möbius covariance only. Thus, the listed structural properties -with the exception of the split property -are already automatic even if diffeomorphism covariance is not assumed.
From the physical point of view, however, diffeomorphism covariance is natural in the low dimensional conformal setting; by an argument of Lüscher and Mack, it should merely be a consequence of the existence of a stress-energy tensor [20]. All important models are diffeomorphism covariant with the exception of some "pathological" counterexamples; see [30,9]. It is worth noting that the example constructed in [9] by infinite tensor products, has neither diffeomorphism symmetry nor the split property. Thus, unlike the mentioned other properties, the split property surely cannot be derived in the Möbius covariant setting. However, as we shall prove it here, the split property is automatic if diffeomorphism covariance is assumed. Note that together with the result of Longo and Xu in [34] regarding strong additivity, this shows that a diffeomorphism covariant local net on S 1 is completely rational if and only if its µ-index is finite.
The crucial points of our proof are the following. We consider a conformal net A on the circle with conformal Hamiltonian L 0 , and fix two (open, proper) intervals I a , I b ∈ I with positive distance from each other. Inspired by the complex analytic argument used in [18] to prove the conformal cluster theorem, for an element X of the * -algebra A(I a ) ∨ alg A(I b ) generated by A(I a ) and A(I b ) with decomposition X = n k=1 A k B k (where n ∈ N, A k ∈ A(I a ), B k ∈ A(I b )), we consider the function on the complex unit disc For every |z| ≤ 1, this defines a functional φ z on A(I a ) ∨ alg A(I b ). For z = 1 this is simply the vacuum state ω, but for z = 0 this is the product vacuum state AB → ω(A)ω(B) (A ∈ A(I a ), B ∈ A(I b )). The split property is essentially equivalent to saying that φ 0 is normal (actually, here some care is needed: in general one needs the product state to be normal and faithful. Fortunately, general results on normality and conormality in a Möbius covariant net [23] of the inclusions A(I 1 ) ⊂ A(I 2 ) for an is a factor; see more details in the preliminaries. It then turns out that the normality of φ 0 is indeed equivalent to the split property).
However, we do not have a direct method to show that φ z is normal at z = 0. On the other hand, we can treat several points inside the disc. Using the positive energy projective representation U of Diff + (S 1 ) given with the theory, for example for any (fixed) r ∈ (0, 1) and I c , I d ∈ I covering the full circle we find a decomposition r L 0 = CD in which C r ∈ A(I c ) and D r ∈ A(I d ). Choosing the intervals I c and I d carefully, C will commute with the A k operators while D will commute with the B k operators and hence showing that for our real r ∈ (0, 1), the functional φ r is normal as it is given by two vectors. Note that the origin of the decomposition r L 0 = CD is the fact that a rotation can be decomposed as a product of local diffeomorphisms; something that using Möbius transformations alone, cannot be achieved (as all nontrivial Möbius transformations are global). However, even using the full diffeomorphism group, the issue is tricky, since we need a decomposition that can be analytically continued over to some imaginary parameters -and of course the words "local" and "analytical" are usually in conflict with each other. Nevertheless, this kind of problem was already treated in [39], and the methods there developed were also used in the proof of [7,Theorem 2.16], so all we needed here was some adaptation of earlier arguments.
We then proceed by "deforming" our decomposition using the work [36] of Neretin, which allows us to access further regions inside the unit disk. In this way we establish normality along a ring encircling the origin, and thus we can use the Cauchy integral formula to conclude normality of φ z at z = 0.
Note that we have really made use of the fact that the conformal Hamiltonian L 0 generates a compact group. Indeed, for a generic complex number z, the very expression z L 0 is meaningful only because Sp(L 0 ) contains integer values only. However, unlike with chiral nets, in the 2-dimensional conformal case the theory does not necessarily extends in a natural way to the compactified spacetime. Thus one might wonder whether our result will remain valid or not: is this compactness of the spacetime just some technicality, or is it an essential ingredient of our proof? The answer turns out to be the latter one.
In fact, we manage to present an example of a diffeomorphism covariant local net on the 2-dimensional spacetime, which does not have the split property. More concretely, we consider a local extension B ⊃ A of the net A = A U (1) ⊗ A U (1) obtained by taking two copies of the U(1)-current net (here considered as "left" and "right" chiral parts). Irreducible sectors of the U(1)-current net are classified by a certain charge q ∈ R. Our construction is such that when considered as a representation of A U (1) ⊗ A U (1) , the net A ⊂ B decomposes as a direct sum ⊕ q∈R (σ q ⊗ σ q ) where σ q is the representation corresponding to the sector with charge q. This model is naturally diffeomorphism covariant, but because its Hilbert space is not separable, it cannot have the split property. Note that here "diffeomorphism covariance" means only that we have an action of Diff + (S 1 ) × Diff + (S 1 ) which factors through the spacelike 2π-rotation, but not that of Diff + (S 1 )×Diff + (S 1 ). This is in complete accordance with our earlier remark on the spectrum of L 0 . This paper is organized as follows. In Section 2 we introduce our operator-algebraic setting for conformal field theory and recall relevant technical results concerning conformal covariance and the split property. Sections 3 and 4 provide our technical ingredients, namely certain decompositions of z L 0 into local elements. In Section 5 we prove our main result, that the split property follows from diffeomorphism covariance, by proving the normality of φ 0 . A two-dimensional counterexample is provided in Section 6. In Section 7 we conclude with open problems.

