# Conformal covariance and the split property

Vincenzo Morinelli111Supported by the ERC advanced grant 669240 QUEST “Quantum
Algebraic Structures and Models”., Yoh Tanimoto222Supported by the JSPS
fellowship for research abroad.
Dipartimento di Matematica, Universitá di Roma Tor Vergata
Via della Ricerca Scientifica, 1, I-00133 Roma, Italy
email<EMAIL_ADDRESS><EMAIL_ADDRESS>
Mihály Weiner333Supported in part by the ERC advanced grant 669240 QUEST
“Quantum Algebraic Structures and Models” and by OTKA grant no. 104206.
Mathematical Institute, Department of Analysis,
Budapest University of Technology & Economics (BME)
Műegyetem rk. 3-9, H-1111 Budapest, Hungary
email<EMAIL_ADDRESS>
###### Abstract

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,
Möbius 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.

##  1 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 Additivity — are 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” counter-examples; 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 $\mu$-index is finite.

The crucial points of our proof are the following. We consider a conformal net
${\mathcal{A}}$ on the circle with conformal Hamiltonian $L_{0}$, and fix two
(open, proper) intervals $I_{a},I_{b}\in{\mathcal{I}}$ 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
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$ generated by
${\mathcal{A}}(I_{a})$ and ${\mathcal{A}}(I_{b})$ with decomposition
$X=\sum_{k=1}^{n}A_{k}B_{k}$ (where
$n\in{\mathbb{N}},\,A_{k}\in{\mathcal{A}}(I_{a}),\,B_{k}\in{\mathcal{A}}(I_{b})$),
we consider the function on the complex unit disc

$z\mapsto\sum_{k=1}^{n}\langle\Omega,A_{k}z^{L_{0}}B_{k}\Omega\rangle.$

For every $|z|\leq 1$, this defines a functional $\phi_{z}$ on
${\mathcal{A}}(I_{a})\vee_{alg}{\mathcal{A}}(I_{b})$. For $z=1$ this is simply
the vacuum state $\omega$, but for $z=0$ this is the product vacuum state
$AB\mapsto\omega(A)\omega(B)$
$(A\in{\mathcal{A}}(I_{a}),\,B\in{\mathcal{A}}(I_{b}))$. The split property is
essentially equivalent to saying that $\phi_{0}$ 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
${\mathcal{A}}(I_{1})\subset{\mathcal{A}}(I_{2})$ for an $I_{1}\subset I_{2}$
imply that ${\mathcal{A}}(I_{a})\vee{\mathcal{A}}(I_{b})$ is a factor; see
more details in the preliminaries. It then turns out that the normality of
$\phi_{0}$ is indeed equivalent to the split property).

However, we do not have a direct method to show that $\phi_{z}$ 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 ${{\rm Diff}^{+}(S^{1})}$
given with the theory, for example for any (fixed) $r\in(0,1)$ and
$I_{c},I_{d}\in{\mathcal{I}}$ covering the full circle we find a decomposition
$r^{L_{0}}=CD$ in which $C_{r}\in{\mathcal{A}}(I_{c})$ and
$D_{r}\in{\mathcal{A}}(I_{d})$. 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

$\displaystyle\phi_{r}(X)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\langle\Omega,A_{k}r^{L_{0}}B_{k}\Omega\rangle=\sum_{k=1}^{n}\langle\Omega,A_{k}CDB_{k}\Omega\rangle$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{n}\langle
C^{*}\Omega,A_{k}B_{k}D\Omega\rangle=\langle C^{*}\Omega,\,X\,D\Omega\rangle$

showing that for our real $r\in(0,1)$, the functional $\phi_{r}$ 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 $\phi_{z}$ 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 ${\rm Sp}(L_{0})$
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
$\mathcal{B}\supset{\mathcal{A}}$ of the net
${\mathcal{A}}={\mathcal{A}}_{U(1)}\otimes{\mathcal{A}}_{U(1)}$ 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 $q\in{\mathbb{R}}$. Our construction is such
that when considered as a representation of
${\mathcal{A}}_{U(1)}\otimes{\mathcal{A}}_{U(1)}$, the net
${\mathcal{A}}\subset\mathcal{B}$ decomposes as a direct sum
$\oplus_{q\in{\mathbb{R}}}\,(\sigma_{q}\otimes\sigma_{q})$ where $\sigma_{q}$
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 $\widetilde{{{\rm
Diff}^{+}(S^{1})}}\times\widetilde{{{\rm Diff}^{+}(S^{1})}}$ which factors
through the spacelike $2\pi$-rotation, but not that of ${{\rm
Diff}^{+}(S^{1})}\times{{\rm Diff}^{+}(S^{1})}$. 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 $\phi_{0}$. A two-dimensional counterexample is provided in
Section 6. In Section 7 we conclude with open problems.

##  2 Preliminaries

Let ${\mathcal{I}}$ be the set of nonempty, nondense, open connected intervals
of the unit circle $S^{1}=\\{z\in{\mathbb{C}}:|z|=1\\}$. A Möbius covariant
net is a map ${\mathcal{A}}$ which assigns to every interval of the circle
$I\in{\mathcal{I}}$ a von Neumann algebra ${\mathcal{A}}(I)$ acting on a fixed
Hilbert space ${\mathcal{H}}$ satisfying the following properties:

  1. 1.

Isotony: if $I_{1},I_{2}\in{\mathcal{I}}$ and $I_{1}\subset I_{2}$, then
${\mathcal{A}}(I_{1})\subset{\mathcal{A}}(I_{2})$;

  2. 2.

Möbius covariance: there exists a strongly continuous, unitary representation
$U$ of the Möbius group ${\mathrm{M\ddot{o}b}}$
($\simeq\mathrm{PSL(2,{\mathbb{R}})}$) on ${\mathcal{H}}$ such that

$U(g){\mathcal{A}}(I)U(g)^{*}={\mathcal{A}}(gI),\qquad
I\in{\mathcal{I}},\,g\in{\mathrm{M\ddot{o}b}};$

  3. 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. 4.

Existence and uniqueness of the vacuum: there exists a unique (up to a phase)
unit $U$-invariant vector $\Omega\in{\mathcal{H}}$, i.e. $U(g)\Omega=\Omega$
for $g\in{\mathrm{M\ddot{o}b}}$;

  5. 5.

Cyclicity: $\Omega$ is cyclic for the von Neumann algebra
$\bigvee_{I\in{\mathcal{I}}}A(I)$.

  6. 6.

Locality: if $I_{1},I_{2}\in{\mathcal{I}}$ and $I_{1}\cap I_{2}=\emptyset$,
then ${\mathcal{A}}(I_{1})\subset{\mathcal{A}}(I_{2})^{\prime}$.

We will denote a Möbius covariant net with the triple
$({\mathcal{A}},U,\Omega)$. Some consequences of the axioms are (see e.g. [19,
18, 22]):

  * 7.

Reeh-Schlieder property: $\Omega$ is a cyclic and separating vector for each
${\mathcal{A}}(I)$, $I\in{\mathcal{I}}$;

  * 8.

Haag duality: ${\mathcal{A}}(I^{\prime})^{\prime}={\mathcal{A}}(I)$, where
$I\in{\mathcal{I}}$ and $I^{\prime}$ is the interior of $S^{1}\backslash I$;

  * 9.

Bisognano-Wichmann property: $U(\delta_{I}(-2\pi
t))=\Delta^{it}_{{\mathcal{A}}(I),\Omega}$ where $\delta_{I}$ is the dilation
subgroup associated to the interval $I$ and
$\Delta_{{\mathcal{A}}(I),\Omega}^{it}$ is the modular group of
${\mathcal{A}}(I)$ with respect to $\Omega$;

  * 10.

Irreducibility:
$\bigvee_{i\in{\mathcal{I}}}{\mathcal{A}}(I)={\mathcal{B}}({\mathcal{H}})$;

  * 11.

Factoriality: algebras ${\mathcal{A}}(I)$ are type $\mathrm{III}_{1}$ factors;

  * 12.

Additivity: let $\\{I_{\kappa}\\}\subset{\mathcal{I}}$ be a covering of $I$,
namely $I\subset\bigcup_{\kappa}I_{\kappa}$, then
${\mathcal{A}}(I)\subset\bigvee_{\kappa}{\mathcal{A}}(I_{\kappa})$.

The following seems relatively less known, yet it follows from Möbius
covariance and has an important implication [23, Theorem 1.6].

  * 13.

Normality and conormality: for any inclusion $I_{1}\subset I_{2}$, it holds
that
${\mathcal{A}}(I_{1})={\mathcal{A}}(I_{2})\cap\left({\mathcal{A}}(I_{1})^{\prime}\cap{\mathcal{A}}(I_{2})\right)^{\prime}$
and
${\mathcal{A}}(I_{2})={\mathcal{A}}(I_{1})\vee\left({\mathcal{A}}(I_{1})^{\prime}\cap{\mathcal{A}}(I_{2})\right)$

From conormality, it follows that two-interval algebras are factors. Indeed,
take $I_{1}\subset I_{2}$ such that they have no common end points. Then
$I_{1}$ and $I_{2}^{\prime}$ are disjoint intervals with a finite distance. By
Haag duality it follows that
$({\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{2}^{\prime}))^{\prime}=A(I_{1})^{\prime}\cap{\mathcal{A}}(I_{2})$,
and by conormality we have

$\displaystyle\left({\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{2}^{\prime})\right)\bigvee\left({\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{2}^{\prime})\right)^{\prime}$
$\displaystyle={\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{2}^{\prime})\vee\left({\mathcal{A}}(I_{1})^{\prime}\cap{\mathcal{A}}(I_{2})\right)$
$\displaystyle={\mathcal{A}}(I_{2})\vee{\mathcal{A}}(I_{2}^{\prime})={\mathcal{B}}({\mathcal{H}}),$

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, ${\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{2})$
is a factor.

Now, we briefly discuss diffeomorphism covariance. Let ${{\rm
Diff}^{+}(S^{1})}$ be the group of orientation preserving diffeomorphisms of
the circle. It is an infinite dimensional Lie group modelled on the real
topological vector space ${{\rm Vect}}(S^{1})$ of smooth real vector fields on
$S^{1}$ with the $C^{\infty}$-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 $f(e^{i\theta})\frac{d}{d\theta}\in{{\rm Vect}}(S^{1})$ with the
corresponding real function $f\in C^{\infty}(S^{1},{\mathbb{R}})$. We denote
with ${{\rm Diff}^{+}(I)}$ the subgroup of ${{\rm Diff}^{+}(S^{1})}$ acting
identically on $I^{\prime}$, namely the diffeomorphisms of $S^{1}$ with
support included in $\overline{I}$.

A strongly continuous, projective unitary representation $U$ of ${{\rm
Diff}^{+}(S^{1})}$ on a Hilbert space ${\mathcal{H}}$ is a strongly continuous
homomorphism of ${{\rm Diff}^{+}(S^{1})}$ into
${\mathcal{U}}({\mathcal{H}})/{\mathbb{T}}$, the quotient of the group of
unitaries in ${\mathcal{B}}({\mathcal{H}})$ by ${\mathbb{T}}$. The restriction
of $U$ to ${\mathrm{M\ddot{o}b}}\subset{{\rm Diff}^{+}(S^{1})}$ always lifts
to a unique strongly continuous unitary representation of the universal
covering group $\widetilde{{\mathrm{M\ddot{o}b}}}$ of ${\mathrm{M\ddot{o}b}}$.
$U$ is said to have positive energy, if the generator $L_{0}$ of rotations,
the conformal Hamiltonian, has a nonnegative spectrum in this lift. Let
$\gamma\in{{\rm Diff}^{+}(S^{1})}$. Note that expressions ${\hbox{\rm
Ad\,}}U(\gamma)$ makes sense as an action on ${\mathcal{B}}({\mathcal{H}})$.
We also write $U(\gamma)\in{\mathcal{M}}$ although $U(\gamma)$ is defined only
up to a scalar.

When one has a strongly continuous projective unitary representation $U$ of
${{\rm Diff}^{+}(S^{1})}$ with positive energy, $e^{i2\pi L_{0}}$ is a
multiple of the identity and therefore $L_{0}$ has a pure point-spectrum. It
follows that the linear span ${\mathcal{D}}_{\mathrm{fin}}$ 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 $f\in
C^{\infty}(S^{1},{\mathbb{R}})$, as a vector field on $S^{1}$, defines a one-
parameter group of diffeomorphisms ${\mathbb{R}}\ni
t\mapsto\gamma_{t}\dot{=}{\rm Exp}(tf)\in{{\rm Diff}^{+}(S^{1})}$, hence, up
to an additive constant, defines the self-adjoint generator $T(f)$ of the
unitary group $t\mapsto U(\gamma_{t})$. For any real smooth function $f$ as
above, $T(f)$ is essentially self-adjoint on the set
$C^{\infty}(L_{0}):=\bigcap_{n\in{\mathbb{N}}_{0}}{\rm Dom\,}(L_{0}^{n})$. $T$
shall be called the stress energy tensor.

