# Positive energy representations of Sobolev diffeomorphism groups of the
circle

Sebastiano Carpi111Supported in part by ERC advanced grant 669240 QUEST
“Quantum Algebraic Structures and Models” and GNAMPA-INDAM.,
Dipartimento di Economia, Università “G. D’Annunzio” di Chieti-Pescara,
Viale Pindaro, 42, I-65127, Pescara, Italy
email<EMAIL_ADDRESS>

Simone Del Vecchio222Supported by ERC advanced grant 669240 QUEST “Quantum
Algebraic Structures and Models”.
Dipartimento di Matematica, Università di Roma Tor Vergata
Via della Ricerca Scientifica 1, I-00133 Roma, Italy
email<EMAIL_ADDRESS>

Stefano Iovieno
Dipartimento di Matematica, Università di Roma La Sapienza
Piazzale Aldo Moro 5, I-00185 Roma, Italy
email<EMAIL_ADDRESS>

Yoh Tanimoto333Supported by Programma per giovani ricercatori, anno 2014 “Rita
Levi Montalcini” of the Italian Ministry of Education, University and
Research.
Dipartimento di Matematica, Università di Roma Tor Vergata
Via della Ricerca Scientifica 1, I-00133 Roma, Italy
email<EMAIL_ADDRESS>
###### Abstract

We show that any positive energy projective representation of ${\rm
Diff}_{+}(S^{1})$ extends to a strongly continuous projective unitary
representation of the fractional Sobolev diffeomorphisms
${\mathcal{D}}^{s}(S^{1})$ for any real $s>3$, and in particular to
$C^{k}$-diffeomorphisms ${\rm Diff}_{+}^{k}(S^{1})$ with $k\geq 4$. A similar
result holds for the universal covering groups provided that the
representation is assumed to be a direct sum of irreducibles.

As an application we show that a conformal net of von Neumann algebras on
$S^{1}$ is covariant with respect to ${\mathcal{D}}^{s}(S^{1})$, $s>3$.
Moreover every direct sum of irreducible representations of a conformal net is
also ${\mathcal{D}}^{s}(S^{1})$-covariant.

##  1 Introduction

The group of (smooth) diffeomorphisms of a manifold has been extensively
studied and there have been many interesting results concerning its algebraic
and topological properties, see e.g. [Mil84]. Among them, the group ${\rm
Diff}_{+}(S^{1})$ of orientation preserving diffeomorphisms of the circle
$S^{1}$ is of particular interest in connection with conformal field theory.
In $(1+1)$-dimensional conformal field theory, the symmetry group of the
chiral components is ${\rm Diff}_{+}({\mathbb{R}})$ and often this can be
extended to ${\rm Diff}_{+}(S^{1})$. As this group contains spacetime
translations, the relevant representations must be positive energy
representations and they act on the space of local observables. The
representation theory of positive energy representations has been exploited
for construction and classification of a certain subclass of conformal field
theories, see e.g. [KL04].

Non-trivial positive energy representations of ${\rm Diff}_{+}(S^{1})$ are
necessarily projective. Any irreducible unitary positive energy representation
of the Virasoro algebra extends to a projective representation of the Lie
algebra ${\rm Vect}(S^{1})$, the Lie algebra of vector fields on $S^{1}$, and
it integrates to a positive energy projective unitary representation of ${\rm
Diff}_{+}(S^{1})$ [GW85, TL99]. It follows from [Car04, Theorem A.2], see also
[CKLW18, Section 3.2], that all irreducible positive energy unitary projective
representations of ${\rm Diff}_{+}(S^{1})$ arise in this way. Accordingly they
are completely classified by the central charge $c$ and the lowest conformal
energy $h$ [KR87]. Related results including reducible representations have
been recently obtained in [NS15, Zel17].

These representations of ${\rm Vect}(S^{1})$ extend to certain non-smooth
vector fields as linear maps [CW05]. Apart from that this fact had many
applications (e.g. the uniqueness of conformal covariance in conformal nets
[CW05], positivity of energy in DHR sectors [Wei06], split property in
conformal nets [MTW18] and covariance of soliton representations [Hen17,
DIT18]), it leads naturally to the question whether the group representations
extend to suitable groups of non-smooth diffeomorphisms. In contrast to the
wide range of results and applications concerning the algebraic, analytic and
topological properties of the group ${\rm Diff}_{+}^{k}(M)$ of $C^{k}$
diffeomorphisms and ${\mathcal{D}}^{s}(M)$ of Sobolev class diffeomorphisms
(see e.g. [EM70, Mis97, Ban97, KW09, Fig10]) and on and some results on
(true) representations [KL02, AM06, Kuz07, Mal08], there appears to be only
few results in the literature on positive energy representations of these
groups. Indeed, ${\mathcal{D}}^{s}(M)$ is an infinite-dimensional manifold
modelled on the space $H^{s}(M)$ of $H^{s}$-vector fields, which is not a Lie
algebra with the usual Lie bracket for ${\rm Vect}^{\infty}(M)$. This makes
the study of representations of ${\mathcal{D}}^{s}(M)$ rather subtle.

In this paper, we show that any positive energy (projective) representations
of the diffeomorphism group extends to ${\mathcal{D}}^{s}(S^{1})$ for $s>3$,
by considering its action on vector fields, and therefore, by exploiting the
representation theory of the Virasoro algebra. We also show that these
representations can be locally made into multiplier representation by fixing
the phase. This allows us to take the direct sum and it turns out that
conformal nets are covariant with respect to this extended action.

For some special representations appearing in Fock space, further extensions
have been done first to $C^{3}$-diffeomorphisms [Vro13], then to
${\mathcal{D}}^{s}(S^{1}),s>2$ [DIT18]. The arguments depend on realizing
these representations in some specific conformal field theory, and it is open
whether the results are valid for general central charge $c$. In contrast, by
our argument, representations extend to ${\mathcal{D}}^{s}(S^{1})$ for any
real $s>3$ and for any $c$. While the extensions to
${\mathcal{D}}^{s}(S^{1}),2<s\leq 3$ do not necessarily act nicely on the Lie
algebra representations, by our construction the extensions to
${\mathcal{D}}^{s}(S^{1}),s>3$ do so and are differentiable.

Indeed, our proof follows in part the strategy in [GW85] for the integrability
of the representations of the Virasoro algebra. The extension to non-smooth
diffeomorphisms then follows from the above mentioned extension to non-smooth
vector fields of the corresponding projective representation of ${\rm
Vect}(S^{1})$ given in [CW05]. Actually, our argument can be used to give a
simpler proof of the results in [GW85], see Remark 3.8.

This paper is organized as follows. In Section 2, we recall the relevant
groups and algebras, their topologies and representations. In Section 3, we
first extend the irreducible projective representations of ${\rm
Diff}_{+}(S^{1})$ to ${\mathcal{D}}^{s}(S^{1})$ with $s>3$. Then we lift them
locally to multiplier representations, and show that the direct sum can make
sense as projective representations. Section 4 demonstrates that two-
dimensional chiral conformal field theories described by conformal nets of von
Neumann algebras have this extended symmetry of ${\mathcal{D}}^{s}(S^{1})$. We
summarize possible further continuation of this work in Section 5.

##  2 Preliminaries

###  2.1 ${\rm Diff}_{+}(S^{1})$ and the Virasoro algebra

##### The diffeomorphism group.

Let us denote by ${\rm Diff}_{+}(S^{1})$ the group of orientation preserving,
smooth diffeomorphisms of the circle
$S^{1}\coloneqq\\{z\in{\mathbb{C}}:|z|=1\\}$ and ${\rm Vect}(S^{1})$ denote
the set of smooth vector fields on $S^{1}.$ ${\rm Diff}_{+}(S^{1})$ is an
infinite dimensional Lie group whose Lie algebra is identified with the real
topological vector space ${\rm Vect}(S^{1})$ of smooth vector fields on
$S^{1}$ with $C^{\infty}$ topology [Mil84]. In the following we identify ${\rm
Vect}(S^{1})$ with $C^{\infty}(S^{1},\mathbb{R})$ and for $f\in
C^{\infty}(S^{1},\mathbb{R})$ we denote by $f^{\prime}$ the derivative of $f$
with respect to the angle $\theta$,

$f^{\prime}(z)=\frac{d}{d\theta}f(e^{i\theta})\bigg{\rvert}_{e^{i\theta}=z}.$

We consider a diffeomorphism $\gamma\in{\rm Diff}_{+}(S^{1})$ as a map from
$S^{1}$ in $S^{1}\subset{\mathbb{C}}$. With this convention, its action on
$f\in{\rm Vect}(S^{1})$ is

$\displaystyle(\gamma_{*}f)(e^{i\theta})=-ie^{-i\theta}\left(\frac{d}{d\varphi}\gamma(e^{i\varphi})\right)\bigg{\rvert}_{e^{i\varphi}=\gamma^{-1}(e^{i\theta})}f(\gamma^{-1}(e^{i\theta})).$
(2.1)

We denote by ${\rm Diff}_{+}^{k}(S^{1})$ the group of $C^{k}$-diffeomorphisms
of $S^{1}$. Note that this is not a Lie group, and indeed, the corresponding
linear space ${\rm Vect}^{k}(S^{1})$ of $C^{k}$-vector fields is not closed
under the natural Lie bracket (see below).

The universal covering group of ${\rm Diff}_{+}(S^{1})$ (resp. ${\rm
Diff}_{+}^{k}(S^{1})$), $\widetilde{{\rm Diff}_{+}(S^{1})}$ (resp.
$\widetilde{{\rm Diff}_{+}^{k}(S^{1})}$), can be identified444The realization
of $\widetilde{{\rm Diff}_{+}^{k}(S^{1})}$ works in the same way as
$\widetilde{{\rm Diff}_{+}(S^{1})}$ as in [TL99, Section 6.1], see also
[Ham82, Example 4.2.6]. with the group of $C^{\infty}$-diffeomorphisms (resp.
$C^{k}$-diffeomorphisms) $\gamma$ of $\mathbb{R}$ which satisfy

$\gamma(\theta+2\pi)=\gamma(\theta)+2\pi.$

If $\gamma\in\widetilde{{\rm Diff}_{+}(S^{1})}$, its image under the covering
map is in the following denoted by $\mathring{\gamma}\in{\rm
Diff}_{+}(S^{1})$, where $\mathring{\gamma}(e^{i\theta})=e^{i\gamma(\theta)}$.
Conversely, if $\gamma\in{\rm Diff}_{+}(S^{1})$, there is an element
$\tilde{\gamma}\in\widetilde{{\rm Diff}_{+}(S^{1})}$ whose image under the
covering map is $\gamma$. Such a $\tilde{\gamma}$ is unique up to $2\pi$ and
called a lift of $\gamma$.

The group ${\rm Diff}_{+}(S^{1})$ admits the Bott-Virasoro cocycle $B:{\rm
Diff}_{+}(S^{1})\times{\rm Diff}_{+}(S^{1})\rightarrow\mathbb{R}$ (see e.g.
[FH05]). The Bott-Virasoro group is then defined as the group with elements

$(\gamma,t)\in{\rm Diff}_{+}(S^{1})\times\mathbb{R}$

and with multiplication

$(\gamma_{1},t_{1})\circ(\gamma_{2},t_{2})=(\gamma_{1}\circ\gamma_{2},t_{1}+t_{2}+B(\gamma_{1},\gamma_{2})).$

Note that, given a true (not projective) unitary irreducible representation
$V$ of the universal covering of the Bott-Virasoro group, one can obtain a
unitary multiplier representation555for the definition of unitary multiplier
representation see section 2.4. $\underline{V}(\gamma):=V(\gamma,0)$ of
$\widetilde{{\rm Diff}_{+}(S^{1})}$ (with respect to the Bott-Virasoro cocycle
$B$). Then the map $\underline{V}:\widetilde{{\rm Diff}_{+}(S^{1})}\rightarrow
U(\mathcal{H})$ satisfies

$\underline{V}(\gamma_{1})\underline{V}(\gamma_{2})=e^{icB(\mathring{\gamma_{1}},\mathring{\gamma_{2}})}\underline{V}(\gamma_{2})\underline{V}(\gamma_{1}),$

where $c\in{\mathbb{R}}$ by irreducibility.

##### The Lie algebra.

The space ${\rm Vect}(S^{1})$ is endowed with the Lie algebra structure with
the Lie bracket given by

$[f,g]=f^{\prime}g-fg^{\prime}.$

As a Lie algebra, ${\rm Vect}(S^{1})$ admits the Gelfand–Fuchs two-cocycle

$\omega(f,g)=\frac{1}{48\pi}\int_{S^{1}}(f(e^{i\theta})g^{\prime\prime\prime}(e^{i\theta})-f^{\prime\prime\prime}(e^{i\theta})g(e^{i\theta}))d\theta.$

The Virasoro algebra ${\rm Vir}$ is the central extension of the
complexification of the algebra generated by the trigonometric polynomials in
${\rm Vect}(S^{1})$ defined by the two-cocycle $\omega$. It can be explicitly
described as the complex Lie algebra generated by $L_{n}$, $n\in\mathbb{Z}$,
and the central element $\mathfrak{1}$, with brackets

$[L_{n},L_{m}]=(n-m)L_{n+m}+\delta_{n+m,0}\frac{n^{3}-n}{12}\mathfrak{1}.$

Consider a representation $\pi:{\rm Vir}\rightarrow{\hbox{End}}(V)$ of ${\rm
Vir}$ on a complex vector space $V$ endowed with a scalar product
$\langle\cdot,\cdot\rangle$. We call $\pi$ a unitary positive energy
representation if the following hold

  1. 1.

Unitarity: $\langle v,\pi(L_{n})w\rangle=\langle\pi(L_{-n})v,w\rangle$ for
every $v,w\in V$ and $n\in{\mathbb{Z}}$;

  2. 2.

Positivity of the energy:
$V=\bigoplus_{\lambda\in{\mathbb{R}}_{+}\cup\\{0\\}}V_{\lambda}$, where
$V_{\lambda}\coloneqq\ker(\pi(L_{0})-\lambda{\mathbbm{1}}_{V})$. The lowest
eigenvalue of $\pi(L_{0})$ is called lowest weight;

  3. 3.

Central charge: $\pi(\mathfrak{1})=c{\mathbbm{1}}_{V}$;

There exists an irreducible unitary positive energy representation with
central charge $c$ and lowest weight $h$ if and only if $c\geq 1$ and $h\geq
0$ (continuous series representation) or $(c,h)=(c(m),h_{p,q}(m))$, where
$c(m)=1-\frac{6}{(m+2)(m+3)}$, $h_{p,q}(m)=\frac{(p(m+1)-qm)^{2}-1}{4m(m+1)}$,
$m=3,4,\cdots$, $p=1,2,\cdots,m-1$, $q=1,2,\cdots,p$, (discrete series
representation) [KR87][DMS97]. In this case the representation space $V$ is
denoted by ${\mathcal{H}}^{\mathrm{fin}}(c,h)$. We denote by
$\mathcal{H}(c,h)$ the Hilbert space completion of the vector space
${\mathcal{H}}^{\mathrm{fin}}(c,h)$ associated with the unique irreducible
unitary positive energy representation of ${\rm Vir}$ with central charge $c$
and lowest weight $h$.

In these representations, the conformal Hamiltonian $\pi(L_{0})$ is
diagonalized, and on the linear span of its eigenvectors
$\mathcal{H}^{\mathrm{fin}}(c,h)$ (the space of finite energy vectors), the
Virasoro algebra acts algebraically as unbounded operators.

##### The stress-energy tensor.

Let $\mathcal{H}(c,h)$ as above and, with abuse of notation, we denote by
$L_{n}$ the elements of ${\rm Vir}$ represented in $\mathcal{H}(c,h)$. For a
smooth complex-valued function $f$ on $S^{1}$ with finitely many non-zero
Fourier coefficients, the (chiral) stress-energy tensor associated with $f$ is
the operator

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

acting on $\mathcal{H}(c,h)$, where

$\hat{f}_{n}=\int_{0}^{2\pi}\frac{d\theta}{2\pi}e^{-in\theta}f(e^{i\theta}).$

by the linear energy bounds, yielding a self-adjoint unbounded operator
$T(f)$. Moreover it can be extended to a particular class of non-smooth
functions [CW05], retaining its self-adjointness. This fact will be used in
this article and will be thus resumed in some detail in Section 2.2.