Preliminaries
Let I be the set of nonempty, nondense, open connected intervals of the unit circle S 1 = {z ∈ C : |z| = 1}. A Möbius covariant net is a map A which assigns to every interval of the circle I ∈ I a von Neumann algebra A(I) acting on a fixed Hilbert space H satisfying the following properties: 1. Isotony: if I 1 , I 2 ∈ I and I 1 ⊂ I 2 , then A(I 1 ) ⊂ A(I 2 ); 2. Möbius covariance: there exists a strongly continuous, unitary representation U of the Möbius group Möb (≃ PSL(2, R)) on H such that U(g)A(I)U(g) * = A(gI), I ∈ I, g ∈ Möb; 3. Positivity of the energy: the conformal Hamiltonian L 0 , i.e. the generator of the rotation one-parameter subgroup has a non negative spectrum.

4.
Existence and uniqueness of the vacuum: there exists a unique (up to a phase) unit U-invariant vector Ω ∈ H, i.e. U(g)Ω = Ω for g ∈ Möb; 5. Cyclicity: Ω is cyclic for the von Neumann algebra I∈I A(I).
We will denote a Möbius covariant net with the triple (A, U, Ω). Some consequences of the axioms are (see e.g. [19,18,22] 13. Normality and conormality: for any inclusion I 1 ⊂ I 2 , it holds that A( From conormality, it follows that two-interval algebras are factors. Indeed, take I 1 ⊂ I 2 such that they have no common end points. Then I 1 and I ′ 2 are disjoint intervals with a finite distance. By Haag duality it follows that (A(I 1 ) ∨ A(I ′ 2 )) ′ = A(I 1 ) ′ ∩ A(I 2 ), and by conormality we have where the last equality is a consequence of Haag duality and factoriality. Let us add this to the list of consequences.
14. Factoriality of two-interval algebras: for disjoint intervals I 1 and I 2 with a finite distance, A(I 1 ) ∨ A(I 2 ) is a factor. Now, we briefly discuss diffeomorphism covariance. Let Diff + (S 1 ) be the group of orientation preserving diffeomorphisms of the circle. It is an infinite dimensional Lie group modelled on the real topological vector space Vect(S 1 ) of smooth real vector fields on S 1 with the C ∞ -topology [35]. Its Lie algebra has to be considered with the negative of the usual bracket on vector fields, in order to have the proper exponentiation of vector fields. We shall identify the vector field f (e iθ ) d dθ ∈ Vect(S 1 ) with the corresponding real function f ∈ C ∞ (S 1 , R). We denote with Diff + (I) the subgroup of Diff + (S 1 ) acting identically on I ′ , namely the diffeomorphisms of S 1 with support included in I.
A strongly continuous, projective unitary representation U of Diff + (S 1 ) on a Hilbert space H is a strongly continuous homomorphism of Diff + (S 1 ) into U(H)/T, the quotient of the group of unitaries in B(H) by T. The restriction of U to Möb ⊂ Diff + (S 1 ) always lifts to a unique strongly continuous unitary representation of the universal covering group Möb of Möb. U is said to have positive energy, if the generator L 0 of rotations, the conformal Hamiltonian, has a nonnegative spectrum in this lift. Let γ ∈ Diff + (S 1 ). Note that expressions Ad U(γ) makes sense as an action on B(H). We also write U(γ) ∈ M although U(γ) is defined only up to a scalar.
When one has a strongly continuous projective unitary representation U of Diff + (S 1 ) with positive energy, e i2πL 0 is a multiple of the identity and therefore L 0 has a pure pointspectrum. It follows that the linear span D fin of eigenvectors of L 0 (the so-called "finite energy vectors") form a dense set. U can then be "differentiated" to obtain a representation at the Lie algebra level [8, Appendix A] (see also [31]). Any smooth function f ∈ C ∞ (S 1 , R), as a vector field on S 1 , defines a one-parameter group of diffeomorphisms R ∋ t → γ t= Exp(tf ) ∈ Diff + (S 1 ), hence, up to an additive constant, defines the selfadjoint generator T (f ) of the unitary group t → U(γ t ). For any real smooth function f as above, T (f ) is essentially self-adjoint on the set C ∞ (L 0 ) := n∈N 0 Dom (L n 0 ). T shall be called the stress energy tensor.
Irreducible, projective, unitary positive energy representation of Diff + (S 1 ) are labelled by certain values of the central charge c > 0 and the lowest weight h ≥ 0. h is the lowest point in the discrete spectrum of the conformal Hamiltonian L 0 . There is a unique (up to a phase) vector Φ ∈ H corresponding to the lowest eigenvalue. See [21,26] for a detailed description of such representations.
One considers particular elements {L n : n ∈ Z}, L n = iT (y n ) − T (x n ), L −n = iT (y n ) + T (x n ) for n ∈ N, where x n (θ) := − 1 n sin nθ and y n (θ) := − 1 n cos nθ (there is a canonical way to fix the scalar part of T (x n ), T (y n ), as L n , L −n and L 0 generate a (projective) representation of Möb). These operators satisfy the so-called Virasoro algebra on finite energy vectors D fin . In particular for all n, m ∈ Z: D fin is an invariant common core for any closed operator L n ; if n > 0 then L n Φ = 0; L −n ⊂ L * n ; the family {L n } n∈Z satisfies the Virasoro algebra relations on D fin : Let f ∈ C ∞ (S 1 , R) be a vector field on S 1 , with Fourier coefficientŝ then, one can recover the stress-energy tensor by and gives the correspondence between the infinitesimal generators and the representation of Diff + (S 1 ) (up to a scalar).