Irreducible, projective, unitary positive energy representation of ${{\rm
Diff}^{+}(S^{1})}$ are labelled by certain values of the central charge $c>0$
and the lowest weight $h\geq 0$. $h$ is the lowest point in the discrete
spectrum of the conformal Hamiltonian $L_{0}$. There is a unique (up to a
phase) vector $\Phi\in{\mathcal{H}}$ corresponding to the lowest eigenvalue.
See [21, 26] for a detailed description of such representations.

One considers particular elements $\\{L_{n}:n\in\mathbb{Z}\\}$,
$L_{n}=iT(y_{n})-T(x_{n}),L_{-n}=iT(y_{n})+T(x_{n})$ for $n\in\mathbb{N}$,
where $x_{n}(\theta):=-\frac{1}{n}\sin n\theta$ and
$y_{n}(\theta):=-\frac{1}{n}\cos n\theta$ (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 $\widetilde{{\mathrm{M\ddot{o}b}}}$). These
operators satisfy the so-called Virasoro algebra on finite energy vectors
${\mathcal{D}}_{\mathrm{fin}}$. In particular for all $n,m\in{\mathbb{Z}}$:
${\mathcal{D}}_{\mathrm{fin}}$ is an invariant common core for any closed
operator $L_{n}$; if $n>0$ then $L_{n}\Phi=0$; $L_{-n}\subset L_{n}^{*}$; the
family $\\{L_{n}\\}_{n\in{\mathbb{Z}}}$ satisfies the Virasoro algebra
relations on ${\mathcal{D}}_{\mathrm{fin}}$:

$[L_{n},L_{m}]=(n-m)L_{n+m}+\frac{c}{12}(n^{3}-n)\delta_{-m,n}{{\mathbbm{1}}}.$

Let $f\in C^{\infty}(S^{1},{\mathbb{R}})$ be a vector field on $S^{1}$, with
Fourier coefficients

$\hat{f}_{n}=\frac{1}{2\pi}\int_{0}^{2\pi}f(\theta)e^{-in\theta}d\theta,\qquad
n\in{\mathbb{Z}},$

then, one can recover the stress-energy tensor by

$T(f)=\overline{\sum_{n\in{\mathbb{Z}}}\hat{f}_{n}L_{n}}$ (1)

and

$e^{iT(f)}=U({\rm Exp}(f))$

gives the correspondence between the infinitesimal generators and the
representation of ${{\rm Diff}^{+}(S^{1})}$ (up to a scalar).

Throughout the next few sections we shall often consider the net of von
Neumann algebras

${\mathcal{A}}_{U}(I)=\\{e^{iT(f)}|\,f\in
C^{\infty}(S^{1},{\mathbb{R}}),\,{\rm supp}(f)\subset
I\\}^{\prime\prime}\qquad(I\in{\mathcal{I}}).$ (2)

Note that when $U$ is a so-called vacuum representation associated to central
charge $c$, ${\mathcal{A}}_{U}$ is nothing else than the well-known Virasoro
net with central charge $c$. In general though, ${\mathcal{A}}_{U}$ 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 ${\mathcal{A}}_{U}(I_{1})\subset{\mathcal{A}}_{U}(I_{2})^{\prime}$
whenever $I_{1}\cap I_{2}=\emptyset$.

The stress energy tensor can be evaluated on a larger set of functions [9].
For a continuous function $f:S^{1}\to{\mathbb{R}}$ with Fourier coefficients
$\\{\hat{f}_{n}\\}_{n\in{\mathbb{Z}}}$ we shall set

$\|f\|_{\frac{3}{2}}=\sum_{n\in{\mathbb{Z}}}|\hat{f}_{n}|\left(1+|n|^{\frac{3}{2}}\right).$

Then $\|\cdot\|_{\frac{3}{2}}$ is a norm on the space $\\{f\in
C(S^{1},{\mathbb{R}})|\,\|f\|_{\frac{3}{2}}<\infty\\}$. By [9], if $f\in
C(S^{1},{\mathbb{R}})$ with $\|f\|_{\frac{3}{2}}<\infty$, then $T(f)$, defined
as in (1), is self-adjoint and moreover if $f_{k}\to f$ in the norm
$\|\cdot\|_{\frac{3}{2}}$, then $T(f_{k})\to T(f)$ in the strong resolvent
sense. In particular, even for a non necessarily smooth function $f$ with
$\|f\|_{\frac{3}{2}}<\infty,{\rm supp\,}f\subset I$, the self-adjoint $T(f)$
is still affiliated to ${\mathcal{A}}_{U}(I)$.

We shall say that a Möbius covariant net $(A,U,\Omega)$ is conformal (or
diffeomorphism covariant) if the ${\mathrm{M\ddot{o}b}}$ representation $U$
extends to a projective unitary representation ${{\rm
Diff}^{+}(S^{1})}\to{\mathcal{U}}({\mathcal{H}})/{\mathbb{T}}$ of ${{\rm
Diff}^{+}(S^{1})}$ (that with a little abuse of notation we continue to
indicate the extension with $U$) and satisfying

  * •

${\hbox{\rm Ad\,}}U(\gamma)({\mathcal{A}}(I))={\mathcal{A}}(\gamma I),\;\text{
for }\gamma\in{{\rm Diff}^{+}(S^{1})}$

  * •

${\hbox{\rm Ad\,}}U(\gamma)(x)=x,\;\text{ for }\gamma\in{{\rm
Diff}^{+}(I)},x\in{\mathcal{A}}(I^{\prime})$

Now we recall the definition of the split property for von Neumann algebra
inclusions and conformal nets.

######  Definition 2.1.

Let $({\mathcal{N}}\subset{\mathcal{M}},\Omega)$ be an standard inclusion of
von Neumann algebras, i.e. $\Omega$ is a cyclic and separating vector for
$N,\,M$ and $N^{\prime}\cap M$.

A standard inclusion $({\mathcal{N}}\subset{\mathcal{M}},\Omega)$ is split if
there exists a type I factor ${\mathcal{R}}$ such that
${\mathcal{N}}\subset{\mathcal{R}}\subset{\mathcal{M}}$.

A Möbius covariant net $({\mathcal{A}},U,\Omega)$ satisfies the split property
if the von Neumann algebra inclusion
${\mathcal{A}}(I_{1})\subset{\mathcal{A}}(I_{2})$ is split, for any inclusion
of intervals $I_{1}\Subset I_{2}$, 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 444If the Hilbert
spaces are not separable, several well-known statements no longer hold. For
example, an isomorphism between type III algebras might be not a unitary
equivalence. nor the faithfulness of the split state in the implication
$2\Rightarrow 1$ below).

######  Proposition 2.2.

Let $({\mathcal{N}}\subset{\mathcal{M}},\Omega)$ be a standard inclusion of
von Neumann algebras. We further assume that that
${\mathcal{N}}\vee{\mathcal{M}}^{\prime}$ is a factor. Then the following are
equivalent.

  1. 1.

${\mathcal{N}}\subset{\mathcal{M}}$ is split;

  2. 2.

there exists a normal state $\phi$ on
${\mathcal{N}}\vee{\mathcal{M}}^{\prime}$ such that the restrictions
$\phi_{\mathcal{N}}$ and $\phi_{{\mathcal{M}}^{\prime}}$ are faithful and
$\phi$ is split, namely,

$\phi(xy)=\phi(x)\phi(y),\qquad x\in{\mathcal{N}},y\in{\mathcal{M}}^{\prime}.$

###### Proof.

If ${\mathcal{N}}\subset{\mathcal{M}}$ is split, namely if there is an
intermediate type I factor ${\mathcal{R}}\simeq{\mathcal{B}}(\mathcal{K})$,
then ${\mathcal{N}}\vee{\mathcal{M}}^{\prime}$ is isomorphic to
${\mathcal{N}}\otimes{\mathcal{M}}^{\prime}$, from which the implication
$1\Rightarrow 2$ follows.

Conversely, let there be a split state as in $2$. First of all, as $\phi$ are
faithful on ${\mathcal{N}}$ and ${\mathcal{M}}^{\prime}$, their GNS
representations $\pi_{{\mathcal{N}}},\pi_{{\mathcal{M}}^{\prime}}$ are
faithful and have a cyclic and separating vector. Next, as $\phi$ is normal on
${\mathcal{N}}\vee{\mathcal{M}}^{\prime}$, its GNS representation
$\pi_{{\mathcal{N}}\vee{\mathcal{M}}^{\prime}}$ is also normal. The Hilbert
space supporting $\pi_{{\mathcal{N}}\vee{\mathcal{M}}^{\prime}}$ is isomorphic
to the closure of ${\mathcal{N}}\vee_{\mathrm{alg}}{\mathcal{M}}^{\prime}$
w.r.t. the scalar product inherited by the normal state $\phi$ as $\langle
x,y\rangle_{\phi}=\phi(x^{*}y)$. By the factorization assumption on $\phi$,
the Hilbert space is the tensor product
$L^{2}({\mathcal{N}},\langle\cdot,\cdot\rangle_{\phi})\otimes
L^{2}({\mathcal{M}}^{\prime},\langle\cdot,\cdot\rangle_{\phi})$ and the GNS
representation $\pi_{{\mathcal{N}}\vee{\mathcal{M}}^{\prime}}$ restricted to
${\mathcal{N}}$ and ${\mathcal{M}}^{\prime}$ are of the form
$\pi_{\mathcal{N}}\otimes{\mathbbm{1}}$ and
${\mathbbm{1}}\otimes\pi_{{\mathcal{M}}^{\prime}}$, respectively. Furthermore,
as both ${\mathcal{N}}$ and ${\mathcal{M}}^{\prime}$ have a cyclic and
separating vector $\Omega$, their GNS representations
$\pi_{{\mathcal{N}}},\pi_{{\mathcal{M}}^{\prime}}$ are actually unitary
equivalences [37, Corollary 10.15]. As a consequence, by normality, we can
assume that
$\pi_{{\mathcal{N}}\vee{\mathcal{M}}^{\prime}}({\mathcal{N}}\vee{\mathcal{M}}^{\prime})={\mathcal{N}}\otimes{\mathcal{M}}^{\prime}$.
Furthermore, by assumption ${\mathcal{N}}\vee{\mathcal{M}}^{\prime}$ is a
factor, hence the GNS representation is an isomorphism. Now, both
${\mathcal{N}}\vee{\mathcal{M}}^{\prime}$ and
${\mathcal{N}}\otimes{\mathcal{M}}^{\prime}$ have a cyclic and separating
vector ($\Omega$ and $\Omega\otimes\Omega$ respectively), therefore, the GNS
representation is actually a unitary equivalence. Then the preimage
${\mathcal{R}}=\pi_{{\mathcal{N}}\vee{\mathcal{M}}^{\prime}}^{-1}\left({\mathcal{B}}({\mathcal{H}})\otimes{\mathbb{C}}{\mathbbm{1}}\right)$
gives the intermediate subfactor
${\mathcal{N}}\subset{\mathcal{R}}\subset{\mathcal{M}}$. ∎

######  Remark 2.3.

The split property implies separability of the Hilbert space. Indeed, if we
have a standard split inclusion of von Neumann algebra on an Hilbert space
${\mathcal{H}}$, then ${\mathcal{H}}$ has to be separable: $\Omega$ is a
cyclic and separating vector for the intermediate type I factor
${\mathcal{R}}$. By considering the cardinality of the basis, either
${\mathcal{R}}$ or ${\mathcal{R}}^{\prime}$ must be isomorphic to
${\mathcal{B}}({\mathcal{H}})$ and $\Omega$ defines a faithful vector state on
it, hence ${\mathcal{B}}({\mathcal{H}})$ is $\sigma$-finite, which is only
possible if ${\mathcal{H}}$ is separable.