It is a crucial fact that the irreducible representations $\mathcal{H}(c,h)$
of ${\rm Vir}$ integrate to irreducible unitary strongly continuous
representations of the universal covering of the Bott-Virasoro group [FH05].
In other words, denoting by $q$ the quotient map
$q:{\mathcal{U}}(\mathcal{H}(c,h))\rightarrow{\mathcal{U}}(\mathcal{H}(c,h))/\mathbb{C}$
(we denote by ${\mathcal{U}}({\mathcal{K}})$ the group of unitary operators on
${\mathcal{K}}$), there is an irreducible, unitary, strongly continuous
multiplier representation $U$ of $\widetilde{{\rm Diff}_{+}(S^{1})}$, the
universal covering of ${\rm Diff}_{+}(S^{1})$, such that

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

for all $f\in{\rm Vect}(S^{1})$, where ${\rm Exp}$ is the Lie-theoretic
exponential map of ${\rm Diff}_{+}(S^{1})$ (see [Mil84]).

For the stress-energy tensor $T$, we have the following covariance [FH05,
Proposition 5.1, Proposition 3.1].

######  Proposition 2.1.

The stress-energy tensor $T$ on $\mathcal{H}(c,h)$ transforms according to

$U(\gamma)T(f)U(\gamma)^{*}=T(\mathring{\gamma}_{*}({f}))+\frac{c}{24\pi}\int^{2\pi}_{0}\\{\mathring{\gamma},z\\}\bigg{\rvert}_{z=e^{i\theta}}f(e^{i\theta})e^{i2\theta}d\theta$

on vectors in $\mathcal{H}^{\mathrm{fin}}(c,h)$, for $f\in{\rm Vect}(S^{1})$
and $\gamma\in\widetilde{{\rm Diff}_{+}(S^{1})}$. Furthermore the commutation
relations

$i[T(g),T(f)]=T(g^{\prime}f-f^{\prime}g)+c\omega(g,f),$

hold for arbitrary $f,g\in C^{\infty}(S^{1})$, on vectors
$\psi\in\mathcal{H}^{\mathrm{fin}}(c,h).$

Here

$\\{\mathring{\gamma},z\\}=\frac{\frac{d^{3}}{dz^{3}}\mathring{\gamma}(z)}{\frac{d}{dz}\mathring{\gamma}(z)}-\frac{3}{2}\left(\frac{\frac{d^{2}}{dz^{2}}\mathring{\gamma}(z)}{\frac{d}{dz}\mathring{\gamma}(z)}\right)^{2}$

is the Schwarzian derivative of $\mathring{\gamma}$ and
$\frac{d}{dz}\mathring{\gamma}(z)=-i\bar{z}\frac{d}{d\theta}\mathring{\gamma}(e^{i\theta})\bigg{\rvert}_{e^{i\theta}=z}$.
Note that

$\beta(\gamma,f)\coloneqq\frac{c}{24\pi}\int_{S^{1}}\\{\mathring{\gamma},z\\}izf(z)dz$

and $\omega(\cdot,\cdot)$ are related by

$\displaystyle\frac{d}{dt}\beta({\rm
Exp}(tf),g)\bigg{\rvert}_{t=0}=-c\omega(f,g).$ (2.2)

###  2.2 The stress-energy tensor on non-smooth vector fields

Let $T$ be the stress-energy tensor on ${\mathcal{H}}(c,h)$. Given a not
necessarily smooth real function $f$ of $S^{1}$ it is possible to evaluate the
stress-energy tensor on $f$ [CW05, Proposition 4.5]. First of all we define
for a real-valued function $f$ of the circle

$\|f\|_{\frac{3}{2}}\coloneqq\sum_{n\in\mathbb{Z}}|{\hat{f}}_{n}|(1+|n|^{\frac{3}{2}}),$

where
$\hat{f}_{n}\coloneqq\frac{1}{2\pi}\int_{0}^{2\pi}e^{-in\theta}f(e^{i\theta})d\theta$
is the $n$-th Fourier coefficient of $f$. We denote666We consider
$\mathcal{S}_{\frac{3}{2}}(S^{1})$ and $H^{s}(S^{1})$ below as the spaces of
nonsmooth vector fields on $S^{1}$, and accordingly, without specification,
they are the spaces of real functions. with $\mathcal{S}_{\frac{3}{2}}(S^{1})$
the class of functions $f\in L^{1}(S^{1},{\mathbb{R}})$ such that
$\|f\|_{\frac{3}{2}}$ is finite endowed with the topology induced by the norm
$\|\cdot\|_{\frac{3}{2}}$.

The following is [CW05, Proposition 4.2, Theorem 4.4, Proposition 4.5].

######  Proposition 2.2.

If $f:S^{1}\rightarrow\mathbb{C}$ is continuous and such that
$\sum_{n\in\mathbb{Z}}|\hat{f}_{n}|(1+|n|^{\frac{3}{2}})<\infty$ then

  1. (1)

the operator $T(f)=\sum_{n\in\mathbb{Z}}L_{n}\hat{f}_{n}$ on the domain
$\mathcal{H}^{\mathrm{fin}}(c,h)$ is well defined, (i.e. the sum is strongly
convergent on the domain);

  2. (2)

$T(f)^{*}$ is an extension of the operator
$T(f)^{+}:=\sum_{n\in\mathbb{Z}}L_{n}\bar{\hat{f}}_{n}$ (this is again
understood as an operator on the domain $\mathcal{H}^{\mathrm{fin}}(c,h)$).

  3. (3)

$T(f)$ is closable and $\overline{T(f)}=(T(f)^{+})^{*}$, where $T(f)$ and
$T(f)^{+}$ are considered as operators on the domain
$\mathcal{H}^{\mathrm{fin}}(c,h)$. In particular, if
$\hat{f}_{n}=\bar{\hat{f}}_{-n}$ for all $n\in\mathbb{Z}$ (i.e. if $f$ is a
real-valued function), then $T(f)$ is essentially self-adjoint on
$\mathcal{H}^{\mathrm{fin}}(c,h)$.

  4. (4)

If $f$ is real, then for every $\xi\in{\mathscr{D}}(L_{0})$ we have the
following energy bounds

$\|T(f)\xi\|\leq r\|f\|_{\frac{3}{2}}\|(1+L_{0})\xi\|$

where $r$ is a positive constant. Consequently,
${\mathscr{D}}(L_{0})\subset{\mathscr{D}}(T(f))$.

  5. (5)

If $\\{f_{n}\\}$ ($n\in\mathbb{N}$) is a sequence777This should be
distinguished from the Fourier coefficients $\hat{f}_{n}$ of a single function
$f$. of continuous real functions on $S^{1}$ in
$\mathcal{S}_{\frac{3}{2}}(S^{1})$ and $\|f-f_{n}\|_{\frac{3}{2}}$ converges
to $0$ as $n$ tends to $\infty$, then

$T(f_{n})\rightarrow T(f)$

in the strong resolvent sense.

It has been also shown that the class $\mathcal{S}_{\frac{3}{2}}(S^{1})$
contains many non-smooth functions [Wei06, Lemma 2.2],[CW05, Lemma 5.3].

######  Proposition 2.3.

If a real-valued function $f$ on the circle is piecewise smooth and once
continuously differentiable on the whole $S^{1}$, then
$f\in\mathcal{S}_{\frac{3}{2}}(S^{1})$.

###  2.3 Groups of diffeomorphisms of Sobolev class $H^{s}(S^{1})$

Let $s>\frac{1}{2}$ be a real number. We introduce (see [EK14, Section 2] and
[EK14, Definition 2.2], respectively)

$\displaystyle H^{s}(S^{1})$ $\displaystyle:=\\{f\in
L^{2}(S^{1},{\mathbb{R}}):\|f\|_{H^{s}}<\infty\\},\text{ where
}\|f\|_{H^{s}}:=\left(\sum_{n\in{\mathbb{Z}}}(1+n^{2})^{s}|\hat{f}_{n}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle H^{s}(S^{1},{\mathbb{C}})$ $\displaystyle:=\\{f\in
L^{2}(S^{1},{\mathbb{C}}):\|f\|_{H^{s}}<\infty\\},\text{ where
}\|f\|_{H^{s}}:=\left(\sum_{n\in{\mathbb{Z}}}(1+n^{2})^{s}|\hat{f}_{n}|^{2}\right)^{\frac{1}{2}}$
$\displaystyle{\mathcal{D}}^{s}(S^{1})$ $\displaystyle:=\\{\gamma\in{\rm
Diff}_{+}^{1}(S^{1}):\tilde{\gamma}-\iota\in H^{s}(S^{1})\\},$

where $\tilde{\gamma}$ is a lift of $\gamma$ to ${\mathbb{R}}$. Actually, in
literature there are various definitions of these Sobolev spaces and their
topologies. Although it is well-known that they coincide, for the convenience
of the reader we recall them and show their equivalence in Appendix. As
$s>\frac{1}{2}$, the space $H^{s}(S^{1})$ is a subspace of
$C(S^{1},{\mathbb{R}})$. The universal covering group
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ of ${\mathcal{D}}^{s}(S^{1})$ is a
subspace of $\widetilde{{\rm Diff}_{+}^{1}(S^{1})}$, namely the space of the
maps $\gamma:{\mathbb{R}}\to{\mathbb{R}}$ satisfying
$\gamma(\theta+2\pi)=\gamma(\theta)+2\pi$ and locally $H^{s}$ (see Appendix),
and this can be identified with an open subset of the subspace of
$H^{s}(S^{1})$.

From these definitions, it is immediate that ${\rm Diff}_{+}^{k}(S^{1})$ is
continuously embedded in ${\mathcal{D}}^{k}(S^{1})$. Conversely, by the
Sobolev-Morrey embedding [IKT13, Proposition 2.2], it holds that
${\mathcal{D}}^{s}\hookrightarrow{\rm Diff}_{+}^{k}(S^{1})$ if
$s>k+\frac{1}{2}$.

The first statement of the following is a straightforward adaptation of
[IKT13, Lemma 2.3]. One can also find various different elementary proofs, for
example [tim, Smy]. The second statement is an adaptation of [IKT13, Lemma
B.4].

######  Lemma 2.4.

Let $s>\frac{1}{2}$. Then $H^{s}(S^{1})$ is an algebra and $\|fg\|_{H^{s}}\leq
C_{s}\|f\|_{H^{s}}\|g\|_{H^{s}}$. If $g\in H^{s}(S^{1})$ and
$\inf_{\theta}(1+g(\theta))>0$, then $\frac{1}{1+g}\in H^{s}(S^{1})$.

The following is a special case of [IKT13, Theorem B.2] and an analogue of
[IKT13, Proposition B.7], see also Appendix. According to [Kol13, P.12], Lemma
2.5(a) for integer $s$ has been first established in [Ebi68].

######  Lemma 2.5.

Let $s>\frac{3}{2}$. Then

  1. (a)

$(\gamma,f)\mapsto f\circ\gamma,\;{\mathcal{D}}^{s}(S^{1})\times
H^{s}(S^{1})\to H^{s}(S^{1})$ is continuous.

  2. (b)

$\gamma\mapsto\gamma^{-1},\;{\mathcal{D}}^{s}(S^{1})\to{\mathcal{D}}^{s}(S^{1})$
is continuous.

  3. (c)

${\mathcal{D}}^{s}(S^{1})$ is a topological group.

By applying these results, we obtain the following

######  Lemma 2.6.

We have the following.

  1. (a)

Let $s>2$. The embedding
$H^{s}(S^{1})\hookrightarrow\mathcal{S}_{\frac{3}{2}}(S^{1})$ is continuous.

  2. (b)

Let $s>\frac{3}{2}$. The map

$\displaystyle{\mathcal{D}}^{s+1}(S^{1})\times H^{s}(S^{1})$
$\displaystyle\rightarrow H^{s}(S^{1})$ $\displaystyle(\gamma,f)$
$\displaystyle\mapsto\gamma_{*}(f),$

where $\gamma_{*}(f)$ is as in (2.1), is continuous.

  3. (c)

Let $s>3$. $\beta(\gamma,f)$ extends continuously to
$\gamma\in{\mathcal{D}}^{s}(S^{1}),f\in L^{2}(S^{1},{\mathbb{R}})$.

###### Proof.

(a) is obtained from the following inequality

$\sum_{k\neq 0}|\hat{f}_{k}||k|^{\frac{3}{2}}=\sum_{k\neq
0}|\hat{f}_{k}||k|^{2+\epsilon}\frac{1}{|k|^{\frac{1}{2}+\epsilon}}\leq\sqrt{\sum_{k\neq
0}\frac{1}{k^{1+2\epsilon}}}\sqrt{\sum_{k\neq
0}|\hat{f}_{k}|^{2}|k|^{4+2\epsilon}}.$

for any $\epsilon>0$.

(b) follows from Lemmas 2.5 and 2.4 and (2.1).

(c) Note that, with $s>3$,
${\mathcal{D}}^{s}(S^{1})\ni\gamma\mapsto\\{\mathring{\gamma},z\\}\in
L^{2}(S^{1},{\mathbb{C}})$ is continuous. To see it, in the definition

$\\{\mathring{\gamma},z\\}=\frac{\frac{d^{3}}{dz^{3}}\mathring{\gamma}(z)}{\frac{d}{dz}\mathring{\gamma}(z)}-\frac{3}{2}\left(\frac{\frac{d^{2}}{dz^{2}}\mathring{\gamma}(z)}{\frac{d}{dz}\mathring{\gamma}(z)}\right)^{2},$

the maps $\gamma\mapsto\frac{d^{3}}{dz^{3}}\mathring{\gamma}(z)\in
L^{2}(S^{1},{\mathbb{C}})$ and
$\gamma\mapsto\frac{1}{\frac{d}{dz}\mathring{\gamma}(z)}\in
H^{s-1}(S^{1},{\mathbb{C}})\subset L^{\infty}(S^{1},{\mathbb{C}})$ are
continuous, hence their product is continuous in $L^{2}(S^{1},{\mathbb{C}})$.
The second derivative
$\gamma\mapsto\frac{d^{2}}{dz^{2}}\mathring{\gamma}(z)\in
H^{s-2}(S^{1},{\mathbb{C}})$ is continuous hence so is
$\gamma\mapsto\left(\frac{\frac{d^{2}}{dz^{2}}\mathring{\gamma}(z)}{\frac{d}{dz}\mathring{\gamma}(z)}\right)^{2}\in
H^{s-2}(S^{1},{\mathbb{C}})$ (by the complexification of Lemma 2.4), hence we
obtain the continuity of $\gamma\mapsto\\{\mathring{\gamma},z\\}$ by the
complexification of Lemma 2.4. Now the claim is immediate because
$\beta(\gamma,f)=\frac{c}{24\pi}\int_{S^{1}}\\{\mathring{\gamma},z\\}izf(z)dz$
∎

###  2.4 Projective and multiplier representations

A strongly continuous unitary projective representation of a topological group
$G$ is a pair $(U,\mathcal{H})$ where $\mathcal{H}$ is a Hilbert space and $U$
is a continuous group homomorphism from $G$ to
$\mathcal{U}(\mathcal{H})/\mathbb{T}$, where ${\mathcal{U}}({\mathcal{H}})$ is
equipped with the strong operator topology and
$\mathcal{U}(\mathcal{H})/\mathbb{T}$ with the quotient topology by the
quotient map $q$. Namely, the subbasis elements which contain $q(u)$ are
$\\{{\mathcal{U}}_{q(u),\xi,\varepsilon}\\}_{\xi\in{\mathcal{H}},\varepsilon>0}$,
where

${\mathcal{U}}_{q(u),\xi,\varepsilon}=\\{q(v):\text{ there are
}u^{\prime},v^{\prime}\in{\mathcal{U}}({\mathcal{H}}),q(u)=q(u^{\prime}),q(v)=q(v^{\prime}),\text{
and }\|(v^{\prime}-u^{\prime})\xi\|<\varepsilon\\}.$

Therefore, it is clear that a net $\\{q(u_{\lambda})\\}$ has limit $q(u)$ if
and only if for each $\xi\in{\mathcal{H}}$ there is
$z_{\xi,\lambda},\hat{z}_{\xi,\lambda}\in\mathbb{T}$ such that
$\|z_{\xi,\lambda}u_{\lambda}\xi-\hat{z}_{\xi,\lambda}u\xi\|\to 0$ if and only
if there is $z_{\xi,\lambda}\in\mathbb{T}$ such that888One can concretely make
the following choice: $z_{\xi,\lambda}=\frac{\overline{\langle
u\xi,u_{\lambda}\xi\rangle}}{|\langle u\xi,u_{\lambda}\xi\rangle|}$, then
$z_{\xi,\lambda}u_{\lambda}\xi$ converges to $u\xi$.
$z_{\xi,\lambda}u_{\lambda}\xi\to u\xi$. Actually, $z_{\xi,\lambda}$ does not
depend on $\xi$ (because, if $z_{\xi,\lambda}u_{\lambda}\eta$ were not
convergent for $\eta\perp\xi$, $z_{\xi,\lambda}u_{\lambda}(\xi+\eta)$ would
not be convergent in ${\mathcal{H}}/\mathbb{T}$, hence convergence holds for
any $\eta$), hence $q(u_{\lambda})$ is convergent if and only if there is a
net $z_{\lambda}\in\mathbb{T}$ such that $z_{\lambda}u_{\lambda}$ is
convergent in the strong operator topology.