Throughout the next few sections we shall often consider the net of von Neumann algebras Note that when U is a so-called vacuum representation associated to central charge c, A U is nothing else than the well-known Virasoro net with central charge c. In general though, A U is not a conformal net in the sense we are introducing them in this preliminary; e.g. we might not have a vacuum vector. Nevertheless, we still have the locality relation The stress energy tensor can be evaluated on a larger set of functions [9]. For a continuous function f : S 1 → R with Fourier coefficients {f n } n∈Z we shall set , then T (f k ) → T (f ) in the strong resolvent sense. In particular, even for a non necessarily smooth function f with f 3 We shall say that a Möbius covariant net (A, U, Ω) is conformal (or diffeomorphism covariant) if the Möb representation U extends to a projective unitary representation Diff + (S 1 ) → U(H)/T of Diff + (S 1 ) (that with a little abuse of notation we continue to indicate the extension with U) and satisfying Now we recall the definition of the split property for von Neumann algebra inclusions and conformal nets. A Möbius covariant net (A, U, Ω) satisfies the split property if the von Neumann algebra inclusion A(I 1 ) ⊂ A(I 2 ) is split, for any inclusion of intervals I 1 ⋐ I 2 , namely when I 1 and I 2 have no common end points.
The following proposition provides an equivalent condition to the split property. Although similar statements are quite well-known to experts (see [11] and [15, Below Definition 1.4]), the precise assumptions we need are difficult to find in the literature (note, for example, that we do not assume neither the separability of the underlying Hilbert space 1 nor the faithfulness of the split state in the implication 2 ⇒ 1 below).
Proposition 2.2. Let (N ⊂ M, Ω) be a standard inclusion of von Neumann algebras. We further assume that that N ∨ M ′ is a factor. Then the following are equivalent.

N ⊂ M is split;
2. there exists a normal state φ on N ∨ M ′ such that the restrictions φ N and φ M ′ are faithful and φ is split, namely, Proof. If N ⊂ M is split, namely if there is an intermediate type I factor R ≃ B(K), then N ∨ M ′ is isomorphic to N ⊗ M ′ , from which the implication 1 ⇒ 2 follows. Conversely, let there be a split state as in 2. First of all, as φ are faithful on N and M ′ , their GNS representations π N , π M ′ are faithful and have a cyclic and separating vector. Next, as φ is normal on N ∨ M ′ , its GNS representation π N ∨M ′ is also normal. The Hilbert space supporting π N ∨M ′ is isomorphic to the closure of N ∨ alg M ′ w.r.t. the scalar product inherited by the normal state φ as x, y φ = φ(x * y). By the factorization assumption on φ, the Hilbert space is the tensor product L 2 (N , ·, · φ ) ⊗ L 2 (M ′ , ·, · φ ) and the GNS representation π N ∨M ′ restricted to N and M ′ are of the form π N ⊗½ and ½⊗π M ′ , respectively. Furthermore, as both N and M ′ have a cyclic and separating vector Ω, their GNS representations π N , π M ′ are actually unitary equivalences [37,Corollary 10.15]. As a consequence, by normality, we can assume that π N ∨M ′ (N ∨ M ′ ) = N ⊗ M ′ . Furthermore, by assumption N ∨ M ′ is a factor, hence the GNS representation is an isomorphism. Now, both N ∨ M ′ and N ⊗ M ′ have a cyclic and separating vector (Ω and Ω ⊗ Ω respectively), therefore, the GNS representation is actually a unitary equivalence. Then the preimage 3 Local decompositions of e −βL 0 Throughout this section, we shall not need a conformal net, as we only work with a strongly continuous projective unitary representation U of Diff + (S 1 ) with positive energy. We shall use the notations introduced in the preliminaries for all associated objects (i.e. L n , (n ∈ N) will stand for the associated Virasoro algebra representation, T for the stressenergy tensor, A U for the system of von Neumann-algebras appearing at (2) etc.).
In what follows, for a β > 0, r = e −β and two open proper arcs (intervals) I c , I d ∈ I that cover the circle: The main idea for producing such a decomposition was already presented and exploited in [39] and in the proof of [7, Theorem 2.16]. Here we shall recall the essential points of the argument presented there and then adjust and refine it to our purposes.
with N being a positive integer such that 6π/N is smaller than the lengths of both arcs that are obtained by taking the intersection I c ∩ I d (note that N must be at least 4).
Proof. Let us fix a positive integer N satisfying the condition of the proposition (see Figure  1). The operators H : Moreover H is diagonalizable with non-negative eigenvalues only, the span of its eigenvectors is exactly D fin which is an invariant core for the operators L ± . It then follows that these operators generate a strongly continuous, positive energy unitary representation of the universal cover Möb of the Möbius group. This construction -both at the Lie algebra as well as the Lie group level -was already considered and used by various authors; see e.g. the work [34]. In particular, for all s > 0. Let us now consider how P andP can be written in terms of the stress-energy T . We have and p andp are the functions defined by the formulas . The function p is nonnegative on S 1 and it has exactly N points where its value is zero: All these null-points are of course local (and also global) minima, where the derivative is zero. We can thus "cut" p into N "nice" pieces: p = p 1 + . . . + p N where the support of the nonnegative function p k is the closure of the arc and p k 3 2 < ∞. This latter follows from the fact that p k is once differentiable and its derivative is of bounded variations; see the similar considerations at [9, Lemma 5.3]. Thus for every k = 1, . . . N, ½ is a well-defined self-adjoint operator affiliated to A U (I k ) and we have P = P 1 + . . . + P N .