##  3 Local decompositions of $e^{-\beta L_{0}}$

Throughout this section, we shall not need a conformal net, as we only work
with a strongly continuous projective unitary representation $U$ of ${{\rm
Diff}^{+}(S^{1})}$ with positive energy. We shall use the notations introduced
in the preliminaries for all associated objects (i.e.
$L_{n},(n\in{\mathbb{N}})$ will stand for the associated Virasoro algebra
representation, $T$ for the stress-energy tensor, ${\mathcal{A}}_{U}$ for the
system of von Neumann-algebras appearing at (2) etc.).

In what follows, for a $\beta>0$, $r=e^{-\beta}$ and two open proper arcs
(intervals) $I_{c},I_{d}\in{\mathcal{I}}$ that cover the circle: $I_{c}\cup
I_{d}=S^{1}$, we shall find a decomposition $e^{-\beta
L_{0}}=r^{L_{0}}=C_{r}D_{r}$ with the bounded operators
$C_{r}\in{\mathcal{A}}_{U}(I_{c})$ and $D_{r}\in{\mathcal{A}}_{U}(I_{d})$. 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.

######  Proposition 3.1.

Let $I_{c},I_{d}\in{\mathcal{I}}$ be two open proper arcs covering the circle:
$I_{c}\cup I_{d}=S^{1}$. Then there exist two norm-continuous families of
operators $(0,1)\ni r\mapsto C_{r}\in{\mathcal{A}}_{U}(I_{c})$ and $(0,1)\ni
r\mapsto D_{r}\in{\mathcal{A}}_{U}(I_{d})$ such that

$r^{L_{0}}=C_{r}D_{r}\;\;\;\;\textrm{and}\;\;\;\;\|C_{r}\|,\|D_{r}\|\leq\frac{1}{r^{q}}$

where the exponent $q=\frac{c}{48}(N^{2}-1)$ with $N$ being a positive integer
such that $6\pi/N$ is smaller than the lengths of both arcs that are obtained
by taking the intersection $I_{c}\cap I_{d}$ (note that $N$ must be at least
$4$).

###### Proof.

$I_{d}$$I_{c}$$I_{k}$$\tilde{I}_{k}$ Figure 1: Intervals $I_{c},I_{d}$
covering $S^{1}$ and $I_{k},\tilde{I}_{k}$ with $N=36$.

Let us fix a positive integer $N$ satisfying the condition of the proposition
(see Figure 1). The operators
$H:=\frac{1}{N}L_{0}+\frac{c}{24}(N-\frac{1}{N}){\mathbbm{1}}$,
$L_{+}:=\frac{1}{N}L_{-N}$ and $L_{-}:=\frac{1}{N}L_{N}$ satisfy the following
relations on ${\mathcal{D}}_{\mathrm{fin}}$:

$[H,L_{\pm}]=\mp
L_{\pm},\;\;\;\;[L_{-},L_{+}]=2H,\;\;\;\;L_{\pm}=L_{\mp}^{*}.$

Moreover $H$ is diagonalizable with non-negative eigenvalues only, the span of
its eigenvectors is exactly ${\mathcal{D}}_{\mathrm{fin}}$ which is an
invariant core for the operators $L_{\pm}$. It then follows that these
operators generate a strongly continuous, positive energy unitary
representation of the universal cover $\widetilde{{\mathrm{M\ddot{o}b}}}$ 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,

$P=\frac{1}{4}\overline{(2H-L_{+}-L_{-})}\;\;\;\textrm{and}\;\;\;\tilde{P}=\frac{1}{4}\overline{(2H+L_{+}+L_{-})}$

are conjugate to each other by the unitary operator $e^{i\pi H}$, with $P$
being the self-adjoint generator of “translations” with spectrum ${\rm
Sp}(P)={\rm Sp}(\tilde{P})={\mathbb{R}}_{+}\cup\\{0\\}$. Moreover, by [4,
Theorem 3.3] we have the relation

$e^{-2sH}=e^{-{\rm tanh}(\frac{s}{2})P}e^{-{\rm sinh}(s)\tilde{P}}e^{-{\rm
tanh}(\frac{s}{2})P}$ (3)

for all $s>0$. Let us now consider how $P$ and $\tilde{P}$ can be written in
terms of the stress-energy $T$. We have

$P=\frac{1}{4n}\overline{(2L_{0}-L_{-N}-L_{N})}+\frac{c}{48}\left(N-\frac{1}{N}\right){\mathbbm{1}}=T(p)+b{\mathbbm{1}}$

and likewise $\tilde{P}=T(\tilde{p})+b{\mathbbm{1}}$, where

$b=\frac{c}{48}\left(N-\frac{1}{N}\right)$

and $p$ and $\tilde{p}$ are the functions defined by the formulas
$p(z)=\frac{1}{4n}(2-z^{N}-z^{-N})$ and
$\tilde{p}(z)=\frac{1}{4N}(2+z^{N}+z^{-N})$.

The function $p$ is nonnegative on $S^{1}$ and it has exactly $N$ points where
its value is zero:

$p(z)=0\;\;\Longleftrightarrow\;\;z=e^{i\frac{2\pi}{N}k}\;\;{\rm
for}\;k=1,\ldots N.$

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:
$p=p_{1}+\ldots+p_{N}$ where the support of the nonnegative function $p_{k}$
is the closure of the arc

$I_{k}=\left\\{e^{i\theta}:\frac{k-1}{N}<\frac{\theta}{2\pi}<\frac{k}{N}\right\\},$

and $\|p_{k}\|_{\frac{3}{2}}<\infty$. 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 $k=1,\ldots
N$,

$P_{k}=T(p_{k})+\frac{b}{N}{\mathbbm{1}}$

is a well-defined self-adjoint operator affiliated to
${\mathcal{A}}_{U}(I_{k})$ and we have $P=\overline{P_{1}+\ldots+P_{N}}$.
Since the terms in this decomposition are affiliated to commuting factors,
just as in the proof [39, Proposition 3.2], we have that

${\rm Sp}(P_{1})+\ldots+{\rm Sp}(P_{N})={\rm
Sp}(P)={\mathbb{R}}_{+}\cup\\{0\\}.$

On the other hand, the spectrum of the operators $P_{k}$ $(k=1,\ldots N)$ 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 $e^{-{\rm
tanh}(\frac{s}{2})P}$ appearing in formula (3), we have the local
decomposition into a product of commuting bounded operators

$e^{-{\rm tanh}(\frac{s}{2})P}=\prod_{k=1}^{N}e^{-{\rm
tanh}(\frac{s}{2})P_{k}}$

where the norm of each term is smaller or equal than $1$.

Let us turn to $\tilde{P}$. As we have $\tilde{P}={\hbox{\rm
Ad\,}}e^{i\frac{\pi}{N}L_{0}}(P)$, the localization of
$\tilde{P}_{k}={\hbox{\rm Ad\,}}e^{i\frac{\pi}{N}L_{0}}(P_{k})$ are different
from that of $P$: $\tilde{P}_{k}$ is affiliated to
${\mathcal{A}}_{U}(\tilde{I}_{k})$ where
$\tilde{I}_{k}=e^{i\frac{\pi}{N}}I_{k}$ and ${\mathcal{A}}_{U}$ is defined in
Section 2 (we are considering the intervals as subsets in ${\mathbb{C}}$).
With this localization, we can still assure the strong commutation between
$P_{k}$ and $\tilde{P}_{j}$ whenever $k\neq j,j+1$ (mod $N$). So in the
decomposition

$\displaystyle e^{-2sH}$ $\displaystyle=$ $\displaystyle e^{-{\rm
tanh}(\frac{s}{2})P}\;\;e^{-{\rm sinh}(s)\tilde{P}}\;\;e^{-{\rm
tanh}(\frac{s}{2})P}$ $\displaystyle=$
$\displaystyle\left(\prod_{k=1}^{N}e^{-{\rm
tanh}(\frac{s}{2})P_{k}}\right)\left(\prod_{k=1}^{N}e^{-{\rm
sinh}(s)\tilde{P}_{k}}\right)\left(\prod_{k=1}^{N}e^{-{\rm
tanh}(\frac{s}{2})\tilde{P}_{k}}\right)$

we can make some rearrangements. Note that $e^{-2sH}=r^{-L_{0}}r^{2q}$, where
$q=\frac{c}{48}(N^{2}-1)$ if we set $r=e^{-2s/N}$. To shorten notations, let
us introduce the self-adjoint contractions $X_{k}=e^{-{\rm
tanh}(\frac{s}{2})P_{k}}$ and $Y_{k}=e^{-{\rm sinh}(s)\tilde{P}_{k}}$. 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 $t>0,x\geq 0$, 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 $l\neq m,m+1$ (mod $N$).
Recall that $\frac{6\pi}{N}$ is smaller than the length of each of the
intervals of $I_{c}\cap I_{d}$. Therefore, by cyclically renaming the
intervals (but keeping the relation between $I_{k}$ and $\tilde{I}_{k}$ and
the corresponding localization of the operators), we may assume that there are
$1\leq k<j\leq N$ such that $I_{k}\cup I_{k+1}$ and $I_{j}\cup I_{j+1}$ are
included in the different connected components of $I_{c}\cap I_{d}$.
Furthermore, to fix the notation, we may assume that $I_{k}\cup\cdots\cup
I_{j+1}\subset I_{c}$, while $I_{j}\cup\cdots\cup I_{N}\cup I_{1}\cdots\cup
I_{k}\subset I_{d}$. Note that
$\tilde{I}_{k}\cup\cdots\cup\tilde{I}_{j}\subset I_{c}$ and
$\tilde{I}_{j}\cup\cdots\cup I_{N}\cup I_{1}\cdots\cup\tilde{I}_{k-1}\subset
I_{d}$ (see Figure 2).

$I_{c}$$I_{d}$$I_{k}$$\tilde{I}_{k}$$I_{k+1}$$I_{j+1}$$\tilde{I}_{j}$$I_{j}$
Figure 2: Localization of the factors of $C_{r}$. The indicated intervals
$I_{\bullet},\tilde{I}_{\bullet}$ correspond to thick segments. The operators
$\prod_{l=k}^{j+1}X_{l},\,\prod_{l=k}^{j}Y_{l},\,\prod_{l=k+1}^{j}X_{l}$ are
localized in the arcs, from the inside, respectively. The corresponding
factors in $D_{r}$ are localized in the complements of these arcs,
respectively.

By the localization explained above, we obtain

$\displaystyle r^{-L_{0}}r^{2q}$ $\displaystyle=$
$\displaystyle\left(\prod_{l=1}^{N}X_{l}\right)\;\;\;\left(\prod_{l=1}^{N}Y_{l}\right)\;\;\;\left(\prod_{l=1}^{N}X_{l}\right)$
(4) $\displaystyle=$
$\displaystyle\left(\prod_{l=1}^{k-1}X_{l}\prod_{l=k}^{j+1}X_{l}\prod_{l=j+2}^{N}X_{l}\right)\,\left(\prod_{l=1}^{k-1}Y_{l}\prod_{l=k}^{j}Y_{l}\prod_{l=j+1}^{N}Y_{l}\right)\,\left(\prod_{l=1}^{k}X_{l}\prod_{l=k+1}^{j}X_{l}\prod_{l=j+1}^{N}X_{l}\right)$
$\displaystyle=$
$\displaystyle\left(\prod_{l=k}^{j+1}X_{l}\prod_{l=k}^{j}Y_{l}\prod_{l=k+1}^{j}X_{l}\right)\;\left(\prod_{l=1}^{k-1}X_{l}\prod_{l=j+2}^{N}X_{l}\prod_{l=1}^{k-1}Y_{l}\prod_{l=j+1}^{N}Y_{l}\prod_{l=1}^{k}X_{l}\prod_{l=j+1}^{N}X_{l}\right).$

Here the first part
$C_{r}=\left(\prod_{l=k}^{j+1}X_{l}\prod_{l=k}^{j}Y_{l}\prod_{l=k+1}^{j}X_{l}\right)$
is an element of ${\mathcal{A}}_{U}(I_{c})$, where whereas the second part
$D_{r}=\left(\prod_{l=1}^{k-1}X_{l}\prod_{l=j+2}^{N}X_{l}\prod_{l=1}^{k-1}Y_{l}\prod_{l=j+1}^{N}Y_{l}\prod_{l=1}^{k}X_{l}\prod_{l=j+1}^{N}X_{l}\right)$
is an element of ${\mathcal{A}}_{U}(I_{d})$.

By construction, $\|C_{r}\|,\|D_{r}\|\leq 1$. Thus, we have obtained the
desired decomposition
$r^{-L_{0}}=(\frac{1}{r^{q}}C_{r})(\frac{1}{r^{q}}D_{r})$.