The above continuity is equivalent to the following, see [Bar54]: whenever
$g_{\lambda}\to g$, it holds for any $x\in{\mathcal{B}}({\mathcal{H}})$ that
${\hbox{\rm Ad\,}}U(g_{\lambda})(x)\to{\hbox{\rm Ad\,}}U(g)(x)$ (note that
${\hbox{\rm Ad\,}}U(g)$ is well-defined for
$U(g)\in{\mathcal{U}}({\mathcal{H}})/\mathbb{T}$). We show it here for the
convenience of the reader. It is straightforward that if $U(g_{\lambda})\to
U(g)$ in ${\mathcal{U}}({\mathcal{H}})/\mathbb{T}$, then one can fix phases of
$U(g_{\lambda})$ and $U(g)$ so that $U(g_{\lambda})\to U(g)$ in the strong
operator topology (with a slight abuse of notation) by the previous paragraph,
hence ${\hbox{\rm Ad\,}}U(g_{\lambda})(x)\to{\hbox{\rm Ad\,}}U(g)(x)$.
Conversely, assume that ${\hbox{\rm Ad\,}}U(g_{\lambda})(x)\to{\hbox{\rm
Ad\,}}U(g)(x)$ for any $x\in{\mathcal{B}}({\mathcal{H}})$, and fix the phases
of $U(g_{\lambda}),U(g)$. For any one-dimensional projection $p_{\xi}$ to a
unit vector $\xi$, we have ${\hbox{\rm
Ad\,}}U(g_{\lambda})(p_{\xi})\to{\hbox{\rm Ad\,}}U(g)(p_{\xi})$, and the
latter is again a one-dimensional projection to, say, ${\mathbb{C}}\xi_{g}$,
and we may assume $\|\xi\|=\|\xi_{g}\|=1$. Since $\langle\xi_{g},{\hbox{\rm
Ad\,}}U(g_{\lambda})(p_{\xi})\xi_{g}\rangle\to\langle\xi_{g},{\hbox{\rm
Ad\,}}U(g)(p_{\xi})\xi_{g}\rangle=1$,
$\|p_{\xi}U(g_{\lambda})^{*}\xi_{g}\|=|\langle\xi,U(g_{\lambda})^{*}\xi_{g}\rangle|\to
1$. Let us assign new phases by
$U^{\prime}(g_{\lambda})=U(g_{\lambda})\cdot\frac{\overline{\langle\xi_{g},U(g_{\lambda})\xi\rangle}}{|\langle\xi_{g},U(g_{\lambda})\xi\rangle|}$.
Then $\langle\xi_{g},U^{\prime}(g_{\lambda})\xi\rangle$ is positive and tends
to $1$, implying that $U^{\prime}(g_{\lambda})\xi\to\xi_{g}$. Since this holds
for an arbitrary $\xi$, it yields the convergence in
${\mathcal{U}}({\mathcal{H}})/\mathbb{T}$ again by the previous paragraph.

We can consider $U(g)$ as an operator acting on $\mathcal{H}$ determined up to
a phase factor. Two projective unitary representations
$(U_{1},\mathcal{H}_{1})$ and $(U_{2},\mathcal{H}_{2})$ are said to be
equivalent if exists an unitary $W:\mathcal{H}_{1}\rightarrow\mathcal{H}_{2}$
such that $WU_{1}(g)=U_{2}(g)W$ for every $g\in G$ up to a phase factor.

A unitary multiplier representation of $G$ is a pair $(U,\mathcal{H})$ were
$U:G\rightarrow\mathcal{U}(\mathcal{H})$ is a map such that
$U(g_{1})U(g_{2})=\omega(g_{1},g_{2})U(g_{1}g_{2})$ and $\omega:G\times
G\rightarrow\mathbb{T}$ is a map which satisfies the equality

$\omega(g_{1},g_{2})\omega(g_{1}g_{2},g_{3})=\omega(g_{1},g_{2}g_{3})\omega(g_{2},g_{3}).$

A unitary multiplier representation $U$ of $G$ is strongly continuous if
$U(g)v$ tends to $U(g_{0})v$ for all $v\in{\mathcal{H}}$ if $g$ tends to
$g_{0}$.

##  3 Extension of the ${\rm Diff}_{+}(S^{1})$ representations to Sobolev
diffeomorphisms

###  3.1 Irreducible case

Our purpose of this section is to extend the (positive energy projective)
representation $U$ on ${\mathcal{H}}(c,h)$ of ${\rm Diff}_{+}(S^{1})$ to
${\mathcal{D}}^{s}(S^{1})$ with $s>3$. In the following $s>3$ will be always
assumed.

An element $\gamma\in{\mathcal{D}}^{s}(S^{1})$ acts on $f\in{\rm Vect}(S^{1})$
via (2.1). If $T$ is the energy-momentum operator associated with a positive
energy unitary representation of the Virasoro algebra ${\rm Vir}$ with central
charge $c$ and lowest weight $h$, we define a new class of operators

$\displaystyle T^{\gamma}(f)\coloneqq T(\gamma_{*}f)-\beta(\gamma,f),$

where $f\in{\rm Vect}(S^{1})$ and
$\beta(\gamma,f)=\frac{c}{24\pi}\int_{S^{1}}\\{\gamma,z\\}izf(z)dz$, which
makes sense for $\gamma\in{\mathcal{D}}^{s}(S^{1})$ by Lemma 2.6 and
Proposition 2.2(1). The fact that $\gamma_{*}f$ is in
$\mathcal{S}_{\frac{3}{2}}(S^{1})$ ensures that $T(\gamma_{*}f)$ is an
essentially self-adjoint operator on $\mathcal{H}^{\mathrm{fin}}(c,h)$ and so
is $T^{\gamma}(f)$ by Proposition 2.2(3). We denote its closure by the same
symbol $T^{\gamma}(f)$, so long as no confusion arises.

Note that, if $\gamma\in{\rm Diff}_{+}(S^{1})$, then we have

$\displaystyle T^{\gamma}(f)={\hbox{\rm Ad\,}}U(\gamma)(T(f)).$ (3.1)

Indeed, by definition $T^{\gamma}(f)=T(\gamma_{*}f)-\beta(\gamma,f)$ and by
Proposition 2.1, (3.1) holds on ${\mathscr{D}}(L_{0})$, and the both operators
are essentially self-adjoint there, hence they must coincide. As they are
unitarily implemented, the energy bound holds as well:

$\displaystyle\|T^{\gamma}(f)\xi\|\leq
r\|f\|_{\frac{3}{2}}\cdot\|(1+L_{0}^{\gamma})\xi\|,$ (3.2)

where $L_{0}^{\gamma}:=T^{\gamma}(1)$.

We define for $\gamma_{1},\gamma_{2}\in{\mathcal{D}}^{s}(S^{1})$

$(T^{\gamma_{1}})^{\gamma_{2}}(f)\coloneqq
T^{\gamma_{1}}((\gamma_{2})_{*}f)-\beta(\gamma_{2},f).$

######  Proposition 3.1.

Let $\gamma_{1},\gamma_{2}\in{\mathcal{D}}^{s}(S^{1})$, $s>3$, and $f\in{\rm
Vect}(S^{1})$. Then
$(T^{\gamma_{1}})^{\gamma_{2}}(f)=T^{\gamma_{1}\circ\gamma_{2}}(f)$.

###### Proof.

Using the properties of the Schwarzian derivative [OT05]

$\left\\{\gamma_{1}\circ\gamma_{2},z\right\\}=\left\\{\gamma_{1},\gamma_{2}(z)\right\\}\left(\frac{d}{dz}\gamma_{2}(z)\right)^{2}+\left\\{\gamma_{2},z\right\\}$

where $y=\gamma_{2}(z)$, we infer that

$\displaystyle\beta(\gamma_{1}\circ\gamma_{2},f)$
$\displaystyle=-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{1}\circ\gamma_{2},z\right\\}\bigg{\rvert}_{z=e^{i\theta}}f(e^{i\theta})e^{i2\theta}d\theta$
$\displaystyle=-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{1},y\right\\}\bigg{\rvert}_{y=\gamma_{2}(e^{i\theta})}\left(\frac{d}{dz}\gamma_{2}(z)\right)^{2}\bigg{\rvert}_{z=e^{i\theta}}f(e^{i\theta})e^{i2\theta}d\theta$
$\displaystyle\qquad-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{2},z\right\\}\bigg{\rvert}_{z=e^{i\theta}}f(e^{i\theta})e^{i2\theta}d\theta$
$\displaystyle=-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{1},y\right\\}\bigg{\rvert}_{y=e^{i\varphi}}\cdot(-i)\frac{d}{d\theta}\left(\gamma_{2}(e^{i\theta})\right)\bigg{\rvert}_{e^{i\theta}=\gamma_{2}^{-1}(e^{i\varphi})}f(\gamma_{2}^{-1}(e^{i\varphi}))e^{i\varphi}d\varphi$
$\displaystyle\qquad-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{2},z\right\\}\bigg{\rvert}_{z=e^{i\theta}}f(e^{i\theta})e^{i2\theta}d\theta$
$\displaystyle=-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{1},y\right\\}\bigg{\rvert}_{y=e^{i\varphi}}\cdot(-i)e^{-i\varphi}\frac{d}{d\theta}\left(\gamma_{2}(e^{i\theta})\right)\bigg{\rvert}_{e^{i\theta}=\gamma_{2}^{-1}(e^{i\varphi})}f(\gamma_{2}^{-1}(e^{i\varphi}))e^{i2\varphi}d\varphi$
$\displaystyle\qquad-\frac{c}{24\pi}\int_{0}^{2\pi}\left\\{\gamma_{2},z\right\\}\bigg{\rvert}_{z=e^{i\theta}}f(e^{i\theta})e^{i2\theta}d\theta$
$\displaystyle=\beta(\gamma_{1},\gamma_{2_{*}}(f))+\beta(\gamma_{2},f),$

where we used the change of variables $e^{i\varphi}=\gamma_{2}(e^{i\theta})$,
hence
$e^{i\theta}\frac{d\theta}{d\varphi}\frac{d\gamma_{2}}{dz}(e^{i\theta})|_{\gamma_{2}(e^{i\theta})=e^{i\varphi}}=e^{i\varphi}$,
$\frac{d\gamma_{2}}{dz}(e^{i\theta})=-ie^{-i\theta}\frac{d}{d\theta}\gamma_{2}(e^{i\theta})$
and (2.1).

So
$(T^{\gamma_{1}})^{\gamma_{2}}(f)=T((\gamma_{1})_{*}((\gamma_{2})_{*}f))-\beta(\gamma_{1},\gamma_{2*}f)-\beta(\gamma_{2},f)=T((\gamma_{1}\circ\gamma_{2})_{*}f)-\beta(\gamma_{1}\circ\gamma_{2},f)=T^{\gamma_{1}\circ\gamma_{2}}(f)$.
∎

######  Lemma 3.2.

Let $s>3$. ${\mathscr{D}}(L_{0})={\mathscr{D}}(L_{0}^{\gamma})$ for every
$\gamma\in{\mathcal{D}}^{s}(S^{1})$, where $L_{0}^{\gamma}\coloneqq
T^{\gamma}(1)$ and here we denote by $1$ the constant function with the value
$1$.

###### Proof.

By Lemma A.4 we can take a sequence $\\{\gamma_{n}\\}$ in ${\rm
Diff}_{+}(S^{1})$ convergent to $\gamma$ in the topology of
${\mathcal{D}}^{s}(S^{1})$. We observe that
$1=\lim_{n}\gamma_{n*}(\gamma^{-1}_{*}(1))$ in the topology of
$\mathcal{S}_{\frac{3}{2}}(S^{1})$. For $\xi\in{\mathscr{D}}(L_{0})$ we know
from Proposition 2.2(5) and (3.2) that

$\displaystyle\|L_{0}\xi\|$
$\displaystyle=\lim_{n\to\infty}\|\left(T^{\gamma_{n}}((\gamma^{-1}_{*})(1))+\beta(\gamma_{n},\gamma_{*}^{-1}(1))\right)\xi\|$
$\displaystyle\leq\left(\lim_{n\to\infty}r\|\gamma^{-1}_{*}(1)\|_{\frac{3}{2}}\cdot\|(1+L_{0}^{\gamma_{n}})\xi\|+|\beta(\gamma_{n},\gamma_{*}^{-1}(1))|\|\xi\|\right)$
$\displaystyle=r\|\gamma^{-1}_{*}(1)\|_{\frac{3}{2}}\cdot\|(1+L_{0}^{\gamma})\xi\|+|\beta(\gamma,\gamma_{*}^{-1}(1))|\|\xi\|,$

Recall that we know that
${\mathscr{D}}(L_{0})\subset{\mathscr{D}}(L^{\gamma}_{0})$ from Proposition
2.2(4) and $L_{0}^{\gamma}$ is essentially self-adjoint on
${\mathscr{D}}(L_{0})$. From the above inequality, we infer that any sequence
$\xi_{n}\in{\mathscr{D}}(L_{0})$ converging to
$\xi\in{\mathscr{D}}(L_{0}^{\gamma})$ in the graph norm of $L_{0}^{\gamma}$ is
also convergent in the graph norm of $L_{0}$, and therefore, we have
${\mathscr{D}}(L_{0}^{\gamma})={\mathscr{D}}(L_{0})$. ∎

######  Proposition 3.3 (energy bounds for $T^{\gamma}$).

Let $\gamma\in{\mathcal{D}}^{s}(S^{1})$, $s>3$. Then

$\|T^{\gamma}(f)\xi\|\leq r\|f\|_{\frac{3}{2}}\|(1+L_{0}^{\gamma})\xi\|$

for all $\xi\in{\mathscr{D}}(L_{0})$.

###### Proof.

Let $\\{\gamma_{n}\\}$ a sequence of elements in ${\rm Diff}_{+}(S^{1})$
converging to $\gamma\in{\mathcal{D}}^{s}(S^{1})$ as in Lemma A.4. By
Proposition 2.2(5) and (3.2),

$\displaystyle\|T^{\gamma}(f)\xi\|$
$\displaystyle=\lim_{n\to\infty}\|T^{\gamma_{n}}(f)\xi\|\leq\lim_{n\to\infty}r\|f\|_{\frac{3}{2}}\|(1+L_{0}^{\gamma_{n}})\xi\|=$
$\displaystyle=r\|f\|_{\frac{3}{2}}\|(1+L_{0}^{\gamma})\xi\|,$

which is the desired inequality. ∎

######  Theorem 3.4.

Let $\gamma\in{\mathcal{D}}^{s}(S^{1})$, $s>3$. $T^{\gamma}$ yields an
irreducible unitary positive energy representation of ${\rm Vir}$ with central
charge $c$ and lowest weight $h$ on ${\mathcal{H}}(c,h)$.