Since the terms in this decomposition are affiliated to commuting factors, just as in the proof [39, Proposition 3.2], we have that On the other hand, the spectrum of the operators P k (k = 1, . . . N) must all coincide, since using rotations one can easily show that they are all unitary conjugate to each other. It then follows that each of them must be a positive operator. Thus for the bounded operator e −tanh( s 2 )P appearing in formula (3), we have the local decomposition into a product of commuting bounded operators where the norm of each term is smaller or equal than 1. Let us turn toP . As we haveP = Ad e i π N L 0 (P ), the localization ofP k = Ad e i π N L 0 (P k ) are different from that of P :P k is affiliated to A U (Ĩ k ) whereĨ k = e i π N I k and A U is defined in Section 2 (we are considering the intervals as subsets in C). With this localization, we can still assure the strong commutation between P k andP j whenever k = j, j + 1 (mod N). So in the decomposition For simplicity, we did not indicate their dependence on r, but note that in the range 0 < r < 1 they depend norm-continuously on r (for t > 0, x ≥ 0, the function e −tx is uniformly continuous in t).
All X-operators and separately, all Y -operators commute between themselves, and moreover [X l , Y m ] = 0 whenever l = m, m + 1 (mod N). Recall that 6π N is smaller than the length of each of the intervals of I c ∩I d . Therefore, by cyclically renaming the intervals (but keeping the relation between I k andĨ k and the corresponding localization of the operators), we may assume that there are 1 ≤ k < j ≤ N such that I k ∪ I k+1 and I j ∪ I j+1 are included in the different connected components of I c ∩ I d . Furthermore, to fix the notation, we Figure 2: Localization of the factors of C r . The indicated intervals I • ,Ĩ • correspond to thick segments. The operators j+1 l=k X l , j l=k Y l , j l=k+1 X l are localized in the arcs, from the inside, respectively. The corresponding factors in D r are localized in the complements of these arcs, respectively. may assume that I k ∪ · · · ∪ I j+1 ⊂ I c , while I j ∪ · · · ∪ I N ∪ I 1 · · · ∪ I k ⊂ I d . Note that I k ∪ · · · ∪Ĩ j ⊂ I c andĨ j ∪ · · · ∪ I N ∪ I 1 · · · ∪Ĩ k−1 ⊂ I d (see Figure 2).
By the localization explained above, we obtain Here the first part By construction, C r , D r ≤ 1. Thus, we have obtained the desired decomposition In the above proposition we specifically worked with L 0 . However, by considering the adjoint actions of U(γ) for all diffeomorphisms γ ∈ Diff + (S 1 ) on the decompositions found above, it is now easy to draw the following conclusion.

Further decompositions
In this section, we shall consider further decompositions of r L 0 , for which the crucial ingredient will be a result of Neretin [36]. Though in his work the relevant theorem is stated for representations which are direct sums of those highest weight ones, as was already mentioned, a positive energy, strongly continuous, projective unitary representation of Diff + (S 1 ) can only be of that form. For better readability, we shall recall the statement that we are going to exploit. We will need the concept of analytic diffeomorphism; we will denote by Diff + a (S 1 ) the set of γ ∈ Diff + (S 1 ) that extends to an annulus around S 1 ⊂ C in a complex analytic manner. Then elements of the form U(γ)r L 0 U(γ) (r ∈ (0, 1], γ,γ ∈ Diff + a (S 1 )) form a projective semigroup: for any γ 1 , γ 2 ,γ 1 ,γ 2 ∈ Diff + a (S 1 ) and r 1 , r 2 ∈ (0, 1] there exist some γ 3 ,γ 3 ∈ Diff + a (S 1 )), r 3 ∈ (0, 1] such that where of course equality is meant in the projective sense. Actually, what we shall really use is a certain adaptation of the above result for the case of true (and not just projective) representations of the Möbius group. Note that every Möbius transformation is of course analytic. To make the necessary modifications, we shall first make an observation 2 . Proof. We need to show the "only if" part; the other direction is true by definition. If γ = id, the statement immediately follows, otherwise, by composing a Möbius element g, we may assume that γ fixes three points on S 1 . Let I 1 , I 2 , I 3 be three intervals with such end points.
By contradiction, let us assume that γ = id. Then there is a point s ∈ S 1 such that γ(s) = s. As γ fixes three points on S 1 , it also preserves each interval bounded by any pair of these points. Say lim n γ n (s) =: s ∞ (for a single element γ ∈ Diff + (S 1 ), S 1 can be decomposed into intervals such that γ is monotone on each interval, hence such a limit exists).
It is not restrictive to assume that I 1 , the closure of I 1 , contains neither {γ n (s)} nor s ∞ . Let us go to the real line picture (which is only necessary below, in order to simplify the conformal distance) and call I 1 := (t 1 , t 2 ), and I := (s, s ∞ ) (or (s ∞ , s) depending on in which direction s is moved, without losing the generality, we may assume the former case). Assume that I 1 and I are bounded intervals on the line. Now they are separated by a finite distance.