∎

In the above proposition we specifically worked with $L_{0}$. However, by
considering the adjoint actions of $U(\gamma)$ for all diffeomorphisms
$\gamma\in{{\rm Diff}^{+}(S^{1})}$ on the decompositions found above, it is
now easy to draw the following conclusion.

######  Corollary 3.2.

Let $I_{c},I_{d}\in{\mathcal{I}}$ be two open proper arcs such that $I_{c}\cup
I_{d}=S^{1}$, and $f$ a strictly positive smooth function on $S^{1}$. Then
there exist two norm-continuous families of operators $(0,1)\ni r\mapsto
C_{r}\in{\mathcal{A}}_{U}(I_{c})$ and $(0,1)\ni r\mapsto
D_{r}\in{\mathcal{A}}_{U}(I_{d})$ such that $r^{T(f)}=C_{r}D_{r}.$

##  4 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 ${{\rm
Diff}^{+}(S^{1})}$ 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 ${\rm Diff}^{+}_{a}(S^{1})$ the set
of $\gamma\in{{\rm Diff}^{+}(S^{1})}$ that extends to an annulus around
$S^{1}\subset{\mathbb{C}}$ in a complex analytic manner.

######  Theorem 4.1.

[36, Theorem 2] Let $U$ be a positive energy, strongly continuous, projective
unitary representation of ${{\rm Diff}^{+}(S^{1})}$ with the associated
conformal Hamiltonian $L_{0}$. Then elements of the form
$U(\gamma)r^{L_{0}}U(\tilde{\gamma})$ $(r\in(0,1],\gamma,\tilde{\gamma}\in{\rm
Diff}^{+}_{a}(S^{1}))$ form a projective semigroup: for any
$\gamma_{1},\gamma_{2},\tilde{\gamma}_{1},\tilde{\gamma}_{2}\in{\rm
Diff}^{+}_{a}(S^{1})$ and $r_{1},r_{2}\in(0,1]$ there exist some
$\gamma_{3},\tilde{\gamma}_{3}\in{\rm Diff}^{+}_{a}(S^{1})),r_{3}\in(0,1]$
such that

$\left(U(\gamma_{1})r_{1}^{L_{0}}U(\tilde{\gamma}_{1})\right)\,\left(U(\gamma_{2})r_{2}^{L_{0}}U(\tilde{\gamma}_{2})\right)=U(\gamma_{3})r_{3}^{L_{0}}U(\tilde{\gamma}_{3}),$

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 observation555Roberto
Longo suggested another idea for the proof of Lemma 4.2: as in the proof in
the main text, we may assume that $\gamma$ preserves three points. As
$U(\gamma)$ preserves the vacuum vector, it commutes with the modular group of
the three intervals between these points, hence with the whole Möbius group.
From this it is straightforward that $\gamma=\mathrm{id}$..

######  Lemma 4.2.

Let $({\mathcal{A}},U,\Omega)$ be a conformal net. Then
$U(\gamma)\Omega\in{\mathbb{C}}\Omega$ if and only if
$\gamma\in{\mathrm{M\ddot{o}b}}$.

###### Proof.

We need to show the “only if” part; the other direction is true by definition.
If $\gamma=\mathrm{id}$, the statement immediately follows, otherwise, by
composing a Möbius element $g$, we may assume that $\gamma$ 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 $\gamma\neq\mathrm{id}$. Then there is a
point $s\in S^{1}$ such that $\gamma(s)\neq s$. As $\gamma$ fixes three points
on $S^{1}$, it also preserves each interval bounded by any pair of these
points. Say $\lim_{n}\gamma^{n}(s)=:s_{\infty}$ (for a single element
$\gamma\in{{\rm Diff}^{+}(S^{1})}$, $S^{1}$ can be decomposed into intervals
such that $\gamma$ is monotone on each interval, hence such a limit exists).

It is not restrictive to assume that $\overline{I}_{1}$, the closure of
$I_{1}$, contains neither $\\{\gamma^{n}(s)\\}$ nor $s_{\infty}$. 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 $I:=(s,s_{\infty})$
(or $(s_{\infty},s)$ 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.

We can pick $x\in{\mathcal{A}}(I_{1})$ and $y\in{\mathcal{A}}(I)$ such that
$\langle x\Omega,y\Omega\rangle\neq 0$ by Reeh-Schlieder property. We may
further assume that
$\langle\Omega,x\Omega\rangle=0=\langle\Omega,y\Omega\rangle$, as we can
subtract their vacuum expectation. By the assumption that
$U(\gamma)\Omega\in{\mathbb{C}}\Omega$ and [18, Conformal cluster theorem], we
obtain

$\displaystyle|\langle x\Omega,y\Omega\rangle|$ $\displaystyle=|\langle
x(U(\gamma)^{*})^{n}\Omega,y(U(\gamma)^{*})^{n}\Omega\rangle|$
$\displaystyle=|\langle{\hbox{\rm Ad\,}}U(\gamma)^{n}(x)\Omega,{\hbox{\rm
Ad\,}}U(\gamma)^{n}(y)\Omega\rangle|$
$\displaystyle\leq\frac{(t_{2}-t_{1})(s_{\infty}-\gamma^{n}(s))}{(\gamma^{n}(s)-t_{1})(s_{\infty}-t_{2})}\|x\|\|y\|\to
0,$

as $\gamma^{n}(s)\to s_{\infty}$, while other distances remain finite. This is
a contradiction, hence $\gamma=\mathrm{id}$ under the assumption that $\gamma$
fixes three points.

∎

######  Corollary 4.3.

Let $V$ be a strongly continuous, unitary representation of
${\mathrm{M\ddot{o}b}}$ with positive energy, with the associated conformal
Hamiltonian $L_{0}$. Then elements of the form
$V(\gamma)r^{L_{0}}V(\tilde{\gamma})$ consist a semigroup: for any
$g_{1},g_{2},\tilde{g}_{1},\tilde{g}_{2}\in{\mathrm{M\ddot{o}b}}$ and
$r_{1},r_{2}\in(0,1]$, there exist some
$g_{3},\tilde{g}_{3}\in{\mathrm{M\ddot{o}b}},r_{3}\in(0,1)$ such that

$\left(V(g_{1})r_{1}^{L_{0}}V(\tilde{g}_{1})\right)\,\left(V(g_{2})r_{2}^{L_{0}}V(\tilde{g}_{2})\right)=V(g_{3})r_{3}^{L_{0}}V(\tilde{g}_{3})$

in the proper (not only projective) sense.

###### Proof.

Consider the well-known conformal net usually referred as the
$\mathrm{U}(1)$-current net [5]. One may restrict its projective unitary
representation $U$ of ${{\rm Diff}^{+}(S^{1})}$ to the Möbius group and
arrange its phase factors in such a way that the vacuum $\Omega$ will be an
invariant vector (see Section 2). In this way we get a positive energy,
strongly continuous, unitary representation $V$ of ${\mathrm{M\ddot{o}b}}$ in
which all such irreducible representations (i.e. every possible integer
highest weight) appear: this is evident because for every $n\geq 1$, the
dimension of $(n+1)^{\rm th}$ energy space is strictly larger than the
dimension of the $n^{\rm th}$ one. Thus, if we can show the statement for our
particular representation $V$, we have proved it for all positive energy
representations of ${\mathrm{M\ddot{o}b}}$. By applying Theorem 4.1 to $U$ of
the $\mathrm{U}(1)$-current net, we obtain that for every
$g_{1},g_{2},\tilde{g}_{1},\tilde{g}_{2}\in{\mathrm{M\ddot{o}b}}$ and
$r_{1},r_{2}\in(0,1]$ there must exist some
$\gamma_{3},\tilde{\gamma}_{3}\in{\rm Diff}^{+}_{a}(S^{1})$ and an
$r_{3}\in(0,1]$ such that

$\left(U(g_{1})r_{1}^{L_{0}}U(\tilde{g}_{1})\right)\,\left(U(g_{2})r_{2}^{L_{0}}U(\tilde{g}_{2})\right)=U(\gamma_{3})r_{3}^{L_{0}}U(\tilde{\gamma}_{3})$

in the projective sense. If $r_{1}=r_{2}=1$, then of course $g_{3}$ and
$\tilde{g_{3}}$ can be chosen to be in ${\mathrm{M\ddot{o}b}}$. On the other
hand, if $r_{1}r_{2}<1$, then they must be in ${\mathrm{M\ddot{o}b}}$. 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 $\Omega$ — i.e.
considering the scalar product $\langle\Omega,\cdot\,\Omega\rangle$ 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 ${\mathbb{C}}\Omega$
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 $\gamma_{3},\tilde{\gamma_{3}}\in{\mathrm{M\ddot{o}b}}$.

Considering the relevant unitary operators rather than projective ones, we
therefore have that for every
$g_{1},g_{2},\tilde{g}_{1},\tilde{g}_{2}\in{\mathrm{M\ddot{o}b}}$ and
$r_{1},r_{2}\in(0,1]$ there must exist some
$g_{3},\tilde{g}_{3}\in{\mathrm{M\ddot{o}b}}$ and an $r_{3}\in(0,1]$ such that
$\left(V(\gamma_{1})r_{1}^{L_{0}}V(\tilde{\gamma}_{1})\right)\,\left(V(\gamma_{2})r_{2}^{L_{0}}V(\tilde{\gamma}_{2})\right)$
and $V(\gamma_{3})r_{3}^{L_{0}}V(\tilde{\gamma}_{3})$ are proportional to each
other. The proof is then finished by evaluating both sides on $\Omega$ and
concluding that this proportion must be $1$. ∎

######  Corollary 4.4.

Let $V$ be a positive energy, strongly continuous, unitary representation of
${\mathrm{M\ddot{o}b}}$ with associated conformal Hamiltonian $L_{0}$. There
exists some $r,r_{1},r_{2}\in(0,1)$ and
$g,g_{1},g_{2}\in{\mathrm{M\ddot{o}b}}$, $g\neq\mathrm{id}$, such that

$r^{L_{0}}=r_{1}^{H_{1}}r_{2}^{H_{2}}V(g)$

in the proper sense, where $H_{j}={\hbox{\rm Ad\,}}U(g_{k})(L_{0})$ ($k=1,2$).

###### Proof.

We choose two elements $\tilde{g}_{1},\tilde{g}_{2}\in{\mathrm{M\ddot{o}b}}$
such that $\tilde{H}_{k}={\hbox{\rm Ad\,}}V(\tilde{g}_{k})(L_{0})$, and
$\tilde{H}_{1}$ and $\tilde{H}_{2}$ do not strongly commute: such choices are
actually abundant, since $L_{0}$ is maximally abelian in the Lie algebra. Then
there must exist some $r_{1},r_{2}\in(0,1)$ such that $r_{1}^{\tilde{H}_{1}}$
and $r_{2}^{\tilde{H}_{2}}$ do not commute (otherwise their generators would
strongly commute by analytic continuation).

Now we apply Corollary 4.3 to
$r_{1}^{\tilde{H}_{1}}r_{2}^{\tilde{H}_{2}}=V(\tilde{g}_{1})r_{1}^{L_{0}}V(\tilde{g}_{1})^{*}V(\tilde{g}_{2})r_{2}{L_{0}}V(\tilde{g}_{2})^{*}$
to obtain $g_{3},\tilde{g}_{3}\in{\mathrm{M\ddot{o}b}}$ and $r\in(0,1)$ such
that

$V(\tilde{g}_{1})r_{1}^{L_{0}}V(\tilde{g}_{1})^{*}V(\tilde{g}_{2})r_{2}^{L_{0}}V(\tilde{g}_{2})^{*}=V(g_{3})r^{L_{0}}V(\tilde{g}_{3})^{*},$

in the proper sense, or equivalently,

$r^{L_{0}}=V(g_{3}^{-1}\tilde{g}_{1})r_{1}^{L_{0}}V(g_{3}^{-1}\tilde{g}_{1})^{*}V(g_{3}^{-1}\tilde{g}_{2})r_{2}^{L_{0}}V(g_{3}^{-1}\tilde{g}_{2})^{*}V(g_{3}\tilde{g}_{3}).$

By defining $g_{k}=g_{3}^{-1}\tilde{g}_{k}$, hence accordingly
$H_{k}:={\hbox{\rm Ad\,}}V(g_{3}^{-1}\tilde{g}_{k})(L_{0})$ and
$g:=g_{3}\tilde{g}_{3}$, we obtain the desired equality. To check that
$g\neq\mathrm{id}$, note that by our choice of $\tilde{H}_{k}$,
$r_{1}^{H_{1}}$ and $r_{2}^{H_{2}}$ do not commute as well. Yet, in the
equality

$r^{L_{0}}=r_{1}^{H_{1}}r_{2}^{H_{2}}V(g),$

the left-hand side is self-adjoint, while if $g=\mathrm{id}$, the right-hand
side would not be self-adjoint. Therefore, $g\neq\mathrm{id}$. ∎

######  Proposition 4.5.

Let ${\mathcal{A}}$ be a conformal net, $U$ be the associated projective
unitary representation of ${{\rm Diff}^{+}(S^{1})}$, and $L_{0}$ the conformal
Hamiltonian. For some $r\in(0,1)$, there exists a Möbius transformation
$g\neq{\rm id}$, such that for any $I_{c},I_{d}\in{\mathcal{I}}$ be two open
proper arcs such that $I_{c}\cup I_{d}=S^{1}$ we have two bounded operators
$C\in{\mathcal{A}}_{U}(I_{c})$ and $D\in{\mathcal{A}}_{U}(I_{d})$ such that we
have the decomposition

$r^{L_{0}}=CDU(g)$

in the proper sense.