###### Proof.

We are going to prove the Virasoro relations on $C^{\infty}(L_{0}^{\gamma})$.
For this purpose, we have to take under control the action of various
exponentiated operators.

##### Computations on ${\mathscr{D}}(L_{0})$.

We start by noting that
$e^{iT^{\gamma}(g)}{\mathscr{D}}(L_{0})\subset{\mathscr{D}}(L_{0})$. Indeed,
using [FH05, Proposition 3.1] we have, for $\xi\in{\mathscr{D}}(L_{0})$ and
$\gamma_{n}\in{\rm Diff}_{+}(S^{1})$ as in Lemma A.4,

$L_{0}e^{iT^{\gamma_{n}}(g)}\xi=e^{iT^{\gamma_{n}}(g)}(T((\gamma_{n}{\rm
Exp}(-g)\gamma_{n}^{-1})_{*}(1))-\beta(\gamma_{n}{\rm
Exp}(-g)\gamma_{n}^{-1},1))\xi,$

and the right-hand side converges as $n\rightarrow\infty$ by Proposition
2.2(5). Therefore, since both $e^{iT^{\gamma_{n}}(g)}\xi$ and
$L_{0}e^{iT^{\gamma_{n}}(g)}\xi$ are convergent, it follows that
$e^{iT^{\gamma}(g)}\xi\in{\mathscr{D}}(L_{0})$ and

$L_{0}e^{iT^{\gamma}(g)}\xi=e^{iT^{\gamma}(g)}(T((\gamma{\rm
Exp}(-g)\gamma^{-1})_{*}(1))-\beta(\gamma{\rm Exp}(-g)\gamma^{-1},1))\xi.$

For vectors $\xi\in{\mathscr{D}}(L_{0})$ and $\gamma_{n}\in{\rm
Diff}_{+}(S^{1})$, by Proposition 2.1 we have the operator equality

$e^{iT^{\gamma_{n}}(g)}T^{\gamma_{n}}(f)e^{-iT^{\gamma_{n}}(g)}=T^{\gamma_{n}}({\rm
Exp}(g)_{*}(f))-\left(\frac{c}{24\pi}\int_{S^{1}}\\{{\rm
Exp}(g),z\\}izf(z)dz\right),$

and we saw above that for $\xi\in{\mathscr{D}}(L_{0})$ and $\gamma_{n}\in{\rm
Diff}_{+}(S^{1})$, it holds that
$e^{-iT^{\gamma_{n}}(g)}\xi\in{\mathscr{D}}(L_{0})\subset{\mathscr{D}}(T^{\gamma_{n}}(f))$,
therefore, we have

$e^{iT^{\gamma_{n}}(g)}T^{\gamma_{n}}(f)e^{-iT^{\gamma_{n}}(g)}\xi=T^{\gamma_{n}}({\rm
Exp}(g)_{*}(f))\xi-\left(\frac{c}{24\pi}\int_{S^{1}}\\{{\rm
Exp}(g),z\\}izf(z)dz\right)\xi.$

We apply to the operator equality the function

$h_{k}:s\in\mathbb{R}\rightarrow s\chi_{(-k,k)}$

where $\chi$ is the characteristic function of the interval
$(-k,k)\subset\mathbb{R}.$ By bounded functional calculus, we obtain for any
$\xi\in{\mathscr{D}}(L_{0})$

$\displaystyle
h_{k}(e^{iT^{\gamma_{n}}(g)}T^{\gamma_{n}}(f)e^{-iT^{\gamma_{n}}(g)})\xi$
$\displaystyle=e^{iT^{\gamma_{n}}(g)}h_{k}(T^{\gamma_{n}}(f))e^{-iT^{\gamma_{n}}(g)}\xi,$
(3.3)

and the right-hand side tends to
$e^{iT^{\gamma}(g)}h_{k}(T^{\gamma}(f))e^{-iT^{\gamma}(g)}\xi$ as
$n\rightarrow\infty$, because we have convergence of $T^{\gamma_{n}}(f)$ to
$T^{\gamma}(f)$ and $T^{\gamma_{n}}(g)$ to $T^{\gamma}(g)$ in the strong
resolvent sense, and their bounded functional calculus
$e^{iT^{\gamma_{n}}(g)},h_{k}(T^{\gamma_{n}}(f))$ converge to
$e^{iT^{\gamma}(g)},h_{k}(T^{\gamma_{n}}(f))$, respectively. On the other
hand, the left-hand side of (3.3) can be rewritten as

$h_{k}\left(T^{\gamma_{n}}({\rm
Exp}(g)_{*}(f))-\frac{c}{24\pi}\int_{S^{1}}\\{{\rm
Exp}(g),z\\}izf(z)dz\right)\xi$

and this converges to

$h_{k}\left(T^{\gamma}({\rm Exp}(g)_{*}(f))-\frac{c}{24\pi}\int_{S^{1}}\\{{\rm
Exp}(g),z\\}izf(z)dz\right)\xi$

as $n\rightarrow\infty$, again by the convergence of $\\{T^{\gamma_{n}}({\rm
Exp}(g)_{*}(f))\\}$ in the strong resolvent sense and bounded functional
calculus with $h_{k}$. Altogether, we know that the following equality holds:

$e^{iT^{\gamma}(g)}h_{k}(T^{\gamma}(f))e^{-iT^{\gamma}(g)}\xi=h_{k}\left(T^{\gamma}({\rm
Exp}(g)_{*}(f))-\frac{c}{24\pi}\int_{S^{1}}\\{{\rm
Exp}(g),z\\}izf(z)dz\right)\xi.$

By taking the limit for $k\rightarrow\infty$, we get for every
$\xi\in{\mathscr{D}}(L_{0})$

$\displaystyle
e^{iT^{\gamma}(g)}T^{\gamma}(f)e^{-iT^{\gamma}(g)}\xi=T^{\gamma}({\rm
Exp}(g)_{*}(f))\xi-\left(\frac{c}{24\pi}\int_{S^{1}}\\{{\rm
Exp}(g),z\\}izf(z)dz\right)\xi.$ (3.4)

Recall that ${\mathscr{D}}(L_{0})={\mathscr{D}}(L^{\gamma}_{0})$. We get in
particular

$\displaystyle
e^{itL_{0}^{\gamma}}T^{\gamma}(f)e^{-itL_{0}^{\gamma}}\xi=T^{\gamma}(f_{t})\xi,$
(3.5)

where $f_{t}(e^{i\theta})=f(e^{i(\theta-t)})$.

##### Computations on $C^{\infty}(L_{0}^{\gamma})$.

The right-hand side of (3.5) is differentiable with respect to $t$ when
$\xi\in{\mathscr{D}}(L_{0})$ since for the right hand side we get

$\lim_{t\rightarrow
0}\frac{1}{t}(T^{\gamma}(f_{t})-T^{\gamma}(f))\xi=\lim_{t\rightarrow
0}T^{\gamma}(\textstyle{\frac{1}{t}}(f_{t}-f))\xi=T^{\gamma}(-f^{\prime})\xi=-T^{\gamma}(f^{\prime})\xi,$

by the continuity of $T^{\gamma}$ in the topology of
$\mathcal{S}_{\frac{3}{2}}(S^{1})$ (Proposition 3.3). Let us specialize it to
$\xi\in
C^{\infty}(L_{0}^{\gamma}):=\bigcap_{n}{\mathscr{D}}((L^{\gamma}_{0})^{n})$.
For the left-hand side of (3.5), we have

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}e^{itL^{\gamma}_{0}}T^{\gamma}(f)e^{-itL_{0}^{\gamma}}\xi$
$\displaystyle=\lim_{t\rightarrow\infty}\left(\frac{1}{t}\left(e^{itL^{\gamma}_{0}}T^{\gamma}(f)e^{-itL_{0}^{\gamma}}-e^{itL^{\gamma}_{0}}T^{\gamma}(f)\right)\xi+\frac{1}{t}\left(e^{itL^{\gamma}_{0}}T^{\gamma}(f)-T^{\gamma}(f)\right)\xi\right).$
(3.6)

The first term converges to $-iT^{\gamma}(f)L_{0}\xi$. Indeed, by Proposition
3.3,

$\displaystyle\left\|\frac{1}{t}\left(e^{itL^{\gamma}_{0}}T^{\gamma}(f)e^{-itL_{0}^{\gamma}}-e^{itL^{\gamma}_{0}}T^{\gamma}(f)\right)\xi+ie^{itL^{\gamma}_{0}}T^{\gamma}(f)L_{0}^{\gamma}\xi\right\|$
$\displaystyle=\left\|\frac{1}{t}\left(T^{\gamma}(f)e^{-itL_{0}^{\gamma}}-T^{\gamma}(f)\right)\xi+iT^{\gamma}(f)L_{0}^{\gamma}\xi\right\|$
$\displaystyle\leq
r\|f\|_{\frac{3}{2}}\left\|(1+L_{0}^{\gamma})\left(\frac{e^{-itL_{0}^{\gamma}}-1}{t}+iL^{\gamma}_{0}\right)\xi\right\|$
$\displaystyle=r\|f\|_{\frac{3}{2}}\left\|\left(\frac{e^{-itL_{0}^{\gamma}}-1}{t}+iL^{\gamma}_{0}\right)(1+L^{\gamma}_{0})\xi\right\|.$

Since $\xi\in C^{\infty}(L_{0}^{\gamma})$, by Stone’s theorem [RS80, Theorem
VIII.7(c)] the above converges to 0 as $t\rightarrow 0$. Thus the limit exists
also for the second term of (3.1), and by applying Stone’s theorem [RS80,
Theorem VIII.7(d)], we get
$T^{{\gamma}}(f)\xi\in{\mathscr{D}}(L^{\gamma}_{0})$, and the second term
converges to $iL_{0}^{\gamma}T^{{\gamma}}(f)\xi$. or in other words,
$T^{{\gamma}}(f)C^{\infty}(L_{0})\subset{\mathscr{D}}(L^{\gamma}_{0})$
(actually, we proved
$T^{{\gamma}}(f){\mathscr{D}}((L_{0}^{\gamma})^{2})\subset{\mathscr{D}}(L^{\gamma}_{0})$).
Thus we have established the following commutation relation on
$C^{\infty}(L^{\gamma}_{0})$:

$\displaystyle[L^{\gamma}_{0},T^{\gamma}(f)]\xi=iT^{\gamma}(f^{\prime})\xi.$
(3.7)

It follows that $C^{\infty}(L^{\gamma}_{0})$ is an invariant domain for every
$T^{\gamma}(f)$ with $f\in C^{\infty}(S^{1},\mathbb{R})$. Indeed, for
$T^{\gamma}(f)\xi$, with $\xi\in C^{\infty}(L_{0}^{\gamma})$ and $f\in
C^{\infty}(S^{1},\mathbb{R})$, (3.7) is equivalent to

$\displaystyle
L^{\gamma}_{0}T^{\gamma}(f)\xi=[L^{\gamma}_{0},T^{\gamma}(f)]\xi+T^{\gamma}(f)L_{0}^{\gamma}\xi=iT^{\gamma}(f^{\prime})\xi+T^{\gamma}(f)L_{0}^{\gamma}\xi.$
(3.8)

Now we go by induction in $k$. Assume that
$T^{\gamma}(f)\xi\in{\mathscr{D}}((L_{0}^{\gamma})^{k})$ and all $f\in
C^{\infty}(S^{1},\mathbb{R})$. It then follows from (3.8) that
$L_{0}^{\gamma}T^{\gamma}(f)\xi\in{\mathscr{D}}((L_{0}^{\gamma})^{k})$, i.e.
$T^{\gamma}(f)\xi\in{\mathscr{D}}((L_{0}^{\gamma})^{k+1})$. We thus get the
desired claim $T^{\gamma}(f)C^{\infty}(L^{\gamma}_{0})\subset
C^{\infty}(L^{\gamma}_{0})$.

##### The Virasoro relations.

Finally we show that the stress-energy tensor $T^{\gamma}$ indeed yields a
representation of ${\rm Vect}(S^{1})$. For $\xi\in
C^{\infty}(L_{0}^{\gamma})$,

$\displaystyle\left.\frac{d}{dt}\right|_{t=0}e^{itT^{\gamma}(g)}T^{\gamma}(f)e^{-itT^{\gamma}(g)}\xi$
$\displaystyle=\lim_{t\rightarrow
0}\left(\frac{1}{t}\left(e^{itT^{\gamma}(g)}T^{\gamma}(f)e^{-itT^{\gamma}(g)}-e^{itT^{\gamma}(g)}T^{\gamma}(f)\right)+\frac{1}{t}\left(e^{itT^{\gamma}(g)}T^{\gamma}(f)-T^{\gamma}(f)\right)\right)\xi.$
(3.9)

As for the left-hand side, from (3.4), we obtain
$(T^{\gamma}(g^{\prime}f-gf^{\prime})+c\omega(g,f))\xi$ by (2.2).

Let us see the right-hand side of (3.1) term by term. As for the first term,
we have

$\displaystyle\left\|\frac{1}{t}\left(e^{itT^{\gamma}(g)}T^{\gamma}(f)e^{-itT^{\gamma}(g)}-e^{itT^{\gamma}(g)}T^{\gamma}(f)\right)\xi+e^{itT^{\gamma}(g)}\cdot
iT^{\gamma}(f)T^{\gamma}(g)\xi\right\|$
$\displaystyle=\left\|\frac{1}{t}\left(T^{\gamma}(f)e^{-itT^{\gamma}(g)}-T^{\gamma}(f)\right)\xi+iT^{\gamma}(f)T^{\gamma}(g)\xi\right\|$
$\displaystyle\leq
r\|f\|_{\frac{3}{2}}\left\|(1+L^{\gamma}_{0})\frac{1}{t}\left(e^{-itT^{\gamma}(g)}-1\right)\xi+(1+L^{\gamma}_{0})\cdot
iT^{\gamma}(g)\xi\right\|$ $\displaystyle\leq
r\|f\|_{\frac{3}{2}}\left(\left\|\left(\frac{1}{t}\left(e^{-itT^{\gamma}(g)}-1\right)+iT^{\gamma}(g)\right)\xi\right\|+\left\|\left(\frac{1}{t}L_{0}^{\gamma}\left(e^{-itT^{\gamma}(g)}-1\right)+iL_{0}^{\gamma}T^{\gamma}(g)\right)\xi\right\|\right).$
(3.10)

The first term of (3.1) goes to $0$ by Stone’s theorem [RS80, Theorem
VIII.7(c)]. The second term can be treated by (3.4) and (3.7) as follows:

$\displaystyle\left\|\frac{1}{t}L^{\gamma}_{0}(e^{-itT^{\gamma}(g)}-1)\xi+iL^{\gamma}_{0}T^{\gamma}(g)\xi\right\|$
$\displaystyle=\left\|\frac{1}{t}\left(e^{-itT^{\gamma}(g)}(T^{\gamma}({\rm
Exp}(tg)_{*}(1))-\beta({\rm
Exp}(tg),1))-L^{\gamma}_{0}\right)\xi+i(iT^{\gamma}(g^{\prime})+T^{\gamma}(g)L^{\gamma}_{0})\xi\right\|$
$\displaystyle\leq\left\|\frac{1}{t}(e^{-itT^{\gamma}(g)}T^{\gamma}({\rm
Exp}(tg)_{*}(1))-e^{-itT^{\gamma}(g)}L_{0}^{\gamma})\xi-T^{\gamma}(g^{\prime})\xi\right\|$
$\displaystyle\qquad\qquad+\left\|\frac{1}{t}(e^{-itT^{\gamma}(g)}L^{\gamma}_{0}-L^{\gamma}_{0})\xi+iT^{\gamma}(g)L^{\gamma}_{0}\xi\right\|+\bigg{|}\frac{1}{t}\beta({\rm
Exp}(tg),1)\bigg{|}\|\xi\|.$

each term can be seen to converge to $0$: the first term is done by noting
that $L_{0}^{\gamma}=T^{\gamma}(1)$, continuity of $T^{\gamma}$ (Proposition
3.3), $[g,1]=g^{\prime}$ and unitarity of $e^{-itT^{\gamma}(g)}$. The second
term vanishes by using Stone’s theorem. The last term also converges to zero
by (2.2) and using the fact that $\omega(g,1)=0$. To summarize, the first term
of the right-hand side of (3.1) tends to $-iT^{\gamma}(f)T^{\gamma}(g)$.