Proof. Consider the well-known conformal net usually referred as the U(1)-current net [5]. One may restrict its projective unitary representation U of Diff + (S 1 ) to the Möbius group and arrange its phase factors in such a way that the vacuum Ω will be an invariant vector (see Section 2). In this way we get a positive energy, strongly continuous, unitary representation V of Möb in which all such irreducible representations (i.e. every possible integer highest weight) appear: this is evident because for every n ≥ 1, the dimension of (n + 1) th energy space is strictly larger than the dimension of the n th one. Thus, if we can show the statement for our particular representation V , we have proved it for all positive energy representations of Möb. By applying Theorem 4.1 to U of the U(1)-current net, we obtain that for every g 1 , g 2 ,g 1 ,g 2 ∈ Möb and r 1 , r 2 ∈ (0, 1] there must exist some γ 3 ,γ 3 ∈ Diff + a (S 1 ) and an r 3 ∈ (0, 1] such that U(g 1 )r L 0 1 U(g 1 ) U(g 2 )r L 0 2 U(g 2 ) = U(γ 3 )r L 0 3 U(γ 3 ) in the projective sense. If r 1 = r 2 = 1, then of course g 3 andg 3 can be chosen to be in Möb. On the other hand, if r 1 r 2 < 1, then they must be in Möb. Indeed, in such a case r must be strictly smaller than 1 (the right hand side cannot be unitary as the left hand side does decrease the length of some vectors). Then "sandwiching" the left hand sides by Ωi.e. considering the scalar product Ω, · Ω for the left hand side (which is of course only defined up-to-phase) -gives a complex number of modulus 1, whereas by Lemma 4.2 and the fact CΩ is the unique eigenvector of L 0 with the eigenvalue 0, the same sandwiching of the right hand side gives a number of modulus 1 if and only if γ 3 ,γ 3 ∈ Möb.
Considering the relevant unitary operators rather than projective ones, we therefore have that for every g 1 , g 2 ,g 1 ,g 2 ∈ Möb and r 1 , r 2 ∈ (0, 1] there must exist some g 3 ,g 3 ∈ Möb and an r 3 ∈ (0, 1] such that V (γ 1 )r L 0 are proportional to each other. The proof is then finished by evaluating both sides on Ω and concluding that this proportion must be 1.

Corollary 4.4. Let V be a positive energy, strongly continuous, unitary representation of
Möb with associated conformal Hamiltonian L 0 . There exists some r, r 1 , r 2 ∈ (0, 1) and g, g 1 , g 2 ∈ Möb, g = id, such that Proof. We choose two elementsg 1 ,g 2 ∈ Möb such thatH k = Ad V (g k )(L 0 ), andH 1 andH 2 do not strongly commute: such choices are actually abundant, since L 0 is maximally abelian in the Lie algebra. Then there must exist some r 1 , r 2 ∈ (0, 1) such that rH 1 1 and rH 2 2 do not commute (otherwise their generators would strongly commute by analytic continuation). Now we apply Corollary 4.3 to rH 1 1 rH 2 2 = V (g 1 )r L 0 1 V (g 1 ) * V (g 2 )r 2 L 0 V (g 2 ) * to obtain g 3 ,g 3 ∈ Möb and r ∈ (0, 1) such that , in the proper sense, or equivalently, . By defining g k = g −1 3g k , hence accordingly H k := Ad V (g −1 3g k )(L 0 ) and g := g 3g3 , we obtain the desired equality. To check that g = id, note that by our choice ofH k , r H 1 1 and r H 2 2 do not commute as well. Yet, in the equality , the left-hand side is self-adjoint, while if g = id, the right-hand side would not be selfadjoint. Therefore, g = id.
Proposition 4.5. Let A be a conformal net, U be the associated projective unitary representation of Diff + (S 1 ), and L 0 the conformal Hamiltonian. For some r ∈ (0, 1), there exists a Möbius transformation g = id, such that for any I c , I d ∈ I be two open proper arcs such that I c ∪ I d = S 1 we have two bounded operators C ∈ A U (I c ) and D ∈ A U (I d ) such that we have the decomposition in the proper sense.
Proof. We apply Corollary 4.4 to obtain r, r 1 , r 2 ∈ (0, 1), g ∈ Möb, g = id and H 1 , H 2 such that r L 0 = r H 1 1 r H 2 2 U(g). Then we apply Corollary 3.2 to H k with the intervals K k,c , K k,d such that K k,c ⊂ I c , K k,d ⊂ I d and K 1,d ∩ K 2,c = ∅ (see Figure 3), to obtain operators C k , D k such that r H k k = C k D k . By the localization, C 2 and D 1 commute.
Hence it holds that

Normality of the product vacuum state
We can now prove our main claim: for a conformal net on S 1 -where by "conformal" we mean that it has the full diffeomorphism covariance (see Section 2) -the split property is automatic. Let (A, U, Ω) be a conformal net, and assume I a , I b ∈ I are two open proper arcs separated by a positive distance.
Consider the * -algebra A(I a ) ∨ alg A(I b ) generated by the commuting factors A(I a ) and A(I b ). We shall now introduce a family {φ z } of functionals on this algebra indexed by a complex number z, |z| ≤ 1. For a generic element X ∈ A(I a ) ∨ alg A(I b ), and a complex number z in the closed unit disk The above quantity is well-defined in the sense that it indeed depends only on z and X, but not on the particular decomposition chosen for X. Note that the expression z L 0 is indeed a well-defined bounded operator for every z ∈ D 1 (for z = 0, we define it by continuity in the strong operator topology, hence to be the projection P 0 onto CΩ): this is because Sp(L 0 ) ⊂ N. That is, we are using not just the positivity of L 0 , but also that elements of its spectrum are all integers (e.g. z 1 2 = √ z would be ambiguous).