###### Proof.

We apply Corollary 4.4 to obtain $r,r_{1},r_{2}\in(0,1)$,
$g\in{\mathrm{M\ddot{o}b}},g\neq\mathrm{id}$ and $H_{1},H_{2}$ such that

$r^{L_{0}}=r_{1}^{H_{1}}r_{2}^{H_{2}}U(g).$

Then we apply Corollary 3.2 to $H_{k}$ with the intervals $K_{k,c},K_{k,d}$
such that $K_{k,c}\subset I_{c},K_{k,d}\subset I_{d}$ and $K_{1,d}\cap
K_{2,c}=\emptyset$ (see Figure 3), to obtain operators $C_{k},D_{k}$ such that
$r_{k}^{H_{k}}=C_{k}D_{k}$. By the localization, $C_{2}$ and $D_{1}$ commute.

Hence it holds that
$r^{L_{0}}=r_{1}^{H_{1}}r_{2}^{H_{2}}U(g)=C_{1}D_{1}C_{2}D_{2}U(g)=C_{1}C_{2}D_{1}D_{2}U(g)$,
and $C:=C_{1}C_{2}$ is localized in $K_{1,c}\cup K_{2,c}\subset I_{c}$, while
$D:=D_{1}D_{2}$ is localized in $K_{1,d}\cup K_{2,d}\subset I_{d}$, as
desired.

$I_{d}$$I_{c}$$K_{1,c}$$K_{1,d}$$K_{2,c}$$K_{2,d}$ Figure 3: Intervals
$I_{c},I_{d},K_{1,c},K_{1,d},K_{2,c},K_{2,d}$.

∎

##  5 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 $({\mathcal{A}},U,\Omega)$
be a conformal net, and assume $I_{a},I_{b}\in{\mathcal{I}}$ are two open
proper arcs separated by a positive distance.

Consider the $*$-algebra
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$ generated by the
commuting factors ${\mathcal{A}}(I_{a})$ and ${\mathcal{A}}(I_{b})$. We shall
now introduce a family $\\{\phi_{z}\\}$ of functionals on this algebra indexed
by a complex number $z$, $|z|\leq 1$. For a generic element
$X\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$,

$X=\sum_{k=1}^{n}A_{k}B_{k}\qquad(n\in{\mathbb{N}},\;A_{k}\in{\mathcal{A}}(I_{a}),\;B_{k}\in{\mathcal{A}}(I_{b}))$
(5)

and a complex number $z$ in the closed unit disk
${\mathcal{D}}_{1}=\\{z\in{\mathbb{C}}:|z|<1\\}$, let

$\phi_{z}(X)=\sum_{k=1}^{n}\langle\Omega,A_{k}z^{L_{0}}B_{k}\Omega\rangle.$

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 $X$. Indeed,
since ${\mathcal{A}}(I_{a})$ and ${\mathcal{A}}(I_{b})$ are commuting factors,
there is a natural isomorphism between the algebraic tensor product
${\mathcal{A}}(I_{a})\odot{\mathcal{A}}(I_{b})$ and
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$, see [38,
Proposition IV.4.20]. In particular, the bilinear form
${\mathcal{A}}(I_{a})\times{\mathcal{A}}(I_{b})\ni(A,B)\mapsto\langle\Omega,Az^{L_{0}}B\Omega\rangle\in{\mathbb{C}}$
extends to a unique linear functional $\phi_{z}$ on
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$.

Note that the expression $z^{L_{0}}$ is indeed a well-defined bounded operator
for every $z\in{\mathcal{D}}_{1}$ (for $z=0$, we define it by continuity in
the strong operator topology, hence to be the projection $P_{0}$ onto
${\mathbb{C}}\Omega$): this is because ${\rm Sp}(L_{0})\subset{\mathbb{N}}$.
That is, we are using not just the positivity of $L_{0}$, but also that
elements of its spectrum are all integers (e.g. $z^{\frac{1}{2}}=\sqrt{z}$
would be ambiguous).

For every $X\in{\mathcal{A}}(I_{1})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{3})$,
the map $z\mapsto\phi_{z}(X)$ is analytic in ${\mathcal{D}}_{1}$. In fact,
denoting by $P_{m}$ the spectral projection of $L_{0}$ associated to the
eigenvalue $m$, we have the power series decomposition of $\phi_{z}(X)$

$\phi_{z}(X)=\sum_{k=1}^{n}\sum_{m=0}^{\infty}\langle\Omega,A_{k}P_{m}B_{k}\Omega\rangle
z^{m}.$

Since $P_{0}=\langle\Omega,\,\cdot\,\rangle\,\Omega$ is the one-dimensional
projection on the vacuum vector, we have that

$\phi_{0}(X)=\sum_{k=1}^{n}\omega(A_{k})\omega(B_{k}),$

i.e. $\phi_{0}$ is the product vacuum state, whereas $\phi_{1}=\omega$. 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
$\phi_{z}$ remains normal. In particular, it would be desirable to obtain
estimates on $\|\phi_{1}-\phi_{0}\|$.

The idea of considering $\phi_{z}$ 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
$z\mapsto\phi_{z}(AB)$ to obtain a bound on $|\phi_{1}(AB)-\phi_{0}(AB)|$ for
a pair of elements $A\in{\mathcal{A}}(I_{1})$ and $B\in{\mathcal{A}}(I_{b})$
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 $\sum_{k}\|A_{k}\|\,\|B_{k}\|$, rather than in terms of the
norm of $X$. Hence their method does not give a useful estimate on
$\|\phi_{1}-\phi_{0}\|$. 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 $\phi_{z}$ depends norm-continuously on $z$ and is
normal (i.e. extends to a normal linear functional of the von Neumann algebra
${\mathcal{A}}(I_{a})\vee{\mathcal{A}}(I_{b})$) 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 $\phi_{0}$:

######  Lemma 5.1.

Let $r_{0}\in(0,1)$ be a fixed radius and suppose that $\phi_{z}$ is normal
whenever $|z|=r_{0}$ and that on the circle with radius $r_{0}$,
$r_{0}S^{1}\ni z\mapsto\phi_{z}$ is norm-continuous. Then $\phi_{0}$ 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 ${\mathcal{M}}$ 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
${\mathcal{M}}$, but by the Kaplansky density theorem, the norm of a normal
functional on ${\mathcal{A}}(I_{a})\vee{\mathcal{A}}(I_{b})$ is equal to the
norm of its restriction to
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$. Therefore, in
the following we do not distinguish them.

Thus one has — e.g. by considering Riemann-sums — that if
$\varphi:[s_{1},s_{2}]\ni t\mapsto\varphi_{t}\in{\mathcal{M}}_{*}$ is a norm-
continuous family of normal linear functionals, then
$\varphi(\cdot)=\int_{s_{1}}^{s_{2}}\varphi_{t}(\cdot)dt$ is also a well-
defined normal functional on ${\mathcal{M}}$.

Since ${\mathcal{D}}_{1}\ni z\mapsto\phi_{z}(X)$ is analytic, by the Cauchy
integral formula we have

$\phi_{0}(X)=\frac{1}{2\pi
i}\oint_{r_{0}S^{1}}\phi_{r_{0}e^{i\theta}}(X)\;\frac{dz}{z}$

for every $X\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$.

∎

Let us now discuss how the decompositions $r^{L_{0}}$ help us out in different
regions of ${\mathcal{D}}_{1}$. Let $I_{c}=I_{a}^{\prime}$, 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
$I_{c}\cup I_{d}=S^{1}$ and we can consider the decomposition of $r^{L_{0}}$
given by Proposition 3.1 with
$\\{C_{r}\\}_{r\in(0,1)}\subset{\mathcal{A}}(I_{c})$ and
$\\{D_{r}\\}_{r\in(0,1)}\subset{\mathcal{A}}(I_{d})$. Let us denote by
$R_{\theta}$ the rotation by $\theta$. Then, as long as $z=re^{i\theta}$ is
such that $r\in(0,1)$ and the angle $\theta$ satisfies the condition

$\overline{I_{d}}\cap R_{\theta}(\overline{I_{b}})=\emptyset,$ (6)

we have that the action of $\rho_{\theta}\equiv{\hbox{\rm Ad\,}}e^{i\theta
L_{0}}$ leaves ${\mathcal{A}}(I_{b})$ inside ${\mathcal{A}}(I_{d})^{\prime}$
and thus for an
$X\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$ with
decomposition (5), we can use locality to rewrite $\phi_{z}(X)$ as

$\displaystyle\phi_{z}(X)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\left\langle\Omega,A_{k}r^{L_{0}}e^{i\theta
L_{0}}B_{k}\Omega\right\rangle=\sum_{k=1}^{n}\left\langle\Omega,A_{k}r^{L_{0}}e^{i\theta
L_{0}}B_{k}e^{-i\theta L_{0}}\Omega\right\rangle$ (7) $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}\left\langle\Omega,A_{k}C_{r}D_{r}\rho_{\theta}(B_{k})\Omega\right\rangle=\left\langle
C_{r}^{*}\Omega,\left(\sum_{k=1}^{n}A_{k}\rho_{\theta}(B_{k})\right)D_{r}\Omega\right\rangle.$

Let $(\theta_{-},\theta_{+})$ be the largest open interval of angles
satisfying our condition (6) and containing $0$, i.e., $\theta_{\pm}$ is the
smallest positive / largest negative angle for which
$R_{\theta_{\pm}}(\overline{I_{b}})$ intersects $\overline{I_{d}}$. We can
obviously find a smooth function $f:S^{1}\to{\mathbb{R}}$ 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 $f(e^{i\theta})\frac{\rm d}{{\rm
d}\theta}$, it gives rise to a one parameter group of diffeomorphisms

${\mathbb{R}}\ni\theta\mapsto\gamma_{\theta}\equiv\,{\rm Exp}(\theta f)$

such that $\gamma_{\theta}$ is “localized” in $I_{a}^{\prime}$, but if
$\theta_{-}<\theta<\theta_{+}$, then the action of $\gamma_{\theta}$ on
$I_{b}$ coincides with that of the rotation by $\theta$. Thus, $e^{i\theta
T(f)}$ commutes with elements of ${\mathcal{A}}(I_{a})$ but for
$\theta\in(\theta_{-},\theta_{+})$, its adjoint action on
${\mathcal{A}}(I_{b})$ coincides with the action of $\rho_{\theta}$ and hence
we can write

$\displaystyle\sum_{k=1}^{n}A_{k}\rho_{\theta}(B_{k})$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{n}A_{k}e^{i\theta T(f)}B_{k}e^{-i\theta
T(f)}=e^{i\theta T(f)}\sum_{k=1}^{n}A_{k}B_{k}e^{-i\theta T(f)}=$
$\displaystyle=$ $\displaystyle e^{i\theta T(f)}Xe^{-i\theta T(f)}.$

Putting this back in (7), we get that for $\theta\in(\theta_{-},\theta_{+})$,

$\phi_{re^{i\theta}}(X)=\langle\eta_{\theta},X\zeta_{\theta}\rangle$

where the vectors $\eta_{\theta}=e^{-i\theta T(f)}C_{r}^{*}\Omega$ and
$\zeta_{\theta}=e^{-i\theta T(f)}D_{r}\Omega$.