The second term of (3.1) is equal to $iT^{\gamma}(g)T^{\gamma}(f)$. Indeed,
since $C^{\infty}(L^{\gamma}_{0})$ is invariant under the action of
$T^{\gamma}(f)$, this follows by Stone’s theorem.

Altogether, we obtained the equality
$i[T^{\gamma}(g),T^{\gamma}(f)]=T^{\gamma}(g^{\prime}f-gf^{\prime})+c\omega(g,f)$
on $C^{\infty}(L_{0}^{\gamma})$, which is the Virasoro commutation relation.

Note that until here we have only used that $T$ is a positive energy
representation of the Virasoro algebra with the central charge $c$ with
diagonalizable $L_{0}$, but not irreducibility. Therefore, one can iterate our
construction for another element in ${\mathcal{D}}^{s}(S^{1})$. In particular,
by taking $\gamma^{-1}$, we obtain by Proposition 3.1

$(T^{\gamma})^{\gamma^{-1}}(f)=T(f).$ (3.11)

We claim that the new representation $T^{\gamma}$ is irreducible and has the
same lowest weight $h$. Indeed, by (3.11), one can approximate $T(f)$ by
$T^{\gamma}(\gamma^{-1}_{n*}f)+\beta(\gamma,(\gamma_{n}^{-1})_{*}(f))$ in the
strong resolvent sense, where $\\{\gamma_{n}\\}\subset{\rm Diff}_{+}(S^{1})$
and $\gamma_{n}\to\gamma$ in the topology of ${\mathcal{D}}^{s}(S^{1})$. As
$\\{e^{iT(f)}:f\in{\rm Vect}(S^{1})\\}$ generates
${\mathcal{B}}({\mathcal{H}}(c,h))$, so does $\\{e^{iT^{\gamma}(f)}:f\in{\rm
Vect}(S^{1})\\}$, and this shows that $T^{\gamma}$ is a irreducible
representation of the Virasoro algebra. Furthermore, the new conformal
Hamiltonian $L^{\gamma}_{0}=T^{\gamma}(1)$ has spectrum which is a subset of
the spectrum of the old conformal Hamiltonian $L_{0}$ since it is obtained as
a limit in the strong resolvent sense of $\\{{\hbox{\rm
Ad\,}}U(\gamma_{n})(L_{0})\\}$ with the same spectrum [RS80, Theorem
VIII.24(a)]. Again by iteration, we have

${\rm sp}\,L_{0}={\rm sp}\,(T^{\gamma})^{\gamma^{-1}}(1)\subset{\rm
sp}\,L^{\gamma}_{0}={\rm sp}\,T^{\gamma}(1)\subset{\rm sp}\,L_{0},$

therefore, all these sets must coincide. In particular, $h$ is the lowest
eigenvalue of $L^{\gamma}_{0}$. ∎

As $T$ and $T^{\gamma}$ are equivalent as irreducible representations of ${\rm
Vect}(S^{1})$ and thus of the Virasoro algebra, there is a unitary intertwiner
$U(\gamma)$, defined up to a scalar such that
$U(\gamma)T(f)=T^{\gamma}(f)U(\gamma)$.

######  Corollary 3.5.

The map $\gamma\mapsto U(\gamma)$ where $\gamma\in{\mathcal{D}}^{s}(S^{1})$,
$s>3$, is a unitary projective representation of ${\mathcal{D}}^{s}(S^{1})$,
i.e. $U(\gamma_{1}\circ\gamma_{2})=U(\gamma_{1})U(\gamma_{2})$ up to a phase
factor.

###### Proof.

We know that for $\gamma_{1},\gamma_{2}\in{\mathcal{D}}^{s}(S^{1})$

$\displaystyle U(\gamma_{1})T(f)$
$\displaystyle=T^{\gamma_{1}}(f)U(\gamma_{1}),$ $\displaystyle
U(\gamma_{2})T(f)$ $\displaystyle=T^{\gamma_{2}}(f)U(\gamma_{2})$

hold for every $f\in{\rm Vect}(S^{1})$. So

$\displaystyle U(\gamma_{1})U(\gamma_{2})T(f)$
$\displaystyle=U(\gamma_{1})T^{\gamma_{2}}(f)U(\gamma_{2})=U(\gamma_{1})(T(\gamma_{2*}f)-\beta(\gamma_{2},f))U(\gamma_{2})=$
$\displaystyle=(T^{\gamma_{1}}(\gamma_{2*}f)-\beta(\gamma_{2},f))U(\gamma_{1})U(\gamma_{2})=$
$\displaystyle=(T((\gamma_{1}\circ\gamma_{2})_{*}f)-\beta(\gamma_{1},\gamma_{2*}f)-\beta(\gamma_{2},f))U(\gamma_{1})U(\gamma_{2}).$

Consequently by the computations of Proposition 3.1

$U(\gamma_{1})U(\gamma_{2})T(f)=T^{\gamma_{1}\circ\gamma_{2}}(f)U(\gamma_{1})U(\gamma_{2}),$

therefore, $U(\gamma_{1}\circ\gamma_{2})=U(\gamma_{1})U(\gamma_{2})$ up to a
phase because we are dealing with irreducible representations of the Virasoro
algebra. ∎

######  Corollary 3.6.

Let $U=U_{(c,h)}$ be the irreducible unitary projective representation of
${\rm Diff}_{+}(S^{1})$ with central charge $c$ and lowest weight $h$. Then
$U$ extends to a strongly continuous irreducible unitary projective
representation of ${\mathcal{D}}^{s}(S^{1})$, $s>3$.

###### Proof.

The only thing that remains to be proven is continuity, namely that the action
$\alpha:{\mathcal{D}}^{s}(S^{1})\rightarrow{\hbox{Aut}}({\mathcal{B}}({\mathcal{H}}(c,h)))$,
$\gamma\mapsto{\hbox{\rm Ad\,}}U(\gamma)$ is pointwise continuous in the
strong operator topology of ${\mathcal{B}}({\mathcal{H}}(c,h))$.

Let $\\{\gamma_{n}\\}\subset{\rm Diff}_{+}(S^{1})$,
$\gamma\in{\mathcal{D}}^{s}(S^{1})$ with $\gamma_{n}\rightarrow\gamma$ in the
topology of ${\mathcal{D}}^{s}(S^{1})$. Then

$\lim_{n\rightarrow\infty}U(\gamma_{n})e^{itT(f)}U(\gamma_{n})^{*}=\lim_{n\rightarrow\infty}e^{itT^{\gamma_{n}}(f)}=e^{itT^{\gamma}(f)}$

where the limit is meant in the strong topology. By taking $f=1$, we obtain
the convergence of $L_{0}^{\gamma_{n}}$ to $L_{0}^{\gamma}$ in the strong
resolvent sense. As they are in the $(c,h)$-representation of the Virasoro
algebra, the lowest eigenprojections $E_{0},E_{0}^{\gamma}$ are one-
dimensional, and it holds that $\lim_{n\to\infty}{\hbox{\rm
Ad\,}}U(\gamma_{n})(E_{0})=E_{0}^{\gamma}$. Let $\Omega,\Omega^{\gamma}$ be
the lowest eigenvectors. By fixing the scalars, we may assume that
$\Omega^{\gamma_{n}}:=U(\gamma_{n})\Omega\to\Omega^{\gamma}$, see the
arguments of Section 2.4.

With this $U(\gamma_{n})$ with fixed phase, the sequence

$U(\gamma_{n})e^{iT(f_{1})}\cdots
e^{iT(f_{k})}\Omega=e^{iT^{\gamma_{n}}(f_{1})}\cdots
e^{iT^{\gamma_{n}}(f_{k})}\Omega^{\gamma_{n}}$

is convergent to $e^{iT^{\gamma}(f_{1})}\cdots
e^{iT^{\gamma}(f_{k})}\Omega^{\gamma}$, because all the operators
$e^{iT^{\gamma_{n}}(f_{1})},\cdots,e^{iT^{\gamma_{n}}(f_{k})}$ are uniformly
bounded and convergent in the strong operator topology. Since vectors of the
form $e^{iT(f_{1})}\cdots e^{iT(f_{k})}\Omega$ span a dense subspace of the
whole Hilbert space ${\mathcal{H}}(c,h)$, together with the uniform
boundedness of $U(\gamma_{n})$, we obtain the convergence of $U(\gamma_{n})$
to $U(\gamma)$ in the strong operator topology.

The claimed continuity is follows from this, because for any
$x\in{\mathcal{B}}({\mathcal{H}})$, ${\hbox{\rm Ad\,}}U(\gamma_{n})(x)$ is
convergent in the strong operator topology, again because $U(\gamma_{n})$ is
uniformly bounded.

∎

######  Corollary 3.7.

Let $U=U_{(c,h)}$ be the irreducible unitary projective representation of
${\rm Diff}_{+}(S^{1})$ with central charge $c$ and lowest weight $h$. Then
$U$ extends to a strongly continuous irreducible unitary projective
representation of ${\rm Diff}_{+}^{k}(S^{1})$ with $k\geq 4$.

###### Proof.

This is an immediate corollary of the continuous embedding ${\rm
Diff}_{+}^{k}(S^{1})\hookrightarrow{\mathcal{D}}^{s}(S^{1})$, $s\leq k$. ∎

######  Remark 3.8.

Our argument for the construction of projective representations of
${\mathcal{D}}^{s}(S^{1})$ can be used to simplify the proof the integrability
of the irreducible unitary positive energy representations of the Virasoro
algebra to strongly continuous projective unitary representations of ${\rm
Diff}_{+}(S^{1})$. Such a proof was first given in by realizing them in the
oscillator algebra [GW85, Section 3, Theorem 4.2]. One can do it now only
within the Virasoro algebra as follows.

Besides the energy-bounds (a priori estimates) in [GW85, Section 2], see also
[BSM90], which are used in [CW05] and are crucial to our proof, we also used
(3.1) coming from [GW85]. More precisely, we used the fact that for every
$\gamma\in{\rm Diff}_{+}(S^{1})$ there is an unitary operator $U(\gamma)$ such
that $U(\gamma)T(f)U(\gamma)^{*}=T^{\gamma}(f)$ for all $f\in{\rm
Vect}(S^{1})$ and $U(\gamma){\mathscr{D}}(L_{0})={\mathscr{D}}(L_{0})$. This
can be proved directly following the strategy in pages 1100-1101 of [CKL08],
see also the proof of [CKLW18, Proposition 6.4]. One only needs some of the
direct consequences of the energy bounds proved in [TL99, Section 2]. We
outline the arguments here:

  * •

Since ${\rm Diff}_{+}(S^{1})$ is simple [Mil84, Remark 1.7], it is generated
by exponentials, because the subgroup generated by exponentials is a normal
subgroup.

  * •

By the proof of Corollary 3.5, the set of $\gamma$ such that an unitary
$U(\gamma)$ with the required properties exists forms a subgroup of ${\rm
Diff}_{+}(S^{1})$. Hence, it is enough to consider the special case where
$\gamma={\rm Exp}(g)$ for $g\in{\rm Vect}(S^{1})$.

  * •

It follows from the linear energy-bounds by [TL99, Proposition 2.1] that
$e^{itT(g)}{\mathscr{D}}(L_{0}^{k})={\mathscr{D}}(L_{0}^{k})$ for all positive
integers $k$ and all $t\in\mathbb{R}$. As a consequence
$e^{itT(g)}C^{\infty}(L_{0})=C^{\infty}(L_{0})$ for all $t\in\mathbb{R}$.

  * •

Now, let $\xi\in C^{\infty}(L_{0})$ and let $\xi(t)=T^{{\rm
Exp}(tg)}(f)e^{itT(g)}\xi$. By [TL99, Corollary 2.2] we have
$\frac{d}{dt}e^{itT(g)}\xi=ie^{itT(g)}T(g)\xi$ in the graph topology of
${\mathscr{D}}(L_{0}^{k})$ for all positive integers $k$. It then follows from
the energy bounds that $\frac{d}{dt}\xi(t)=iT(g)\xi(t)$. Hence,
$\xi(t)=e^{itT(g)}T(f)\xi$ for all $\xi\in C^{\infty}(L_{0})$ so that $T^{{\rm
Exp}(tg)}(f)=e^{itT(g)}T(f)e^{-itT(g)}$ which is the required relation.
Continuity of $U$ follows as in Corollary 3.6.

###  3.2 Direct sum of irreducible representations

Here we prove that every positive energy projective unitary representation of
${\rm Diff}_{+}(S^{1})$ extends to a unitary projective representation of
${\mathcal{D}}^{s}(S^{1})$ for $s>3$. A similar result holds for the universal
covering groups provided that the representation is assumed to be a direct sum
of irreducibles. This is not an immediate consequence of Corollary 3.6,
because, in general, the direct sum of projective representations does not
make sense: ${\mathcal{U}}({\mathcal{H}}_{j})/{\mathbb{C}}$ is not a linear
space. On the other hand, if we have multiplier representations of a group $G$
with the same cocycle,
$U_{j}(g_{1})U_{j}(g_{2})=\omega(g_{1},g_{2})U_{j}(g_{1}g_{2})$ where
$\omega(g_{1},g_{2})$ is a 2-cocycle $H^{2}(G,{\mathbb{C}})$ of G, then the
direct sum $\bigoplus_{j}U_{j}(g)$ is again a multiplier representation with
the same cocycle $\omega$. If we are interested in a projective representation
of a certain quotient $G/H$ by a normal subgroup $H$ we have to make sure that
the direct sum $\bigoplus U_{j}(h)$ reduces to a scalar when $h\in H$.

##### Continuous fragmentation of $\widetilde{{\mathcal{D}}^{s}(S^{1})}$.

Let $I$ be a proper open interval of $S^{1}$ and $I^{\prime}=(S^{1}\setminus
I)^{\circ}$ the interior of its complement. We denote by $\overline{I}$ the
closure of $I$. ${\rm Diff}_{+}(I)$ (resp. ${\mathcal{D}}^{s}(I)$) denotes
the subgroup of diffeomorphisms ${\rm Diff}_{+}(S^{1})$ (resp.
${\mathcal{D}}^{s}(S^{1})$) such that $\gamma(x)=x$ for $x\in I^{\prime}$. We
also say that $\gamma\in{\rm Diff}_{+}(I)$ (resp.
$\gamma\in{\mathcal{D}}^{s}(I)$) is supported in $I$.

Let $\\{I_{j}\\}_{j=1,2,3}$ be a cover of the unit circle as Fig. 1. Let us
name the end points of the intervals: $I_{k}=(a_{k},b_{k})$. We also take a
slightly smaller interval $\hat{I}_{k}=(\hat{a}_{k},\hat{b}_{k})\subset I_{k}$
which still consist a cover of $S^{1}$ points $\breve{a}_{1},\breve{b}_{1}$,
c.f. [DFK04]. Furthermore, we take $\hat{b}_{2},\check{b}_{2}$ such that
$\hat{a}_{1}<\hat{b}_{2}<\check{b}_{2}<b_{2}$.

$I_{1}$$I_{2}$$I_{3}$

$I_{1}$$($$)$$\hat{I}_{1}$$($$)$$a_{1}$$b_{1}$$\breve{a}_{1}$$\hat{a}_{1}$$\hat{b}_{1}$$\breve{b}_{1}$$I_{2}$$b_{2}$$($$)$$I_{3}$$a_{3}$

Figure 1: The covering of the unit circle.