For every X ∈ A(I 1 ) ∨ alg A(I 3 ), the map z → φ z (X) is analytic in D 1 . In fact, denoting by P m the spectral projection of L 0 associated to the eigenvalue m, we have the power series decomposition of φ z (X) Since P 0 = Ω, · Ω is the one-dimensional projection on the vacuum vector, we have that i.e. φ 0 is the product vacuum state, whereas φ 1 = ω. Thus, in view of Proposition 2.2, in order to prove the split property, we need to show that while "changing" the parameter z from 1 to 0, the functional φ z remains normal. In particular, it would be desirable to obtain estimates on φ 1 − φ 0 .
The idea of considering φ z not only at the points z = 1 and z = 0, but on a larger area (so that its analytic dependence on z can be exploited) comes from [18]. There the authors work with the function z → φ z (AB) to obtain a bound on |φ 1 (AB) − φ 0 (AB)| for a pair of elements A ∈ A(I 1 ) and B ∈ A(I b ) thereby proving the conformal cluster theorem for a Möbius covariant net. However, their estimate involves the product of norms A B ; when it is reformulated for an element X of the considered form (5), we get some bounds in terms of k A k B k , rather than in terms of the norm of X. Hence their method does not give a useful estimate on φ 1 − φ 0 . In fact, they cannot obtain anything that would imply the split property: this is because they only use Möbius covariance, and as was mentioned in the introduction, counterexamples to the split property exist when diffeomorphism covariance is not assumed [9,Section 6].
Instead, our idea is the following: using diffeomorphism covariance and in particular the decompositions of r L 0 established in the previous sections, we can show that φ z depends norm-continuously on z and is normal (i.e. extends to a normal linear functional of the von Neumann algebra A(I a ) ∨ A(I b )) when z is in a certain region. Unfortunately, the region directly obtainable by such decompositions do not contain the desired point z = 0. However, if this region contains a ring encircling the point z = 0 (and as we shall see, this will exactly be the case) we can use general complex analytic arguments (essentially the Cauchy theorem) to deduce normality of φ 0 : Lemma 5.1. Let r 0 ∈ (0, 1) be a fixed radius and suppose that φ z is normal whenever |z| = r 0 and that on the circle with radius r 0 , r 0 S 1 ∋ z → φ z is norm-continuous. Then φ 0 is also normal.
Proof. We shall use some well-known technical facts. In particular, we shall exploit that the norm-limit of a sequence of normal functionals on a von Neumann algebra M is normal (see e.g. [27,Corollary 7.1.13]). To apply this fact, one should note that the norm is defined on the von Neumann algebra M, but by the Kaplansky density theorem, the norm of a normal functional on A(I a )∨A(I b ) is equal to the norm of its restriction to A(I a )∨ alg A(I b ). Therefore, in the following we do not distinguish them.
Thus one has -e.g. by considering Riemann-sums -that if ϕ : [s 1 , s 2 ] ∋ t → ϕ t ∈ M * is a norm-continuous family of normal linear functionals, then ϕ(·) = s 2 s 1 ϕ t (·)dt is also a well-defined normal functional on M.
Since D 1 ∋ z → φ z (X) is analytic, by the Cauchy integral formula we have Let us now discuss how the decompositions r L 0 help us out in different regions of D 1 . Let I c = I ′ a , and I d be an (open) interval containing the closure of I a but not intersecting I b .
(Such an "enlargement" of I b exists as I a and I b were assumed to have a positive distance from each other). We then have that I c ∪ I d = S 1 and we can consider the decomposition of r L 0 given by Proposition 3.1 with {C r } r∈(0,1) ⊂ A(I c ) and {D r } r∈(0,1) ⊂ A(I d ). Let us denote by R θ the rotation by θ. Then, as long as z = re iθ is such that r ∈ (0, 1) and the angle θ satisfies the condition we have that the action of ρ θ ≡ Ad e iθL 0 leaves A(I b ) inside A(I d ) ′ and thus for an X ∈ A(I a ) ∨ alg A(I b ) with decomposition (5), we can use locality to rewrite φ z (X) as Let (θ − , θ + ) be the largest open interval of angles satisfying our condition (6) and containing 0, i.e., θ ± is the smallest positive / largest negative angle for which R θ ± (I b ) intersects I d .
We can obviously find a smooth function f : S 1 → R such that f on I a is zero, but is constant 1 on the complement of I d (i.e. on the complement of the "enlarged" version of I a ). Viewing f as the vector field on S 1 formally written as f (e iθ ) d dθ , it gives rise to a one parameter group of diffeomorphisms such that γ θ is "localized" in I ′ a , but if θ − < θ < θ + , then the action of γ θ on I b coincides with that of the rotation by θ. Thus, e iθT (f ) commutes with elements of A(I a ) but for θ ∈ (θ − , θ + ), its adjoint action on A(I b ) coincides with the action of ρ θ and hence we can write Putting this back in (7), we get that for θ ∈ (θ − , θ + ), r Ω and ζ θ = e −iθT (f ) D r Ω. Corollary 5.2. {φ z } is a norm-continuous family of normal functionals in the region {re iθ |r ∈ (0, 1) and I d ∩ R θ (I b ) = ∅}.