######  Corollary 5.2.

$\\{\phi_{z}\\}$ is a norm-continuous family of normal functionals in the
region $\\{re^{i\theta}|r\in(0,1)\;\;{\rm and}\;\;\overline{I_{d}}\cap
R_{\theta}(\overline{I_{b}})=\emptyset\\}.$

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

$\|\phi_{re^{i\theta}}\|\leq\|C_{r}^{*}\Omega\|\;\|D_{r}\Omega\|\leq\frac{1}{r^{2q}}.$

Unfortunately, though this estimate is nicely uniform in $\theta$, it “blows
up” at $r\to 0$ and hence in itself it does not show that $\phi_{r}$ converges
to $\phi_{0}$ in norm as $r\to 0$.

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
${r_{0}}\in(0,1)$ such that the decomposition in Proposition 4.5 holds; that
is, we have two bounded elements $C\in{\mathcal{A}}(I_{c})$ and
$D\in{\mathcal{A}}(I_{d})$ and $g\neq{\rm id}$ a Möbius transformation such
that $r_{0}^{L_{0}}=CDU(g)$ (recall that this is valid in the proper sense,
namely one can fix the phase of $U(g)$ for $g\in{\mathrm{M\ddot{o}b}}$
unambiguously). Then, repeating the steps we did before with our previous
decomposition and setting instead of (7), this time we get

$\phi_{r_{0}e^{i\theta}}(X)=\sum_{k=1}^{n}\left\langle\Omega,A_{k}CDU(g)e^{i\theta
L_{0}}B_{k}\Omega\right\rangle=\left\langle
C^{*}\Omega,\left(\sum_{k=1}^{n}A_{k}\rho_{\theta}(B_{k})\right)D\Omega\right\rangle$

whenever the disjointness condition

$\overline{I_{d}}\cap g\circ R_{\theta}(\overline{I_{b}})=\emptyset$

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.

$\\{\phi_{z}\\}$ is a norm-continuous family of normal functionals in the
region $\\{r_{0}e^{i\theta}|\overline{I_{d}}\cap g\circ
R_{\theta}(\overline{I_{b}})=\emptyset\\}.$

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
$I_{d}\supset\overline{I_{a}}$ might lead to nowhere, it is enough for us to
show that there is a “right” choice.

######  Theorem 5.4.

A conformal net $({\mathcal{A}},U,\Omega)$ on the circle automatically has the
split property.

###### Proof.

By conformal covariance, we may assume that $I_{a},I_{b}$ and even the
enlargement $I_{d}\supset\overline{I_{a}}$ 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 ${\mathcal{D}}_{1}$ minus a slightly
enlarged version of the half-line $\\{te^{i\alpha}|t\geq 0\\}$ where $\alpha$
is the angle for which $R_{\alpha}(b)=a$. 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 $r_{0}e^{i\tilde{\alpha}}$, where
$\tilde{\alpha}$ is the angle for which $g\circ R_{\tilde{\alpha}}(b)=a$,
which is of course equivalent to saying that
$R_{\tilde{\alpha}}(b)=g^{-1}(a)$. Since $g$ is a certain fixed, non trivial
Möbius transformation, we might even assume that our choice of $a$ is such
that $g^{-1}(a)\neq a$. Then $\alpha\neq\tilde{\alpha}$ and the union of the
two regions covers the circle $r_{0}S^{1}$.

Lemma 5.1 shows that $\phi_{0}$ is a normal linear functional on
${\mathcal{A}}(I_{1})\vee{\mathcal{A}}(I_{3})$. Now our claim is concluded by
Proposition 2.2 and a technical Lemma 5.5 below, by noting that

  * •

${\mathcal{A}}(I_{a})\vee{\mathcal{A}}(I_{b})$ is a factor (Section 2,
Factoriality of two-interval algebras)

  * •

The restrictions of $\phi_{0}$ to ${\mathcal{A}}(I_{a})$ and
${\mathcal{A}}(I_{b})$ are equal to the vacuum state, hence faithful.

∎

######  Lemma 5.5.

$\phi_{0}$ is a positive normal functional on
${\mathcal{A}}(I_{a})\vee{\mathcal{A}}(I_{b})$.

###### Proof.

We first consider $\phi_{0}$ on
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$. By [38,
Proposition 4.20], the map $\tau:\sum_{k}x_{k}y_{k}\mapsto\sum x_{k}\otimes
y_{k}$ is well-defined and is a $*$-isomorphism from
${\mathcal{A}}(I_{1})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{3})$ onto
${\mathcal{A}}(I_{a})\odot{\mathcal{A}}(I_{b})$. Now the linear functional
$\phi_{0}(\sum_{k}x_{k}y_{k})$ translates into
${\mathcal{A}}(I_{a})\odot{\mathcal{A}}(I_{b})$ as
$\langle\Omega\otimes\Omega,\cdot\;\Omega\otimes\Omega\rangle$. Namely,
$\phi_{0}=(\omega\otimes\omega)\circ\tau^{-1}$. Now, $\omega\otimes\omega$ is
clearly positive, and $\tau^{-1}(x^{*}x)=\tau^{-1}(x)^{*}\tau^{-1}(x)$,
therefore, $\phi_{0}(x^{*}x)\geq 0$ for any
$x\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$.

We claim that, also on
${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$, $\phi_{0}$ is
positive 666${\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$ is
not a $C^{*}$-algebra, in particular, a positive element
$a\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$ in the sense
of ${\mathcal{B}}({\mathcal{H}})$ is not necessarily of the form $x^{*}x$,
where $x\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$..
Indeed, take a positive element
$a\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$. The
function $f(x)=x^{\frac{1}{2}}$, $x\in\left[0,\|a\|\right]$ can be arbitrarily
approximated by polynomials $f_{n}$ with real coefficients, uniformly on
$\left[0,\|a\|\right]$ and $f_{n}(a)^{2}$ tends to $a$ in norm. We saw that
$\phi_{0}$ is a normal linear functionals, hence it is in particular
continuous in norm. Since $\phi_{0}(x^{*}x)\geq 0$ for
$x\in{\mathcal{A}}(I_{a})\vee_{\mathrm{alg}}{\mathcal{A}}(I_{b})$ then
$\phi_{0}(f_{n}(a)^{2})\geq 0$, hence $\phi_{0}(a)\geq 0$ by norm continuity
of $\phi_{0}$.

Now, by the Kaplansky density theorem and the normality of $\phi_{0}$,
$\phi_{0}$ is a positive functional. ∎

##  6 A non-split conformal net in two-dimensions

Conformal nets on $S^{1}$ constitute the building blocks of two-dimensional
conformal nets. Let us recall the relevant definitions. A locally normal,
positive energy, Möbius covariant representation $\rho$ of a conformal net
$({\mathcal{A}},U,\Omega)$ on $S^{1}$ is a family of normal representations
$\\{\rho_{I}:I\in{\mathcal{I}}\\}$ of the von Neumann algebras
$\\{{\mathcal{A}}(I):I\in{\mathcal{I}}\\}$ on a fixed Hilbert space
${\mathcal{H}}_{\rho}$ and a unitary, positive energy unitary representation
$U^{\rho}$ on ${\mathcal{H}}_{\rho}$ of the universal covering group of the
Möbius group $\widetilde{{\mathrm{M\ddot{o}b}}}$:

  1. 1.

Compatibility: if $I_{1},I_{2}\in{\mathcal{I}}$ and $I_{1}\subset I_{2}$ then
$\rho_{I_{2}}|_{{\mathcal{A}}(I_{1})}=\rho_{I_{1}}$

  2. 2.

Covariance: ${\hbox{\rm
Ad\,}}U^{\rho}(g)\circ\rho_{I}=\rho_{gI}\circ{\hbox{\rm Ad\,}}U(g)$,
$g\in\widetilde{\mathrm{M\ddot{o}b}}$

A representation $\rho$ is irreducible if
$\bigvee_{I\in{\mathcal{I}}}\rho({\mathcal{A}}(I))={\mathcal{B}}({\mathcal{H}}_{\rho})$.
The defining representation $\\{\mathrm{id}_{{\mathcal{A}}(I)}\\}$ is called
the vacuum representation.

A representation of a conformal net $\rho$ is said to be localizable in
$I_{0}$ if $\rho_{I_{0}^{\prime}}\simeq\mathrm{id}$, where $\simeq$ means
unitary equivalence. The unitary equivalence class of $\rho$ defines a
superselection sector, also called a DHR (Doplicher-Haag-Roberts) sector [14].
By Haag duality we have that $\rho({\mathcal{A}}(I))\subset{\mathcal{A}}(I)$
if $I_{0}\subset I$. Thus we can always choose, within the sector of $\rho$, a
representation $\rho_{0}$ on the defining Hilbert space ${\mathcal{H}}$ such
that $\rho_{0,I_{0}}$ is an endomorphism of ${\mathcal{A}}(I_{0})$. If each
$\rho_{I}$ is an automorphism of ${\mathcal{A}}(I)$, we call $\rho$ an
automorphism of $({\mathcal{A}},U,\Omega)$. Automorphisms can be composed in a
natural way.

Let $({\mathcal{A}},U,\Omega)$ be the $\mathrm{U}(1)$-current net [5]. The
main ingredients are (see [40] for a more detailed review):

  * •

The Weyl operators $W(f)$ parametrized by real smooth functions $f$ on $S^{1}$
which satisfy the commutation relations
$W(f)W(g)=e^{\frac{i}{2}\mathrm{Im}(f,g)}W(f+g)$, where
$(f,g):=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,f^{\prime}(e^{i\theta})g(e^{i\theta})$.

  * •

There is a distinguished realization (“vacuum representation”) of the Weyl
operators (which we denote again by $W(f)$) with a unitary positive energy
representation of ${\mathrm{M\ddot{o}b}}$ which extends to a projective
unitary representation $U$ of ${{\rm Diff}^{+}(S^{1})}$, and the vacuum vector
$\Omega$ such that ${\hbox{\rm Ad\,}}U(\gamma)(W(f))=W(f\circ\gamma)$ and
$U(g)\Omega=\Omega$ if $g\in{\mathrm{M\ddot{o}b}}$.

  * •

The $\mathrm{U}(1)$-current net ${\mathcal{A}}(I):=\\{W(f):{\rm
supp\,}f\subset I\\}^{\prime\prime}$.

  * •

Irreducible sectors parametrized by $q\in{\mathbb{R}}$: we fix a real smooth
function $\varphi$ such that
$\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,\varphi(e^{i\theta})=1$. The map
$W(f)\to e^{iq\varphi(f)}W(f)$ extends to an automorphism $\sigma_{q,I}$ of
${\mathcal{A}}(I)$, where ${\rm supp\,}f\subset I$ and
$\varphi(f)=\frac{1}{2\pi}\int_{0}^{2\pi}d\theta\,f(e^{i\theta})\varphi(e^{i\theta})$.
We call this automorphism of the net $\sigma_{q}$. Different functions
$\varphi$ with the conditions above with the same $q$ give the equivalent
sectors, while sectors with different $q$ are inequivalent. It holds that
$\sigma_{q}\circ\sigma_{q^{\prime}}=\sigma_{q+q^{\prime}}.$

  * •

Each irreducible sector is covariant: the projective representation
$\gamma\mapsto U_{q}(\gamma):=\sigma_{q}(U(\gamma))$ of local diffeomorphisms
extends to $\widetilde{{{\rm Diff}^{+}(S^{1})}}$, hence makes the automorphism
$\sigma_{q}$ covariant [10, Proposition 2] (in an irreducible representation
$\sigma_{q}$, the choice of $U_{q}(\gamma)$ is unique up to a scalar [10,
Remark after Proposition 2]): ${\hbox{\rm
Ad\,}}U_{q}(\gamma)(\sigma_{q}(x))=\sigma_{q}({\hbox{\rm Ad\,}}U(\gamma)(x))$.
Furthermore, we can fix the phase of $U_{q}(\gamma)$ and consider them as
unitary operators (see [17, Proposition 5.1], where the phase does not depend
on $h$, hence one can take the direct sum of multiplier representations
(projective representations with fixed phases)). In this case, it holds that
$U_{q}(\gamma_{1})U_{q}(\gamma_{2})=c(\gamma_{1},\gamma_{2})U(\gamma_{1},\gamma_{2})$
where $c(\gamma_{1},\gamma_{2})\in{\mathbb{C}}{\mathbbm{1}}$.
$c(\gamma_{1},\gamma_{2})$ can be chosen without dependence on $q$, and
continuous in a neighborhood of the unit element. This projective
representation (restricted to $\widetilde{{\mathrm{M\ddot{o}b}}}$) has
positive energy [9].