Any given diffeomorphism $\gamma$ can be written as a product of elements
supported in $I_{k}$. This is known as fragmentation (see [Man15] and
references therein). We need a slightly refined version of it, namely, if
$\gamma$ is in a small neighborhood ${\mathcal{V}}$ of the unit element
$\iota$, then we can take the fragments $\gamma_{k}$ also in a small, but
larger neighborhood $\hat{\mathcal{V}}$. The precise statement is the
following.

######  Lemma 3.9.

Let $s>3/2$. There is a neighborhood ${\mathcal{V}}$ of the unit element
$\iota$ of $\widetilde{{\mathcal{D}}^{s}(S^{1})}$ and continuous localizing
maps $\chi_{k}:{\mathcal{V}}\to$ $\widetilde{{\mathcal{D}}^{s}(I_{k})}$ with

$\gamma=\chi_{1}(\gamma)\chi_{2}(\gamma)\chi_{3}(\gamma)$

and $\chi_{k}(\iota)=\iota$, ${\rm supp\,}\chi_{k}(\gamma)\subset I_{k}$,
where ${\rm supp\,}\gamma:=\overline{\\{\theta\in
S^{1}:\gamma(\theta)\neq\theta\\}}$. If ${\rm
supp\,}\gamma\subset\breve{I}_{k}\cup\breve{I}_{k+1}$, then
$\chi_{k+2}(\gamma)=\iota$, where $k=1,2,3\mod 3$.

###### Proof.

We may assume without loss of generality that
$0<a_{1}<\breve{a}_{1}<\hat{a}_{1}<b_{2}<a_{3}<\hat{b}_{1}<\breve{b}_{1}<b_{1}<2\pi$,
(see Figure 1).

Let us take a smooth $2\pi$-periodic function $D_{\mathrm{c},1}$ with
$D_{\mathrm{c},1}(t)=1$ for $t\in\hat{I}_{1}=[\hat{a}_{1},\hat{b}_{1}]$ and
$D_{\mathrm{c},1}(t)=0$ for $t\in[0,\breve{a}_{1}]\cup[\breve{b}_{1},2\pi]$
and $0\leq D_{\mathrm{c},1}(t)\leq 1$ everywhere. Let $0\leq
D_{\mathrm{l},1}(t)\leq 1$ be another smooth $2\pi$-periodic function with
support in $(a_{1},\breve{a}_{1})$ and with
$\int_{0}^{2\pi}D_{\mathrm{l},1}(t)dt=\int_{a_{1}}^{\breve{a}_{1}}D_{\mathrm{l},1}(t)dt=\frac{1}{2}(\breve{a}_{1}-a_{1})$
(which is possible because the interval $(a_{1},\breve{a}_{1})$ is longer than
$\frac{1}{2}(a_{1},\breve{a}_{1})$). Similarly, let $0\leq
D_{\mathrm{r},1}(t)\leq 1$ be a smooth $2\pi$-periodic function with support
in $(\breve{b}_{1},b_{1})$ and with
$\int_{0}^{2\pi}D_{\mathrm{r},1}(t)dt=\frac{1}{2}(b_{1}-\breve{b}_{1})$.

We consider the following neighborhood of the unit element of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$

${\mathcal{V}}_{\varepsilon}\coloneqq\left\\{\gamma\in\widetilde{{\mathcal{D}}^{s}(S^{1})}:|\gamma(\theta)-\iota(\theta)|<\varepsilon,|\gamma^{\prime}(\theta)-1|<\varepsilon\;\text{
for }\theta\in[0,2\pi]\right\\}.$

Note that ${\mathcal{V}}_{\varepsilon}$ is open since $s>3/2$, by the Sobolev-
Morrey embedding theorem.

Suppose $\gamma\in{\mathcal{V}}_{\varepsilon}$. We set

$\displaystyle M$
$\displaystyle:=\max\left\\{D_{\mathrm{c},1}(t),t\in[0,2\pi]\right\\}$

and define the constant $\alpha(\gamma)$ by

$\displaystyle\alpha_{1}(\gamma)=\frac{2}{\breve{a}_{1}-a_{1}}\left(\gamma(\hat{a}_{1})-\hat{a}_{1}-\int_{0}^{\hat{a}_{1}}(\gamma^{\prime}(t)-1)D_{\mathrm{c},1}(t)dt\right).$
(3.12)

It follows that

$|\alpha_{1}(\gamma)|\leq\frac{2}{|\breve{a}_{1}-a_{1}|}\varepsilon(1+\hat{a}_{1}M)$
(3.13)

by the definition of ${\mathcal{V}}_{\varepsilon}$ and

$\gamma(\hat{a}_{1})=\int_{0}^{\hat{a}_{1}}((\gamma^{\prime}(t)-1)D_{\mathrm{c},1}(t)+1+\alpha_{1}(\gamma)D_{\mathrm{l},1}(t))dt.$

Similarly, set the constant $\beta_{1}(\gamma)$ by

$\displaystyle\beta_{1}(\gamma)$
$\displaystyle\;\;=\frac{-2}{b_{1}-\breve{b}_{1}}\left(\int_{0}^{2\pi}((\gamma^{\prime}(t)-1)D_{\mathrm{c},1}(t)+\alpha_{1}(\gamma)D_{\mathrm{l},1}(t))dt\right)$
(3.14)
$\displaystyle\left(=\frac{2}{b_{1}-\breve{b}_{1}}\left(\hat{b}_{1}-\gamma(\hat{b}_{1})-\int_{\hat{b}_{1}}^{b_{1}}(\gamma^{\prime}(t)-1)D_{\mathrm{c},1}(t)\right)\right),$

then it follows that

$\displaystyle|\beta_{1}(\gamma)|\leq\frac{2}{|b_{1}-\breve{b}_{1}|}\varepsilon(|\hat{b}_{1}-b_{1}|M+1)$
(3.15)

and

$\displaystyle
b_{1}=\int_{0}^{b_{1}}((\gamma^{\prime}(t)-1)D_{\mathrm{c},1}(t)+1+\alpha_{1}(\gamma)D_{\mathrm{l},1}(t)+\beta_{1}(\gamma)D_{\mathrm{r},1}(t))dt.$

Now, the function

$\displaystyle\gamma_{1}(\theta)=\int_{0}^{\theta}((\gamma^{\prime}(t)-1)D_{\mathrm{c},1}(t)+1+\alpha_{1}(\gamma)D_{\mathrm{l},1}(t)+\beta_{1}(\gamma)D_{\mathrm{r},1}(t))dt$
(3.16)

is $2\pi$-periodic, the first derivative

$\displaystyle\gamma^{\prime}_{1}(\theta)=(\gamma^{\prime}(\theta)-1)D_{\mathrm{c},1}(\theta)+1+\alpha_{1}(\gamma)D_{\mathrm{l},1}(\theta)+\beta_{1}(\gamma)D_{\mathrm{r},1}(\theta)$

is positive by (3.13), (3.15) if $\varepsilon$ is taken sufficiently small and
$\gamma_{1}^{\prime}-1\in H^{s-1}(S^{1})$ (by Lemma 2.4, using that
$\gamma-\iota\in H^{s}$), therefore, $\gamma_{1}$ can be regarded as an
element in $\widetilde{D^{s}(S^{1})}$. It also has the desired properties,
namely $\gamma_{1}(\theta)=\theta$ for $\theta\in I_{1}^{\prime}$ and
$\gamma_{1}(\theta)=\gamma(\theta)$ for $\theta\in\hat{I}_{1}$. Note that the
assignment
${\mathcal{V}}_{\varepsilon}\rightarrow\widetilde{{\mathcal{D}}^{s}(S^{1})}$,
$\gamma\rightarrow\gamma_{1}$ is continuous by (3.16)(3.12)(3.14) and Lemma
A.1.

We choose $\varepsilon$ such that $\gamma_{1}^{\prime}$ is positive for
$\gamma\in{\mathcal{V}}_{\varepsilon}$. Now the assignment
${\mathcal{V}}_{\varepsilon}\rightarrow\widetilde{{\mathcal{D}}^{s}(S^{1})}$,
$\gamma\rightarrow\gamma\gamma_{1}^{-1}$ is continuous by Lemma 2.5. We take
${\mathcal{V}}\subset{\mathcal{V}}_{\varepsilon}$ to be the neighborhood of
the identity of $\widetilde{D^{s}(S^{1})}$ such that for
$\gamma\in{\mathcal{V}}$ we have
$\gamma\gamma_{1}^{-1}\in{\mathcal{V}}_{\varepsilon_{1}}$ where
$\varepsilon_{1}$ is small enough that we obtain
$\gamma_{2}\in\widetilde{D^{s}(S^{1})}$ (in particular $\gamma_{2}^{\prime}$
is positive) if we do an analogous construction on $I_{2}$ for
$\gamma\gamma_{1}^{-1}$.

For $\gamma\in{\mathcal{V}}$ we set $\chi_{1}(\gamma)=\gamma_{1}$. The
continuity of the map $\chi_{1}$ in the topology of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ is clear from (3.16) and (3.12)(3.14).

Next we construct $\chi_{2}(\gamma)$. By construction
$(\gamma\gamma_{1}^{-1})(\theta)=\theta$ for $\theta\in\hat{I}_{1}$, therefore
, ${\rm supp\,}\gamma\gamma_{1}^{-1}\subset I_{2}\cup I_{3}$. We can apply an
analogous construction to $I_{2}$ and $\gamma\gamma_{1}^{-1}$ to obtain
$\gamma_{2}$ such that ${\rm
supp\,}\gamma_{2}\subset\hat{I}_{2},\gamma_{2}(\theta)=(\gamma\gamma_{1}^{-1})(\theta)$
for $\theta\in\hat{I}_{2}$. In this way we obtain the continuous map
$\chi_{2}(\gamma):=\gamma_{2}$. Furthermore, by our choice
$\hat{a}_{1}<\hat{b}_{2}<\check{b}_{2}<b_{2}$,
$\gamma_{2}(\theta)=(\gamma\gamma_{1}^{-1})(\theta)$ for
$\theta\in\hat{I}_{1}$ where both are equal to $\theta$, hence for
$\hat{I}_{1}\cup\hat{I}_{2}$.

Now we have $(\gamma\gamma_{1}^{-1}\gamma_{2}^{-1})(\theta)=\theta$ for
$\theta\in\hat{I}_{1}\cup\hat{I}_{2}$, and as $\\{\hat{I}_{k}\\}$ is a cover
of $S^{1}$, $(\hat{I}_{1}\cup\hat{I}_{2})^{\prime}\subset\hat{I}_{3}$.
Therefore, if we set $\chi_{3}(\gamma)=\gamma\gamma_{1}^{-1}\gamma_{2}^{-1}$,
it is supported in $\hat{I}_{3}\subset I_{3}$ and the map $\chi_{3}$ is
continuous because it is a composition of continuous maps (Lemma 2.5). ∎

If $\gamma$ is already localized, we can have the following improvement.

######  Lemma 3.10.

Let $k\in\\{1,2,3\\}\mod 3$ and $\tilde{I}_{k}=I_{k}\cup I_{k+1}$. There is a
neighborhood ${\mathcal{V}}$ of the unit element $\iota$ of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$, $s>3/2$, and continuous localizing
maps

$\displaystyle\chi^{(k)}_{k}$
$\displaystyle:{\mathcal{V}}\cap\widetilde{{\mathcal{D}}^{s}(\tilde{I}_{k})}\to\widetilde{{\mathcal{D}}^{s}(I_{k})},$
$\displaystyle\chi^{(k)}_{k+1}$
$\displaystyle:{\mathcal{V}}\cap\widetilde{{\mathcal{D}}^{s}(\tilde{I}_{k})}\to\widetilde{{\mathcal{D}}^{s}(I_{k+1})}$

with $\gamma=\chi^{(k)}_{k}(\gamma)\chi^{(k)}_{k+1}(\gamma)$ and
$\chi^{(k)}_{k}(\iota)=\chi^{(k)}_{k+1}(\iota)=\iota$.

###### Proof.

Without loss of generality, we may assume $k=2$. This is done by applying the
steps of constructing $\chi_{2}$ and $\chi_{3}$ in the proof of Lemma 3.9 to
slightly enlarged $I_{2}$ and $\hat{I}_{2}$, so that
$\chi^{(2)}_{2}(\gamma)(\theta)=\gamma(\theta)$ for $\theta\in
I_{3}^{\prime}$. ∎

######  Lemma 3.11.

Let $U_{(c,h_{1})},U_{(c,h_{2})}$ be irreducible, projective representations
of $\widetilde{{\mathcal{D}}^{s}(S^{1})}$ with central charge $c$ and lowest
weight $h_{1},h_{2}$ respectively, constructed as in Section 3. Let $I$ be a
proper interval of $S^{1}$. Then the projective representations
$U_{(c,h_{1})}$ and $U_{(c,h_{2})}$ restricted to ${\mathcal{D}}^{s}(I)$ are
unitarily equivalent. Furthermore, a unitary $U$ intertwines $U_{(c,h_{1})}$
and $U_{(c,h_{2})}$ restricted to ${\mathcal{D}}^{s}(I)$ if and only if it
intertwines $T_{(c,h_{1})}(f)$ and $T_{(c,h_{2})}(f)$ for every $f\in{\rm
Vect}(S^{1})$ with support in $I$.

###### Proof.

Let $\tilde{I}$ an open proper interval of $S^{1}$ such that
$\tilde{I}\supset\overline{I}$. By [Wei17, Theorem 5.6] there exists a unitary
$W$ which intertwines the representations $U_{(c,h_{1})},U_{(c,h_{2})}$ when
restricted to ${\rm Diff}_{+}(\tilde{I})$. Let
$\gamma\in{\mathcal{D}}^{s}(I)$, then by Lemma A.4 there exists a sequence of
$C^{\infty}$-diffeomorphisms $\\{\gamma_{n}\\}\subset{\rm
Diff}_{+}(\tilde{I})$ converging to $\gamma$. By Corollary 3.6,

$\displaystyle{\hbox{\rm Ad\,}}WU_{(c,h_{1})}(\gamma)W^{*}$
$\displaystyle={\hbox{\rm
Ad\,}}\lim_{n\rightarrow\infty}WU_{(c,h_{1})}(\gamma_{n})W^{*}={\hbox{\rm
Ad\,}}\lim_{n\rightarrow\infty}U_{(c,h_{2})}(\gamma_{n})={\hbox{\rm
Ad\,}}U_{(c,h_{2})}(\gamma).$

The last assertion follows from [Wei17, Lemma 2.1]. ∎

We are going to show that we can take the direct sum of irreducible projective
representations of ${\mathcal{D}}^{s}(S^{1})$, $\\{U_{(c,h_{j})}\\}$, with the
same central charge $c$ but possibly different lowest weights $\\{h_{j}\\}$
where differences $h_{j}-h_{j^{\prime}}$ are integers. We split the proof into
two steps. First, we make $U_{(c,h_{j})}$ into continuous multiplier
representations with the same cocycle in some neighborhood ${\mathcal{V}}$ of
the identity diffeomorphism $\iota\in\widetilde{{\mathcal{D}}^{s}(S^{1})}$.
Then it is straightforward to take the direct sum. Next, we show that the
direct sum representation reduced to a projective representation of
${\mathcal{D}}^{s}(S^{1})$ if the differences $h_{j}-h_{j^{\prime}}$ are
integers.

Let $G$ and $G^{\prime}$ be two topological groups. Given a neighborhood
${\mathcal{V}}$ of the identity in $G$, a continuous map
$\mu:{\mathcal{V}}\rightarrow G^{\prime}$ is a local homomorphism if
$\mu(g_{1})\mu(g_{2})=\mu(g_{1}g_{2})$ for all $g_{1},g_{2}\in{\mathcal{V}}$
and $g_{1}g_{2}\in{\mathcal{V}}$.