Note that Proposition 3.1 gives some bounds on the norms of C r and D r , and so actually with the constant q > 0 defined there, in the discussed region we have Unfortunately, though this estimate is nicely uniform in θ, it "blows up" at r → 0 and hence in itself it does not show that φ r converges to φ 0 in norm as r → 0. However, so far we have only used our first decomposition of r L 0 . We shall now exploit the second one derived in Section 4. Consider a radius r 0 ∈ (0, 1) such that the decomposition in Proposition 4.5 holds; that is, we have two bounded elements C ∈ A(I c ) and D ∈ A(I d ) and g = id a Möbius transformation such that r L 0 0 = CDU(g) (recall that this is valid in the proper sense, namely one can fix the phase of U(g) for g ∈ Möb unambiguously). Then, repeating the steps we did before with our previous decomposition and setting instead of (7), this time we get whenever the disjointness condition holds. It is an easy exercise to show, that we can continue exactly as in the first case, and hence this time obtain the following.

Corollary 5.3. {φ z } is a norm-continuous family of normal functionals in the region
Does the union of the two treated regions encircle the point 0? This might not be the case. However, note that the Möbius transformation g given by Proposition 4.5 is an "absolute" one; i.e. it does not depend on the intervals I c and I d (whereas of course the elements C and D obviously do). And though for some choices of I a , I b and I d ⊃ I a might lead to nowhere, it is enough for us to show that there is a "right" choice. Proof. By conformal covariance, we may assume that I a , I b and even the enlargement I d ⊃ I a are "tiny"; almost point-like intervals around two points which we will conveniently call a and b. Then the region guaranteed by Corollary 5.2 is D 1 minus a slightly enlarged version of the half-line {te iα |t ≥ 0} where α is the angle for which R α (b) = a. On the other hand, the region guaranteed by Corollary 5.3 is the circle r 0 S 1 minus a slightly enlarged version of the point r 0 e iα , whereα is the angle for which g • Rα(b) = a, which is of course equivalent to saying that Rα(b) = g −1 (a). Since g is a certain fixed, non trivial Möbius transformation, we might even assume that our choice of a is such that g −1 (a) = a. Then α =α and the union of the two regions covers the circle r 0 S 1 .
Lemma 5.1 shows that φ 0 is a normal linear functional on A(I 1 ) ∨A(I 3 ). Now our claim is concluded by Proposition 2.2 and a technical Lemma 5.5 below, by noting that is a factor (Section 2, Factoriality of two-interval algebras) • The restrictions of φ 0 to A(I a ) and A(I b ) are equal to the vacuum state, hence faithful.
Lemma 5.5. φ 0 is a positive normal functional on A(I a ) ∨ A(I b ).
Proof. We first consider φ 0 on A(I a ) ∨ alg A(I b ). By [38,Proposition 4.20], the map τ : We claim that, also on A(I a ) ∨ alg A(I b ), φ 0 is positive 3 . Indeed, take a positive element a ∈ A(I a ) ∨ alg A(I b ). The function f (x) = x 1 2 , x ∈ [0, a ] can be arbitrarily approximated by polynomials f n with real coefficients, uniformly on [0, a ] and f n (a) 2 tends to a in norm. We saw that φ 0 is a normal linear functionals, hence it is in particular continuous in norm.
Now, by the Kaplansky density theorem and the normality of φ 0 , φ 0 is a positive functional. 1. Compatibility: if I 1 , I 2 ∈ I and I 1 ⊂ I 2 then ρ I 2 | A(I 1 ) = ρ I 1

Covariance: Ad
A representation of a conformal net ρ is said to be localizable in I 0 if ρ I ′ 0 ≃ id, where ≃ means unitary equivalence. The unitary equivalence class of ρ defines a superselection sector, also called a DHR (Doplicher-Haag-Roberts) sector [14]. By Haag duality we have that ρ(A(I)) ⊂ A(I) if I 0 ⊂ I. Thus we can always choose, within the sector of ρ, a representation ρ 0 on the defining Hilbert space H such that ρ 0,I 0 is an endomorphism of A(I 0 ). If each ρ I is an automorphism of A(I), we call ρ an automorphism of (A, U, Ω). Automorphisms can be composed in a natural way.
Let (A, U, Ω) be the U(1)-current net [5]. The main ingredients are (see [40] for a more detailed review): 2π 0 dθ f (e iθ )ϕ(e iθ ). We call this automorphism of the net σ q . Different functions ϕ with the conditions above with the same q give the equivalent sectors, while sectors with different q are inequivalent. It holds that σ q • σ q ′ = σ q+q ′ .
• For two equivalent automorphisms ρ,ρ localized in I,Ĩ, respectively, an operator which intertwines them is called a charge transporter. In the present case, as both ρ,ρ are irreducible, such a charge transporter is unique up to a scalar. A charge transporter acts trivially on A((I ∪Ĩ) ′ ), hence belongs to A((I ∪Ĩ) ′ ) ′ . In particular, it can be considered as an element in a local algebra containing I andĨ.