  * •

For two equivalent automorphisms $\rho,\tilde{\rho}$ localized in
$I,\tilde{I}$, respectively, an operator which intertwines them is called a
charge transporter. In the present case, as both $\rho,\tilde{\rho}$ are
irreducible, such a charge transporter is unique up to a scalar. A charge
transporter acts trivially on ${\mathcal{A}}((I\cup\tilde{I})^{\prime})$,
hence belongs to ${\mathcal{A}}((I\cup\tilde{I})^{\prime})^{\prime}$. In
particular, it can be considered as an element in a local algebra containing
$I$ and $\tilde{I}$.

  * •

The operator $z_{q}(\gamma):=U(\gamma)U_{q}(\gamma)^{*}$ is a charge
transporter between $\sigma_{q}$ and
$\alpha_{\gamma}\sigma_{q}\alpha_{\gamma^{-1}}$.

  * •

For a given pair of automorphisms $\rho_{1},\rho_{2}$, one defines the
braiding $\epsilon_{\rho_{1},\rho_{2}}$: one chooses equivalent automorphisms
$\tilde{\rho}_{1},\tilde{\rho}_{2}$ localized in
$\tilde{I}_{1},\tilde{I}_{2}$, respectively, such that
$\tilde{I}_{1}\cap\tilde{I}_{2}=\emptyset$ and charge transporters
$V_{1},V_{2}$ between $\rho_{1}$ and $\tilde{\rho}_{1}$, and $\rho_{2}$ and
$\tilde{\rho}_{2}$, respectively. Define
$\epsilon_{\rho_{1},\rho_{2}}^{\pm}:=\rho_{2}(V_{1}^{*})V_{2}^{*}V_{1}\rho_{1}(V_{2}^{*})$,
where $+$ or $-$ depends on the choice whether $\tilde{I}_{1}$ is on the
left/right of $\tilde{I}_{2}$ (which results from the choice of localization
of the charge transporter above), but $\epsilon_{\rho_{1},\rho_{2}}^{\pm}$ do
not depend on the choice of $\tilde{\rho}_{k},V_{k}$ under such a
configuration.

  * •

For our concrete automorphisms $\sigma_{q},\sigma_{q^{\prime}}$ on the
$\mathrm{U}(1)$-current net, one can take the charge transporters
$V_{q},V_{q^{\prime}}$ as Weyl operators and finds that the braiding satisfies
$\epsilon_{\sigma_{q},\sigma_{q^{\prime}}}^{\pm}\in{\mathbb{C}}{\mathbbm{1}}$,
$\overline{\epsilon_{\sigma_{q},\sigma_{q^{\prime}}}^{+}}=\epsilon_{\sigma_{q},\sigma_{q^{\prime}}}^{-}$.

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
$z_{q}(\gamma)\sigma_{q}(z_{q^{\prime}}(\gamma))=z_{q+q^{\prime}}(\gamma)$.

###### Proof.

First recall that $z_{q}(\gamma)$ is an intertwiner between $\sigma_{q}$ and
$\alpha_{\gamma}\sigma_{q}\alpha_{\gamma^{-1}}$, hence the product
$z_{q}(\gamma)\sigma_{q}(z_{q^{\prime}}(\gamma))$ is an intertwiner between
$\sigma_{q}\sigma_{q^{\prime}}=\sigma_{q+q^{\prime}}$ and
$\alpha_{\gamma}\sigma_{q}\alpha_{\gamma^{-1}}\circ\alpha_{\gamma}\sigma_{q^{\prime}}\alpha_{\gamma^{-1}}=\alpha_{\gamma}\sigma_{q+q^{\prime}}\alpha_{\gamma^{-1}}$.
$z_{q+q^{\prime}}(\gamma)$ also intertwines $\sigma_{q+q^{\prime}}$ and
$\alpha_{\gamma}\sigma_{q+q^{\prime}}\alpha_{\gamma^{-1}}$. As they are
automorphisms, hence irreducible, the difference between
$z_{q}(\gamma)\sigma_{q}(z_{q^{\prime}}(\gamma))$ and
$z_{q+q^{\prime}}(\gamma)$ must be a scalar.

Next we show that
$U^{\prime}_{q+q^{\prime}}(\gamma):=(z_{q}(\gamma)\sigma_{q}(z_{q^{\prime}}(\gamma)))^{*}U(\gamma)$
is a multiplier representation of $\widetilde{{{\rm Diff}^{+}(S^{1})}}$ such
that
$U_{q+q^{\prime}}^{\prime}(\gamma_{1})U_{q+q^{\prime}}^{\prime}(\gamma_{2})=c(\gamma_{1},\gamma_{2})U_{q+q^{\prime}}^{\prime}(\gamma_{1}\gamma_{2})$,
namely it has the same $2$-cocycle $c$ as $U_{q+q^{\prime}}$. Indeed,

$\displaystyle
U^{\prime}_{q+q^{\prime}}(\gamma_{1})U^{\prime}_{q+q^{\prime}}(\gamma_{2})$
$\displaystyle=(z_{q}(\gamma_{1})\sigma_{q}(z_{q^{\prime}}(\gamma_{1})))^{*}U(\gamma_{1})(z_{q}(\gamma_{2})\sigma_{q}(z_{q^{\prime}}(\gamma_{2})))^{*}U(\gamma_{2})$
$\displaystyle=\sigma_{q}(z_{q^{\prime}}(\gamma_{1}))^{*}U_{q}(\gamma_{1})\sigma_{q}(z_{q^{\prime}}(\gamma_{2}))^{*}U_{q}(\gamma_{2})$
$\displaystyle=\sigma_{q}(z_{q^{\prime}}(\gamma_{1}))^{*}\sigma_{q}(\alpha_{\gamma_{1}}(z_{q^{\prime}}(\gamma_{2})))^{*}\cdot
c(\gamma_{1},\gamma_{2})U_{q}(\gamma_{1}\gamma_{2})$
$\displaystyle=\sigma_{q}(z_{q^{\prime}}(\gamma_{1})^{*}\alpha_{\gamma_{1}}(z_{q^{\prime}}(\gamma_{2}))^{*})\cdot
c(\gamma_{1},\gamma_{2})U_{q}(\gamma_{1}\gamma_{2})$
$\displaystyle=\sigma_{q}\left(U_{q^{\prime}}(\gamma_{1})U(\gamma_{1})^{*}U(\gamma_{1})U_{q^{\prime}}(\gamma_{2})U(\gamma_{2})^{*}U(\gamma_{1})^{*}\right)\cdot
c(\gamma_{1},\gamma_{2})U_{q}(\gamma_{1}\gamma_{2})$
$\displaystyle=\sigma_{q}(U_{q^{\prime}}(\gamma_{1}\gamma_{2})U(\gamma_{1}\gamma_{2})^{*})\cdot
c(\gamma_{1},\gamma_{2})z_{q}(\gamma_{1}\gamma_{2})^{*}U(\gamma_{1}\gamma_{2})$
$\displaystyle=c(\gamma_{1},\gamma_{2})U_{q+q^{\prime}}^{\prime}(\gamma_{1}\gamma_{2}),$

where in the 3rd and 6th equalities we used that $U$ and $U_{q}$ share the
same $2$-cocycle $c$.

Now let us define
$U^{\prime\prime}(\gamma):=U_{q}^{\prime}(\gamma)^{*}U_{q}(\gamma)$. As the
difference between $U_{q}^{\prime}(\gamma)$ and $U_{q}(\gamma)$ is just a
phase and they share the same $2$-cocycle $c$, it is easy to show that
$U^{\prime\prime}$ is a ${\mathbb{C}}$-valued true (with trivial multiplier)
representation of $\widetilde{{{\rm Diff}^{+}(S^{1})}}$. It is well-known that
then $U^{\prime\prime}$ must be trivial,
$U^{\prime\prime}(\gamma)={\mathbbm{1}}$. From this the claim immediately
follows. ∎

Let $G$ be the quotient of
$\widetilde{\mathrm{M\ddot{o}b}}\times\widetilde{\mathrm{M\ddot{o}b}}$ by the
normal subgroup generated by $(R_{2\pi},R_{-2\pi})$, where
${\mathrm{M\ddot{o}b}}$ naturally includes the universal covering
${\mathbb{R}}$ of the rotation subgroup $S^{1}$ and $R_{2\pi},R_{-2\pi}$ are
the elements corresponding to $2\pi,-2\pi$ rotations, respectively. We call
${\mathbb{R}}\times S^{1}$ the Einstein cylinder $\mathcal{E}$, where the
Minkowski space is identified with a maximal square
$(-\pi,\pi)\times(-\pi,\pi)$ (see [1]) 777Here the segments
$(-\pi,\pi)\times\\{0\\}$ and $\\{0\\}\times(-\pi,\pi)$ are identified with
the time and space axis, respectively.. The group $G$ acts naturally on it.
Furthermore, let $\mathrm{Diff}_{0}({\mathbb{R}})$ be the group of
diffeomorphisms of the real line ${\mathbb{R}}$ with compact support. Then
$\mathrm{Diff}_{0}({\mathbb{R}})\times\mathrm{Diff}_{0}({\mathbb{R}})$ acts
naturally on the Minkowski space as the product of two lightrays 888The
lightray decomposition ${\mathbb{R}}^{2}={\mathbb{R}}\times{\mathbb{R}}$ is
not compatible with the above identification of ${\mathbb{R}}$ with
$(-\pi,\pi)\times(-\pi,\pi)$, where the components correspond to the time and
space axis., and its action naturally extends to $\mathcal{E}$ by periodicity.
Let us denote by $\mathrm{Conf}(\mathcal{E})$ the group generated by $G$ and
$\mathrm{Diff}_{0}({\mathbb{R}})\times\mathrm{Diff}_{0}({\mathbb{R}})$. A two-
dimensional conformal net $(\tilde{\mathcal{A}},\tilde{U},\tilde{\Omega})$
consists of a family $\\{\tilde{\mathcal{A}}(O)\\}$ of von Neumann algebras
parametrized by double cones $\\{O\\}$ in the Minkowski space
${\mathbb{R}}^{2}$, a strongly-continuous unitary representation of $G$ which
extends to a projective unitary representation of
$\mathrm{Conf}(\mathcal{E})$, and a vector $\tilde{\Omega}$ such that the
following axioms are satisfied [28, Section 2]:

  * •

Isotony. If $O_{1}\subset O_{2}$, then
$\tilde{\mathcal{A}}(O_{1})\subset\tilde{\mathcal{A}}(O_{2})$.

  * •

Locality. If $O_{1}$ and $O_{2}$ are spacelike separated, then
$\tilde{\mathcal{A}}(O_{1})$ and $\tilde{\mathcal{A}}(O_{2})$ commute.

  * •

Covariance. For a double cone $O$, it holds that ${\hbox{\rm
Ad\,}}\tilde{U}(\gamma)(\tilde{\mathcal{A}}(O))=\tilde{\mathcal{A}}(\gamma O)$
for $\gamma\in\mathcal{V}\subset\mathrm{Conf}(\mathcal{E})$, where
$\mathcal{V}$ is a neighborhood of the unit element of
$\mathrm{Conf}(\mathcal{E})$ such that $\gamma O\subset{\mathbb{R}}^{2}$ for
$\gamma\in\mathcal{V}$. For $x\in\tilde{\mathcal{A}}(O)$ and if
$\gamma\in\mathrm{Diff}_{0}({\mathbb{R}})\times\mathrm{Diff}_{0}({\mathbb{R}})$
acts identically on $O$, then ${\hbox{\rm Ad\,}}\tilde{U}(\gamma)(x)=x$.

  * •

Existence and uniqueness of vacuum. $\tilde{\Omega}$ is a unique (up to a
scalar) invariant vector for $\tilde{U}|_{G}$.

  * •

Cyclicity. $\tilde{\Omega}$ is cyclic for
$\bigvee_{O\subset{\mathbb{R}}^{2}}\tilde{\mathcal{A}}(O)$.

  * •

Positivity of energy. The restriction of $\tilde{U}$ to the group of
translations has the spectrum contained in
$V_{+}:=\\{(x_{0},x_{1}):x_{0}\geq|x_{1}|\\}$.