We say that a map $U$ is a local unitary multiplier representation of a
topological group $G$ on a neighborhood ${\mathcal{V}}$ of the identity if $U$
is a map from ${\mathcal{V}}$ to the unitary group
${\mathcal{U}}({\mathcal{H}})$ of a Hilbert space ${\mathcal{H}}$ which
satisfies the equality $U(g_{1})U(g_{2})=\omega(g_{1},g_{2})U(g_{1}g_{2})$,
where $\omega:{\mathcal{V}}\times{\mathcal{V}}\rightarrow\mathbb{T}$ and
$\omega(g_{1},g_{2})\omega(g_{1}g_{2},g_{3})=\omega(g_{1},g_{2}g_{3})\omega(g_{2},g_{3})$
whenever $g_{1},g_{2},g_{3}$, $g_{1}g_{2}$ and $g_{2}g_{3}$ are in
${\mathcal{V}}$. The following is obtained by reversing the idea of [Tan18].

######  Proposition 3.12.

Let $s>3$. For a family $\\{(c,h_{j})\\}$ of pairs with the same central
charge $c$, there is a neighborhood ${\mathcal{V}}$ of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ such that the irreducible unitary
projective representations $U_{(c,h_{j})}$ lift to local multiplier
representations of ${\mathcal{V}}$ with the same cocycle $c(\cdot,\cdot)$.

###### Proof.

Let us take $h_{1}$. By [Bar54][Mor17, Proposition 12.44], in a neighborhood
$\hat{{\mathcal{V}}}$ of the identity $\iota\in\widetilde{{\rm
Diff}_{+}^{4}(S^{1})}$, $U_{(c,h_{1})}$ lifts to a continuous multiplier
representation, with some continuous cocycle $c(\cdot,\cdot)$, which we will
denote by $U_{1}$.

Because $\widetilde{{\mathcal{D}}^{s}(S^{1})}$ is a topological group, and by
Lemmas 3.9, 3.10, for each neighborhood ${\mathcal{W}}$, there is a smaller
neighborhood $p({\mathcal{W}})$ such that
$p({\mathcal{W}})^{2}\subset{\mathcal{W}}$ and
$\chi_{k}(\gamma),\chi^{(k)}_{k}(\gamma),\chi^{(k)}_{k+1}(\gamma)\subset{\mathcal{W}}$
for $\gamma\in p({\mathcal{W}})$. We take
${\mathcal{V}}=p^{11}(\hat{\mathcal{V}})=\underset{11\text{-times}}{\underbrace{p(p(p(\cdots\hat{\mathcal{V}}\cdots)))}}$.

##### Construction of multiplier representations $U_{j}$.

We show that we can take $U_{j}$ with the same cocycle $c(\cdot,\cdot)$.

We fix a covering $\\{I_{k}\\}$ of $S^{1}$ as in Lemma 3.9. For $\gamma\in
p(\hat{\mathcal{V}})$, we define $U_{j}$ as follows: By Lemma 3.11, there are
unitary intertwiners $\\{V_{j,k}\\}$ between $U_{(c,h_{1})}$ and
$U_{(c,h_{j})}$ restricted to ${\mathcal{D}}^{s}(I_{k})$. We set

$U_{j}(\chi_{k}(\gamma))={\hbox{\rm Ad\,}}V_{j,k}(U_{1}(\gamma_{k})),$

which makes sense because $p(\hat{\mathcal{V}})\subset\hat{\mathcal{V}}$. Note
that $U_{j}(\chi_{k}(\gamma))$ does not depend on the choice of unitary
intertwiner $V_{j,k}$, since, if $V_{j,k}$ and $\hat{V}_{j,k}$ are both
unitary intertwiners, then by Lemma 3.11

${\hbox{\rm
Ad\,}}V_{j,k}^{*}\hat{V}_{j,k}(U_{j}(\chi_{k}(\gamma)))=U_{j}(\chi_{k}(\gamma))$

for $\gamma$ smooth, and by continuity of $U_{1}$ for
$\chi_{k}(\gamma)\in{\mathcal{D}}^{s}(I_{k})\cap\hat{\mathcal{V}}$.

Let us denote $\gamma_{k}=\chi_{k}(\gamma)$ for simplicity. Now, since
$\gamma=\gamma_{1}\gamma_{2}\gamma_{3}$ with
$\gamma_{k}\in{\mathcal{D}}^{s}(I_{k})\cap\hat{\mathcal{V}}$, we can define
$U_{j}(\gamma)$ by

$\displaystyle
U_{j}(\gamma)=U_{j}(\gamma_{1})U_{j}(\gamma_{2})U_{j}(\gamma_{3})c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1},$
(3.17)

and note that the corresponding equation holds for $U_{1}$.

##### Well-definedness.

We used a particular set of maps $\chi_{k}$ to define $U_{j}$, but actually
they do not depend on the choice of such map $\chi_{k}$ if $\gamma$ satisfies
certain properties and is sufficiently close to $\iota$. Namely, we take two
decompositions
$\gamma=\gamma_{1}\gamma_{2}\gamma_{3}=\gamma^{\prime}_{1}\gamma^{\prime}_{2}\gamma^{\prime}_{3}$
where $\gamma_{k},\gamma_{k}^{\prime}\in{\mathcal{D}}^{s}(I_{k})\cap
p^{5}(\hat{\mathcal{V}})$.

It holds that
$\gamma_{3}^{-1}\gamma_{2}^{-1}\gamma_{1}^{-1}\gamma^{\prime}_{1}\gamma^{\prime}_{2}\gamma^{\prime}_{3}=\iota$
in $\widetilde{{\mathcal{D}}^{s}(S^{1})}$ and
$U_{1}(\gamma_{1})^{*}=c(\gamma_{1},\gamma_{1}^{-1})U_{1}(\gamma_{1}^{-1})$,
hence we have

$c(\gamma_{1},\gamma_{2},\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime}_{2},\gamma^{\prime}_{3}):=U_{1}(\gamma_{3})^{*}U_{1}(\gamma_{2})^{*}U_{1}(\gamma_{1}^{-1}\gamma^{\prime}_{1})U_{1}(\gamma^{\prime}_{2})U_{1}(\gamma^{\prime}_{3})\in\mathbb{C}.$

Furthermore, as $U_{1}$ is a multiplier representation in $\hat{\mathcal{V}}$,
we have

$\displaystyle U_{1}(\gamma)$
$\displaystyle=U_{1}(\gamma_{1})U_{1}(\gamma_{2})U_{1}(\gamma_{3})c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}$
$\displaystyle=U_{1}(\gamma^{\prime}_{1})U_{1}(\gamma^{\prime}_{2},)U_{1}(\gamma^{\prime}_{3})c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma_{2}^{\prime},\gamma^{\prime}_{3})^{-1}.$

By putting all factors in one side, we obtain

$\displaystyle
c(\gamma_{1},\gamma_{2},\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime}_{2},\gamma^{\prime}_{3})c(\gamma_{1}^{-1},\gamma^{\prime}_{1})c(\gamma_{1},\gamma_{1}^{-1})c(\gamma_{1},\gamma_{2})c(\gamma_{1}\gamma_{2},\gamma_{3})c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma_{2}^{\prime},\gamma^{\prime}_{3})^{-1}=1.$
(3.18)

Note that $U_{j}$ is unitarily equivalent to $U_{1}$ on any proper interval,
therefore,
$U_{j}(\gamma_{1})^{*}U_{j}(\gamma^{\prime}_{1})=c(\gamma_{1}^{-1},\gamma^{\prime}_{1})c(\gamma_{1},\gamma_{1}^{-1})U_{j}(\gamma_{1}^{-1}\gamma^{\prime}_{1})$,
and
$\gamma_{1}^{-1}\gamma^{\prime}_{1}=\gamma_{2}\gamma_{3}\gamma_{3}^{\prime-1}\gamma_{2}^{\prime-1}$
has support in $I_{2}\cup I_{3}$. Then we can again use the unitary
equivalence between $U_{j}$ and $U_{1}$ on $I_{2}\cup I_{3}$ to obtain

$U_{j}(\gamma_{3})^{*}U_{j}(\gamma_{2})^{*}U_{j}(\gamma_{1}^{-1}\gamma^{\prime}_{1})U_{j}(\gamma^{\prime}_{2})U_{j}(\gamma^{\prime}_{3})=c(\gamma_{1},\gamma_{2},\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime}_{2},\gamma^{\prime}_{3}),$

which is, by (3.18), equivalent to the equality

$\displaystyle
U_{j}(\gamma_{1})U_{j}(\gamma_{2})U_{j}(\gamma_{3})c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}$
$\displaystyle=\;$ $\displaystyle
U_{j}(\gamma^{\prime}_{1})U_{j}(\gamma^{\prime}_{2})U_{j}(\gamma^{\prime}_{3})c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma_{2}^{\prime},\gamma^{\prime}_{3})^{-1}.$

In other words, $U_{j}$ is well-defined on $p^{6}(\hat{\mathcal{V}})$.

##### Cocycle relations.

Next we show that $U_{j}$ is a local multiplier representation on
${\mathcal{V}}$. Let
$\gamma,\gamma^{\prime}\in{\mathcal{V}}=p^{11}(\hat{\mathcal{V}})$ and we take
decompositions
$\gamma=\gamma_{1}\gamma_{2}\gamma_{3},\gamma^{\prime}=\gamma^{\prime}_{1}\gamma^{\prime}_{2}\gamma^{\prime}_{3}$.
We first look at the product $\gamma_{3}\gamma^{\prime}_{1}$. This is
supported in $I_{1}\cup I_{3}$, and we can find another decomposition
$\gamma_{3}\gamma^{\prime}_{1}=\gamma^{\prime\prime}_{1}\gamma^{\prime\prime}_{3}$
using Lemma 3.10, where
$\gamma^{\prime\prime}_{j}\in{\mathcal{D}}^{s}(I_{j})\cap
p^{8}(\hat{\mathcal{V}})$. By repeating such operations and taking new
decompositions in proper intervals, we find

$\displaystyle\gamma\gamma^{\prime}$
$\displaystyle=\gamma_{1}\gamma_{2}\gamma_{3}\gamma^{\prime}_{1}\gamma^{\prime}_{2}\gamma^{\prime}_{3}$
$\displaystyle=\gamma_{1}\gamma_{2}\gamma^{\prime\prime}_{1}\gamma^{\prime\prime}_{3}\gamma^{\prime}_{2}\gamma^{\prime}_{3}$
$\displaystyle=\gamma_{1}\gamma^{\prime\prime\prime}_{1}\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3},$

where $\gamma_{j}^{(k)}\in{\mathcal{D}}^{s}(I_{j})\cap
p^{6}(\hat{\mathcal{V}})$.

Again, by considering the multiplier representation $U_{1}$, we can prove the
following relations

$\displaystyle\begin{array}[]{rl}U_{1}(\gamma_{3})U_{1}(\gamma^{\prime}_{1})&=U_{1}(\gamma^{\prime\prime}_{1})U_{1}(\gamma^{\prime\prime}_{3})c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3}),\\\
U_{1}(\gamma_{2})U_{1}(\gamma^{\prime\prime}_{1})&=U_{1}(\gamma^{\prime\prime\prime}_{1})U_{1}(\gamma^{\prime\prime\prime}_{2})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}),\\\
U_{1}(\gamma^{\prime\prime}_{3})U_{1}(\gamma^{\prime}_{2})&=U_{1}(\gamma^{\prime\prime\prime\prime}_{2})U_{1}(\gamma^{\prime\prime\prime\prime}_{3})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3}),\end{array}$
(3.22)

where
$c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3}),c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}),c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3})\in\mathbb{C}$
are defined through these equalities. Therefore, as $U_{1}$ has the cocycle
$c$,

$\displaystyle c(\gamma,\gamma^{\prime})U_{1}(\gamma\gamma^{\prime})$
$\displaystyle\,=U_{1}(\gamma)U_{1}(\gamma^{\prime})$
$\displaystyle\begin{array}[]{l}=c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}\\\
\quad\times
U_{1}(\gamma_{1})U_{1}(\gamma_{2})U_{1}(\gamma_{3})U_{1}(\gamma^{\prime}_{1})U_{1}(\gamma^{\prime}_{2})U_{1}(\gamma^{\prime}_{3})\end{array}$
$\displaystyle\text{ by }\eqref{eq:defpi}$
$\displaystyle\begin{array}[]{l}=c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}\\\
\quad\times
U_{1}(\gamma_{1})U_{1}(\gamma^{\prime\prime\prime}_{1})U_{1}(\gamma^{\prime\prime\prime}_{2})U_{1}(\gamma^{\prime\prime\prime\prime}_{2})U_{1}(\gamma^{\prime\prime\prime\prime}_{3})U_{1}(\gamma^{\prime}_{3})\\\
\quad\times
c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3})\end{array}$
$\displaystyle\text{ by }\eqref{eq:defc4}$
$\displaystyle\begin{array}[]{l}=c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}\\\
\quad\times
c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3})\\\
\quad\times
c(\gamma_{1},\gamma^{\prime\prime\prime}_{1})c(\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\cdot
U_{1}(\gamma_{1}\gamma^{\prime\prime\prime}_{1})U_{1}(\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})U_{1}(\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\end{array}$
$\displaystyle\begin{array}[]{l}=c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}\\\
\quad\times
c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3})\\\
\quad\times
c(\gamma_{1},\gamma^{\prime\prime\prime}_{1})c(\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\cdot
c(\gamma_{1}\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma_{1}\gamma^{\prime\prime\prime}_{1}\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})U_{1}(\gamma\gamma^{\prime})\end{array}$

or equivalently, the following relation between scalars:

$\displaystyle c(\gamma,\gamma^{\prime})=$
$\displaystyle\;c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}$
$\displaystyle\times
c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3})$
(3.23) $\displaystyle\times
c(\gamma_{1},\gamma^{\prime\prime\prime}_{1})c(\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\cdot
c(\gamma_{1}\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma_{1}\gamma^{\prime\prime\prime}_{1}\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3}).$

Since $U_{j}$ is locally equivalent to $U_{1}$, the following also follows
from (3.22):

$\displaystyle\begin{array}[]{rl}U_{j}(\gamma_{3})U_{j}(\gamma^{\prime}_{1})&=U_{j}(\gamma^{\prime\prime}_{1})U_{j}(\gamma^{\prime\prime}_{3})c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3}),\\\
U_{j}(\gamma_{2})U_{j}(\gamma^{\prime\prime}_{1})&=U_{j}(\gamma^{\prime\prime\prime}_{1})U_{j}(\gamma^{\prime\prime\prime}_{2})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}),\\\
U_{j}(\gamma^{\prime\prime}_{3})U_{j}(\gamma^{\prime}_{2})&=U_{j}(\gamma^{\prime\prime\prime\prime}_{2})U_{j}(\gamma^{\prime\prime\prime\prime}_{3})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3}),\end{array}$
(3.27)

Now, in order to show that $U_{j}$ is a local multipler representation with
the cocycle $c$, we only have to compute