• For a given pair of automorphisms ρ 1 , ρ 2 , one defines the braiding ǫ ρ 1 ,ρ 2 : one chooses equivalent automorphismsρ 1 ,ρ 2 localized inĨ 1 ,Ĩ 2 , respectively, such thatĨ 1 ∩Ĩ 2 = ∅ and charge transporters V 1 , V 2 between ρ 1 andρ 1 , and ρ 2 andρ 2 , respectively. Define where + or − depends on the choice whetherĨ 1 is on the left/right ofĨ 2 (which results from the choice of localization of the charge transporter above), but ǫ ± ρ 1 ,ρ 2 do not depend on the choice ofρ k , V k under such a configuration. • For our concrete automorphisms σ q , σ q ′ on the U(1)-current net, one can take the charge transporters V q , V q ′ as Weyl operators and finds that the braiding satisfies The following is may be well known to experts, but it is difficult to find the right reference (for example, [32,Proposition 1.4] is proved for Möbius covariance). We note that a systematic formulation, closer to our needs, is to appear in [13]. Nevertheless, in part because we deal with multiplier representations, and in part for better readability, we include a formal statement with a proof. Proposition 6.1 (Tensoriality of cocycles). It holds that z q (γ)σ q (z q ′ (γ)) = z q+q ′ (γ).
Next we show that , namely it has the same 2-cocycle c as U q+q ′ . Indeed, where in the 3rd and 6th equalities we used that U and U q share the same 2-cocycle c. Now let us define U ′′ (γ) := U ′ q (γ) * U q (γ). As the difference between U ′ q (γ) and U q (γ) is just a phase and they share the same 2-cocycle c, it is easy to show that U ′′ is a C-valued true (with trivial multiplier) representation of Diff + (S 1 ). It is well-known that then U ′′ must be trivial, U ′′ (γ) = ½. From this the claim immediately follows.
Let G be the quotient of Möb × Möb by the normal subgroup generated by (R 2π , R −2π ), where Möb naturally includes the universal covering R of the rotation subgroup S 1 and R 2π , R −2π are the elements corresponding to 2π, −2π rotations, respectively. We call R×S 1 the Einstein cylinder E, where the Minkowski space is identified with a maximal square (−π, π) × (−π, π) (see [1]) 4 . The group G acts naturally on it. Furthermore, let Diff 0 (R) be the group of diffeomorphisms of the real line R with compact support. Then Diff 0 (R) × Diff 0 (R) acts naturally on the Minkowski space as the product of two lightrays 5 , and its action naturally extends to E by periodicity. Let us denote by Conf(E) the group generated by G and Diff 0 (R) × Diff 0 (R). A two-dimensional conformal net (Ã,Ũ ,Ω) consists of a family {Ã(O)} of von Neumann algebras parametrized by double cones {O} in the Minkowski space R 2 , a strongly-continuous unitary representation of G which extends to a projective unitary representation of Conf(E), and a vectorΩ such that the following axioms are satisfied [28, Section 2]: • Existence and uniqueness of vacuum.Ω is a unique (up to a scalar) invariant vector forŨ | G .
• Positivity of energy. Each component U q ⊗ U q has positive energy.
• Existence and uniqueness of the vacuum. Only U 0 ⊗ U 0 contains the vacuum vector.
• Cyclicity. The fields ψ q brings H 0 ⊗ H 0 to any H q ⊗ H q , while the local algebrã σ(A(I) ⊗ A(I)) acts irreducibly on each H q ⊗ H q .
• Locality. In the two-dimensional situation, the spacelike separation of I × I and I + × I − means either I + sits on the left of I and I − on the right, or vice versa. We may assume the former case, as the latter is parallel.
This net cannot satisfy the split property. Namely, if there were a type I factor R such thatÃ(D 1 ) ⊂ R ⊂Ã(D 2 ), Ω, · Ω would define a faithful normal state on R, as it is separating for R. As the full Hilbert spaceH is non-separable, by conformal covariance R must be isomorphic to B(H). But this is impossible because the existence of a faithful normal state implies that B(H) is σ-finite, while it is not whenH is non separable.

Outlook
In general, a standard technique to prove the split property is to verify certain nuclearity conditions for the dynamics. In the Möbius covariant case, the most handy one is the trace class condition of the conformal Hamiltonian e −βL 0 [4]. The split property in turn implies certain compactness conditions [3]. With our result, one is lead to conjecture that the trace class property should be also automatic.
The existence of an intermediate type I factor does not depend on the sector. Assume A to be a Möbius covariant net satisfying split property (for instance A is a conformal net) and I 1 ⊂ I 2 an inclusion of intervals with no common end points. Any representation π of A is a family of local algebra faithful isomorphisms onto their image, as any local algebra is a factor. Then an intermediate type I factor A(I 1 ) ⊂ R ⊂ A(I 2 ) is mapped through ρ onto an intermediate type I factor ρ I 2 (A(I 1 )) ⊂ ρ I 2 (R) ⊂ ρ I 2 (A(I 2 )) as ρ I 2 restricts to an isomorphism of R on ρ I 2 (R). Furthermore, when ρ is localizable, then ρ I 1 (A(I 1 )) ⊂ ρ I 2 (A(I 2 )) is a standard split inclusion acting on a separable Hilbert space (we can unitarily identify the Hilbert spaces). At this point it is also natural to expect that the trace class property of L ρ 0 in irreducible or factorial sectors should be automatic. While the split property has important implications in algebraic QFT, it is almost never seen in other approaches to CFT, such as vertex operator algebras (VOAs). On the other hand, the trace class property, or even the finite-dimensionality of the eigenspaces of L 0 would be useful for the study of VOAs.
We are grateful to Marcel Bischoff, Sebastiano Carpi, Luca Giorgetti and Roberto Longo for various interesting discussions on two-dimensional CFT.