Now we construct a two-dimensional conformal net as follows, following the
ideas of [16, 33]. Let us fix an interval $I\subset{\mathbb{R}}\subset S^{1}$
and a real smooth function $\varphi$ as above. On the Hilbert space
${\mathcal{H}}_{q}={\mathcal{H}}$, we take the automorphism $\sigma_{q}$ of
the $\mathrm{U}(1)$-current net ${\mathcal{A}}$. The full Hilbert space is the
non-separable direct sum
$\tilde{\mathcal{H}}=\bigoplus_{q\in{\mathbb{R}}}{\mathcal{H}}_{q}\otimes{\mathcal{H}}_{q}$.
The observable net ${\mathcal{A}}\otimes{\mathcal{A}}$ acts on
$\tilde{\mathcal{H}}$ as the direct sum $\tilde{\sigma}(x\otimes
y)=\bigoplus_{q}\sigma_{q}(x)\otimes\sigma_{q}(y)$. We can also define a
multiplier representation of $\widetilde{{{\rm
Diff}^{+}(S^{1})}}\times\widetilde{{{\rm Diff}^{+}(S^{1})}}$ by
$\tilde{U}(\gamma_{+},\gamma_{-}):=\bigoplus_{q}U_{q}(\gamma_{+})\otimes
U_{q}(\gamma_{-})$. The representation $\tilde{U}$ actually factors through
$\mathrm{Conf}(\mathcal{E})$. This can be seen by noting that in each
component $U_{q}\otimes U_{q}$ the generator of spacelike rotations is
$L_{0}^{\sigma_{q}}\otimes{\mathbbm{1}}-{\mathbbm{1}}\otimes
L_{0}^{\sigma_{q}}$ whose spectrum is included in ${\mathbb{Z}}$, since the
spectrum of $L_{0}^{\sigma_{q}}$ is included in
${\mathbb{N}}+\frac{q^{2}}{2}$.

As all the components are the same
${\mathcal{H}}_{q}\otimes{\mathcal{H}}_{q}={\mathcal{H}}\otimes{\mathcal{H}}$,
the shift operators $\\{\psi^{q}\\}$ (“fields”) act naturally on
$\tilde{\mathcal{H}}$: for $\Psi\in{\mathcal{H}}$, where
$(\Psi)_{q}\in{\mathcal{H}}_{q}\otimes{\mathcal{H}}_{q}$,

$(\psi^{q^{\prime}}\Psi)_{q}=(\Psi)_{q+q^{\prime}}.$

It is useful to note how they behave under covariance:

$\displaystyle({\hbox{\rm
Ad\,}}\tilde{U}(\gamma_{+},\gamma_{-})(\psi^{q^{\prime}})\Psi)_{q}$
$\displaystyle=U_{q}(\gamma_{+})\otimes
U_{q}(\gamma_{-})(\psi^{q^{\prime}}\cdot\tilde{U}(\gamma_{+},\gamma_{-})^{*}\Psi)_{q}$
$\displaystyle=U_{q}(\gamma_{+})\otimes
U_{q}(\gamma_{-})(\tilde{U}(\gamma_{+},\gamma_{-})^{*}\Psi)_{q+q^{\prime}}$
$\displaystyle=\left(U_{q}(\gamma_{+})\otimes
U_{q}(\gamma_{-})\right)\cdot\left(U_{q+q^{\prime}}(\gamma_{+})^{*}\otimes
U_{q+q^{\prime}}(\gamma_{-})^{*}\right)(\Psi)_{q+q^{\prime}}$
$\displaystyle=(z_{q}(\gamma_{+})^{*}z_{q+q^{\prime}}(\gamma_{+}))\otimes(z_{q}(\gamma_{-})^{*}z_{q+q^{\prime}}(\gamma_{-}))(\Psi)_{q+q^{\prime}}$
$\displaystyle=\left(\sigma_{q}(z_{q^{\prime}}(\gamma_{+})))\otimes(\sigma_{q}(z_{q^{\prime}}(\gamma_{-}))\right)(\Psi)_{q+q^{\prime}}$
$\displaystyle=(\tilde{\sigma}(z_{q^{\prime}}(\gamma_{+})\otimes
z_{q^{\prime}}(\gamma_{-}))\psi^{q^{\prime}}\Psi)_{q}$

where we used tensoriality of cocycles in the 5th equality.

We define the local algebra, first for $I\times
I\subset{\mathbb{R}}\times{\mathbb{R}}\subset{\mathbb{R}}^{2}$, where the real
lines are identified with the lightrays $x_{0}\pm x_{1}=0$, by

$\tilde{\mathcal{A}}(I\times I)=\\{\tilde{\sigma}(x\otimes
y),\psi^{q}:x,y\in{\mathcal{A}}(I),q\in{\mathbb{R}}\\}^{\prime\prime},$

and for other bounded regions by covariance: take
$\gamma_{\pm}\in\mathrm{Diff}_{0}({\mathbb{R}})$ such that
$\gamma_{\pm}I=I_{\pm}$ and

$\tilde{\mathcal{A}}(I_{+}\times I_{-})=\\{\tilde{\sigma}(x\otimes
y),\psi^{q}:x,y\in{\mathcal{A}}(I),q\in{\mathbb{R}}\\}^{\prime\prime}.$

This does not depend on the choice of $\gamma_{\pm}$. Indeed, if
$\gamma_{\pm}$ preserves $I$, then $z_{q^{\prime}}(\gamma_{+})\otimes
z_{q^{\prime}}(\gamma_{-})\in{\mathcal{A}}(I)\otimes{\mathcal{A}}(I)$ and
${\hbox{\rm
Ad\,}}\tilde{U}(\gamma_{+},\gamma_{-})(\psi^{q^{\prime}})\in\tilde{\mathcal{A}}(I\times
I)$ by above computation.

  * •

Covariance. ${\hbox{\rm
Ad\,}}\tilde{U}(\gamma_{+},\gamma_{-})(\tilde{\mathcal{A}}(O))=\tilde{\mathcal{A}}((\gamma_{+},\gamma_{-})\cdot
O)$ holds by definition. If
$(\gamma_{+},\gamma_{-})\in\mathrm{Diff}_{0}({\mathbb{R}})\times\mathrm{Diff}_{0}({\mathbb{R}})$
acts trivially on $I\times I$, then
$\tilde{U}(\gamma_{+},\gamma_{-})=\tilde{\sigma}(U(\gamma_{+})\otimes
U(\gamma_{-}))$ and this commutes with $\tilde{\mathcal{A}}(I\times I)$, as
${\rm supp\,}\gamma_{\pm}$ are disjoint from $I$.

  * •

Isotony. By covariance, we may assume that $I_{\pm}\supset I$. Take
$\gamma_{\pm}$ such that $\gamma_{\pm}I=I_{\pm}$. From the expression

${\hbox{\rm
Ad\,}}\tilde{U}(\gamma_{+},\gamma_{-})(\psi^{q^{\prime}})=(\tilde{\sigma}(z_{q^{\prime}}(\gamma_{+})\otimes
z_{q^{\prime}}(\gamma_{-}))\psi^{q^{\prime}}\Psi)_{q}$

and from the fact that
$z_{q^{\prime}}(\gamma_{\pm})\in{\mathcal{A}}(I_{\pm})$, the isotony follows.

  * •

Positivity of energy. Each component $U_{q}\otimes U_{q}$ has positive energy.

  * •

Existence and uniqueness of the vacuum. Only $U_{0}\otimes U_{0}$ contains the
vacuum vector.

  * •

Cyclicity. The fields $\psi^{q}$ brings
${\mathcal{H}}_{0}\otimes{\mathcal{H}}_{0}$ to any
${\mathcal{H}}_{q}\otimes{\mathcal{H}}_{q}$, while the local algebra
$\tilde{\sigma}({\mathcal{A}}(I)\otimes{\mathcal{A}}(I))$ acts irreducibly on
each ${\mathcal{H}}_{q}\otimes{\mathcal{H}}_{q}$.

  * •

Locality. In the two-dimensional situation, the spacelike separation of
$I\times I$ and $I_{+}\times I_{-}$ 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.

The commutativity between the observables $\tilde{\sigma}(x\otimes y)$ is
trivial. As for the observables and the fields $\\{\psi^{q}\\}$, if
$x,y\in{\mathcal{A}}(I_{\pm})$ respectively, as $I_{\pm}$ are disjoint from
$I$ and $\sigma_{q}$ are localized in $I$, we have $\tilde{\sigma}(x\otimes
y)=\bigoplus_{q}x\otimes y$ and this commutes with shifts $\psi^{q}$. Finally,
we need to check the commutativity between fields $\psi^{q_{1}},{\hbox{\rm
Ad\,}}U(\gamma_{+})\otimes U(\gamma_{-})(\psi^{q_{2}})$, where
$\gamma_{\pm}I=I_{\pm}$. We can compute the commutator explicitly:

$\displaystyle([\psi^{q_{1}},({\hbox{\rm
Ad\,}}\tilde{U}(\gamma_{+})\otimes\tilde{U}(\gamma_{-})(\psi^{q_{2}})]\Psi)_{q}$
$\displaystyle=(\psi^{q_{1}}\tilde{\sigma}(z_{q_{2}}(\gamma_{+})\otimes
z_{q_{2}}(\gamma_{-}))\psi^{q_{2}}\Psi-\tilde{\sigma}(z_{q_{2}}(\gamma_{+})\otimes
z_{q_{2}}(\gamma_{-}))\psi^{q_{2}}\psi^{q_{1}}\Psi)_{q}$
$\displaystyle=(\tilde{\sigma}(\sigma_{q_{1}}(z_{q_{2}}(\gamma_{+}))\otimes\sigma_{q_{1}}(z_{q_{2}}(\gamma_{-})))\psi^{q_{1}+q_{2}}\Psi-\tilde{\sigma}(z_{q_{2}}(\gamma_{+})\otimes
z_{q_{2}}(\gamma_{-}))\psi^{q_{1}+q_{2}}\Psi)_{q},$

and this vanishes because
$z_{q_{2}}(\gamma_{+})^{*}\sigma_{q_{1}}(z_{q_{2}}(\gamma_{+}))\otimes
z_{q_{2}}(\gamma_{-})^{*}\sigma_{q_{1}}(z_{q_{2}}(\gamma_{-}))=\epsilon_{q_{1},q_{2}}^{+}\otimes\epsilon_{q_{1},q_{2}}^{-}={\mathbbm{1}}$,
as the braidings $\epsilon_{q_{1},q_{2}}^{\pm}$ are scalar and conjugate to
each other.

This net cannot satisfy the split property. Namely, if there were a type I
factor ${\mathcal{R}}$ such that
$\tilde{\mathcal{A}}(D_{1})\subset{\mathcal{R}}\subset\tilde{\mathcal{A}}(D_{2})$,
$\langle\Omega,\cdot\,\Omega\rangle$ would define a faithful normal state on
${\mathcal{R}}$, as it is separating for ${\mathcal{R}}$. As the full Hilbert
space $\tilde{\mathcal{H}}$ is non-separable, by conformal covariance
${\mathcal{R}}$ must be isomorphic to ${\mathcal{B}}(\tilde{\mathcal{H}})$.
But this is impossible because the existence of a faithful normal state
implies that ${\mathcal{B}}(\tilde{\mathcal{H}})$ is $\sigma$-finite, while it
is not when $\tilde{\mathcal{H}}$ is non separable.

##  7 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
$e^{-\beta L_{0}}$ [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 ${\mathcal{A}}$ to be a Möbius covariant net satisfying split property
(for instance ${\mathcal{A}}$ is a conformal net) and $I_{1}\subset I_{2}$ an
inclusion of intervals with no common end points. Any representation $\pi$ of
${\mathcal{A}}$ is a family of local algebra faithful isomorphisms onto their
image, as any local algebra is a factor. Then an intermediate type I factor
${\mathcal{A}}(I_{1})\subset{\mathcal{R}}\subset{\mathcal{A}}(I_{2})$ is
mapped through $\rho$ onto an intermediate type I factor
$\rho_{I_{2}}({\mathcal{A}}(I_{1}))\subset\rho_{I_{2}}({\mathcal{R}})\subset\rho_{I_{2}}({\mathcal{A}}(I_{2}))$
as $\rho_{I_{2}}$ restricts to an isomorphism of ${\mathcal{R}}$ on
$\rho_{I_{2}}({\mathcal{R}})$. Furthermore, when $\rho$ is localizable, then
$\rho_{I_{1}}({\mathcal{A}}(I_{1}))\subset\rho_{I_{2}}({\mathcal{A}}(I_{2}))$
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 $L_{0}^{\rho}$ 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.

### Acknowledgment

We would like to thank James Tener for calling our attention to the article
[36] of Neretin, which allowed us to bridge the gap in the concept of the
proof we previously had. We are grateful to Marcel Bischoff, Sebastiano Carpi,
Luca Giorgetti and Roberto Longo for various interesting discussions on two-
dimensional CFT.

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