$\displaystyle U_{j}(\gamma)U_{j}(\gamma^{\prime})$
$\displaystyle\begin{array}[]{l}=c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}\\\
\quad\times
U_{j}(\gamma_{1})U_{j}(\gamma_{2})U_{j}(\gamma_{3})U_{j}(\gamma^{\prime}_{1})U_{j}(\gamma^{\prime}_{2})U_{j}(\gamma^{\prime}_{3})\end{array}$
$\displaystyle\text{ by }\eqref{eq:defpi}$
$\displaystyle\begin{array}[]{l}=c(\gamma_{1},\gamma_{2})^{-1}c(\gamma_{1}\gamma_{2},\gamma_{3})^{-1}c(\gamma^{\prime}_{1},\gamma^{\prime}_{2})^{-1}c(\gamma^{\prime}_{1}\gamma^{\prime}_{2},\gamma^{\prime}_{3})^{-1}\\\
\quad\times
U_{j}(\gamma_{1})U_{j}(\gamma^{\prime\prime\prime}_{1})U_{j}(\gamma^{\prime\prime\prime}_{2})U_{j}(\gamma^{\prime\prime\prime\prime}_{2})U_{j}(\gamma^{\prime\prime\prime\prime}_{3})U_{j}(\gamma^{\prime}_{3})\\\
\quad\times
c(\gamma_{3},\gamma^{\prime}_{1},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime}_{3})c(\gamma_{2},\gamma^{\prime\prime}_{1},\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2})c(\gamma^{\prime\prime}_{3},\gamma^{\prime}_{2},\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3})\end{array}$
$\displaystyle\text{ by }\eqref{eq:c4-j}$
$\displaystyle\begin{array}[]{l}=c(\gamma,\gamma^{\prime})\left(c(\gamma_{1},\gamma^{\prime\prime\prime}_{1})c(\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\cdot
c(\gamma_{1}\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma_{1}\gamma^{\prime\prime\prime}_{1}\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\right)^{-1}\\\
\quad\times
U_{j}(\gamma_{1})U_{j}(\gamma^{\prime\prime\prime}_{1})U_{j}(\gamma^{\prime\prime\prime}_{2})U_{j}(\gamma^{\prime\prime\prime\prime}_{2})U_{j}(\gamma^{\prime\prime\prime\prime}_{3})U_{j}(\gamma^{\prime}_{3})\end{array}$
$\displaystyle\text{ by }\eqref{eq:c246}$
$\displaystyle\begin{array}[]{l}=c(\gamma,\gamma^{\prime})\left(c(\gamma_{1}\gamma^{\prime\prime\prime}_{1},\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})c(\gamma_{1}\gamma^{\prime\prime\prime}_{1}\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2},\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\right)^{-1}\\\
\quad\times
U_{j}(\gamma_{1}\gamma^{\prime\prime\prime}_{1})U_{j}(\gamma^{\prime\prime\prime}_{2}\gamma^{\prime\prime\prime\prime}_{2})U_{j}(\gamma^{\prime\prime\prime\prime}_{3}\gamma^{\prime}_{3})\end{array}$
$\displaystyle=\,c(\gamma,\gamma^{\prime})U_{j}(\gamma\gamma^{\prime}),$

where we used local equivalence between $U_{j}$ and $U_{1}$ in and 4th
equalities, and the well-definedness (independence of the partition of a group
element into ${\mathcal{D}}^{s}(I_{k})\cap p^{5}(\hat{\mathcal{V}})$) in the
5th equality. Namely, $U_{j}$ has the cocycle $c$ on
${\mathcal{V}}=p^{11}(\hat{\mathcal{V}})$. ∎

##### Direct sum of multiplier representations.

Since all the projective representations $U_{j}$ can be made into the local
multiplier representations with the same cocycle $c$, the direct sum
$U:=\bigoplus_{j}U_{j}$ is again a local multiplier representation of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ on ${\mathcal{V}}$. By forgetting the
phase, we can interpret that $U$ is a local projective representation of
${\mathcal{V}}\subset\widetilde{{\mathcal{D}}^{s}(S^{1})}$, or in other words,
a continuous local group homomorphism from ${\mathcal{V}}$ into
${\mathcal{U}}({\mathcal{H}})/\mathbb{T}$ (see Section 2.4), where
${\mathcal{H}}=\bigoplus_{j}{\mathcal{H}}(c,h_{j})$. As
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ is simply connected and locally
connected, $U$ extends to a continuous projective representation of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ [Pon46, Theorem 63].

######  Theorem 3.13.

Let $s>3$. For a family $\\{(c,h_{j})\\}$ of pairs with the same central
charge $c$ such that $h_{j}-h_{j^{\prime}}\in{\mathbb{N}}$, the direct sum
projective representation $U$ of $\widetilde{{\mathcal{D}}^{s}(S^{1})}$ as
above satisfies $U(R(2\pi))\in{\mathbb{C}}$, where $R(\cdot)$ is the lift of
rotations to $\widetilde{{\mathcal{D}}^{s}(S^{1})}$, or in other words, $U$ is
a projective representation of ${\mathcal{D}}^{s}(S^{1})$.

###### Proof.

Let $\tilde{U}_{(c,h_{j})}$ the irreducible global multiplier representation
of $\widetilde{{\rm Diff}_{+}(S^{1})}$ with central charge $c$ and lowest
weight $h_{j}$ associated to the Bott-Virasoro cocycle. As a projective
representation, we have $U\big{|}_{\widetilde{{\rm
Diff}_{+}(S^{1})}}=\bigoplus_{j}\tilde{U}_{(c,h_{j})}$: this is because, by
definition of $U$, they agree on a neighborhood of the identity of
$\widetilde{{\rm Diff}_{+}(S^{1})}$, and since $\widetilde{{\rm
Diff}_{+}(S^{1})}$ is simply connected they agree globally. Since
$\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}$ is a simple Lie group,
$U\big{|}_{\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}}$ extends to a true
representation of $\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}$ by changing
$U(\gamma)$ only by a scalar [Bar54][Mor17, Theorem 12.72] (see also [Mor17,
Example 12.77]). The lift to a true representation of
$\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}$ is unique, since if $V_{1}$ and
$V_{2}$ are true representations which give rise to the same projective
representation, we have that $V_{1}(g)=\chi(g)V_{2}(g)$ for all
$g\in\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}$, where $\chi$ is a character.
Since $\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}$ is a perfect group,
$\chi(g)=1$ for all $g$. By the uniqueness of the lift of
$U\big{|}_{\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}}$ to a true representation
$V$, we have that $V=\bigoplus_{j}iV_{(c,h_{j})}$, where $V_{(c,h_{j})}$ is
the lift of
$\tilde{U}_{(c,h_{j})}\big{|}_{\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}}$ to a
true representation. As we assume that $h_{j}-h_{j^{\prime}}$ are integers,
$V(R(2\pi))\in{\mathbb{C}}$.

∎

From the previous theorem, it follows that every positive energy projective
unitary representation of ${\rm Diff}_{+}(S^{1})$ extends to a unitary
projective representation of ${\mathcal{D}}^{s}(S^{1})$ using the following
well-known fact that we here prove for completeness.

######  Proposition 3.14.

Let $U$ be a positive energy unitary projective representation of ${\rm
Diff}_{+}(S^{1})$ on the Hilbert space ${\mathcal{H}}$. Then $U$ is unitarily
equivalent to a direct sum of irreducible positive energy unitary projective
representation of ${\rm Diff}_{+}(S^{1})$ and extends to
${\mathcal{D}}^{s}(S^{1})$, $s>3$.

###### Proof.

As in the proof of Theorem 3.13, we have that
$U\big{\rvert}_{\mathrm{PSL}(2,{\mathbb{R}})}$ can be lifted to a true
representation of $\widetilde{\mathrm{PSL}(2,{\mathbb{R}})}$. Thus we can take
the generator of rotations $L_{0}$ and, since $e^{i2\pi
L_{0}}\in{\mathbb{C}}{\mathbbm{1}}$ from the fact that $U$ is a projective
representation of ${\rm Diff}_{+}(S^{1})$, it follows that $L_{0}$ is
diagonalizable with spectrum Sp$(L_{0})\subset\\{h_{1}+\mathbb{N}\\}$ with
$h_{1}\in\mathbb{R}$, $h_{1}\geq 0$. Let ${\mathcal{H}}^{\mathrm{fin}}$ be the
dense subspace of ${\mathcal{H}}$ generated by the eigenvectors of $L_{0}$. We
can apply [CKLW18, Theorem 3.4] to conclude that there exists a positive
energy unitary representation $\pi_{U}$ of ${\rm Vir}$ on
${\mathcal{H}}^{\mathrm{fin}}$.

The representation of ${\rm Vir}$ on ${\mathcal{H}}^{\mathrm{fin}}$ is
equivalent to an algebraic orthogonal direct sum of multiples of irreducible
positive energy representations of ${\rm Vir}$ in the following sense. Let
$V_{1}$ be the smallest $\pi_{U}$-invariant subspace of
${\mathcal{H}}^{\mathrm{fin}}$ which contains
$\ker(L_{0}-h_{1}{\mathbbm{1}}_{{\mathcal{H}}^{\mathrm{fin}}})$ where $h_{1}$
is the smallest eigenvalue of $L_{0}$. By induction let $V_{n}$ be the
smallest $\pi_{U}$-invariant subspace of $\left(V_{1}\oplus
V_{2}\oplus\cdots\oplus
V_{n-1}\right)^{\perp}\cap{\mathcal{H}}^{\mathrm{fin}}$ which contains
$\left(V_{1}\oplus V_{2}\oplus\cdots\oplus
V_{n-1}\right)^{\perp}\cap\ker(L_{0}-h_{n}{\mathbbm{1}}_{{\mathcal{H}}^{\mathrm{fin}}})$
where $h_{n}$ is the smallest eigenvalue of $L_{0}$ restricted to
$\left(V_{1}\oplus V_{2}\oplus\cdots\oplus
V_{n-1}\right)^{\perp}\cap{\mathcal{H}}^{\mathrm{fin}}$. It is straightforward
to see that ${\mathcal{H}}^{\mathrm{fin}}=\bigoplus_{n}V_{n}$ in the algebraic
sense. Now choose an orthonormal basis $\\{e^{n}_{j}\\}$ of $W_{n}\coloneqq
V_{n}\cap\ker(L_{0}-h_{n}{\mathbbm{1}}_{{\mathcal{H}}^{\mathrm{fin}}})$. We
define $H_{j}^{n}$ to be the smallest $\pi_{U}$-invariant subspace of $W_{n}$
which contains the vector $e^{n}_{j}$. By construction $H_{j}^{n}$ has no
proper $\pi_{U}$-invariant subspaces, $H_{j}^{n}$ and $H_{k}^{n}$ are
orthogonal subspaces for $j\neq k$ and
$\overline{V_{n}}=\bigoplus_{j}\overline{H^{n}_{j}}$. Let $T$ be the stress-
energy tensor associated to the representation $\pi_{U}$ of ${\rm Vir}$. By
construction $T(f)|_{H_{j}^{n}}$ is essentially self-adjoint on $H^{n}_{j}$.

To conclude the decomposition of $U$, we have to show that
$e^{iT(f)}\overline{H^{n}_{j}}\subset\overline{H^{n}_{j}}$ for all $f\in{\rm
Vect}(S^{1})$. We note that
${\mathscr{D}}\left(\left(\overline{(T(f)|_{H_{j}^{n}})}\right)^{\ell}\right)\subset{\mathscr{D}}(T(f)^{\ell})$
and if
$\xi\in{\mathscr{D}}\left(\left(\overline{(T(f)|_{H_{j}^{n}})}\right)^{\ell}\right)$
then $\left(\overline{T(f)|_{H_{j}^{n}}}\right)^{\ell}\xi=(T(f))^{\ell}\xi$.
Thus the analytic vectors for $\overline{(T(f)|_{H_{j}^{n}})}$ are also
analytic for $T(f)$ and $e^{i\overline{(T(f)|_{H_{j}^{n}})}}\xi=e^{iT(f)}\xi$.
Using the density of the analytic vectors in $\overline{H_{j}^{n}}$, we obtain
that $e^{i\overline{(T(f)|_{H_{j}^{n}})}}=e^{iT(f)}\big{|}_{H^{n}_{j}}$.
Irreducibility of $U|_{\overline{H^{n}_{j}}}$ follows because $T|_{H^{n}_{j}}$
is irreducible.

The extension to ${\mathcal{D}}^{s}(S^{1})$ is now a mere corollary of Theorem
3.13. ∎

######  Corollary 3.15.

Let $U$ be a positive energy unitary projective representation of ${\rm
Diff}_{+}(S^{1})$ on the Hilbert space ${\mathcal{H}}$. Then $U$ is unitarily
equivalent to a direct sum of irreducible positive energy unitary projective
representation of ${\rm Diff}_{+}(S^{1})$ and extends to ${\rm
Diff}_{+}^{k}(S^{1})$ with $k\geq 4$.

###### Proof.

This again follows from Proposition 3.14 and the continuous embedding ${\rm
Diff}_{+}^{k}(S^{1})\hookrightarrow{\mathcal{D}}^{s}(S^{1}),s\leq k$. ∎

We do not know whether our local multiplier representations can be extended to
a global multiplier representation of $\widetilde{{\mathcal{D}}^{s}(S^{1})}$.
It is also open whether the global multiplier representation of ${\rm
Diff}_{+}(S^{1})$ with the Bott-Virasoro cocycle [FH05, Proposition 5.1]
extends to $\widetilde{{\mathcal{D}}^{s}(S^{1})}$ by continuity.

##  4 Conformal nets and diffeomorphism covariance

Let $\mathrm{PSL}(2,\mathbb{R})$ be the Möbius group and $\mathcal{I}$ be the
set of nonempty, non-dense, open intervals of the unit circle $S^{1}$.
$I^{\prime}$ denotes the interior of the complement of the interval
$I\in\mathcal{I}$, namely $I^{\prime}=(S^{1}\setminus I)^{\circ}$. A Möbius
covariant net $({\mathcal{A}},U,\Omega)$ on $S^{1}$ is a triple of a family
$\mathcal{A}=\left\\{\mathcal{A}(I),I\in\mathcal{I}\right\\}$ of von Neumann
algebras, a strongly continuous unitary representation $U$ of
$\mathrm{PSL}(2,\mathbb{R})$ acting on a separable complex Hilbert space
$\mathcal{H}$ and $\Omega\in{\mathcal{H}}$, satisfying the following
properties:

  1. (1)

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

  2. (2)

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

  3. (3)

Möbius covariance: for $g\in\mathrm{PSL}(2,\mathbb{R})$, $I\in\mathcal{I}$,

$U(g)\mathcal{A}(I)U(g)^{-1}=\mathcal{A}(gI)$

where $\mathrm{PSL}(2,\mathbb{R})$ acts on $S^{1}$ by Möbius transformations.

  4. (4)

Positivity of energy: the representation $U$ has positive energy, i.e. the
conformal Hamiltonian $L_{0}$ (the generator of rotations) has non-negative
spectrum.

  5. (5)

Vacuum vector: there exists a unique (up to scalar) vector
$\Omega\in\mathcal{H}$ with the property $U(g)\Omega=\Omega$ for
$g\in\mathrm{PSL}(2,\mathbb{R})$. Additionally $\Omega$ is cyclic for the
algebra $\bigvee_{I\in\mathcal{I}}\mathcal{A}(I)$.

With these assumptions, the following automatically hold [GF93, Theorem
2.19(ii)][FJ96, Section 3]

  1. (6)

Reeh-Schlieder property: $\Omega$ is cyclic and separating for
${\mathcal{A}}(I)$.

  2. (7)

Haag duality: for every $I\in\mathcal{I}$,
$\mathcal{A}(I^{\prime})=\mathcal{A}(I)^{\prime}$ where
$\mathcal{A}(I)^{\prime}$ is the commutant of $\mathcal{A}(I)$.

  3. (8)

Additivity: if $\\{I_{\alpha}\\}_{\alpha\in A}$ is a covering of
$I\in\mathcal{I}$, with $I_{\alpha}\in\mathcal{I}$ for every $\alpha$, then
${\mathcal{A}(I)\subset\bigvee_{\alpha}\mathcal{A}(I_{\alpha})}$.

  4. (9)

Semicontinuity: if $I_{n}\in\mathcal{I}$ is a decreasing family of intervals
and $I=\left(\bigcap_{n}I_{n}\right)^{\circ}$ then
$\mathcal{A}(I)=\bigwedge_{n}\mathcal{A}(I_{n})$.

By a conformal net (or diffeomorphism covariant net) we shall mean a Möbius
covariant net which satisfies the following:

  1. (10)

The representation $U$ extends to a projective unitary representation of ${\rm
Diff}_{+}(S^{1})$ such that for all $I\in\mathcal{I}$ we have

$\displaystyle U(\gamma)\mathcal{A}(I)U(\gamma)^{*}$
$\displaystyle=\mathcal{A}(\gamma I),\quad\gamma\in{\rm Diff}_{+}(S^{1}),$
$\displaystyle U(\gamma)xU(\gamma)^{*}$ $\displaystyle=x,\hskip
8.53581ptx\in\mathcal{A}(I),\quad\gamma\in{\rm Diff}_{+}(I^{\prime})$

where ${\rm Diff}_{+}(I^{\prime})$ denotes the subgroup of diffeomorphisms
$\gamma$ such that $\gamma(z)=z$ for all $z\in I$.

A positive energy representation $U$ of ${\rm Diff}_{+}(S^{1})$ is equivalent
to a direct sum of irreducible representations, see Proposition 3.14. Every
irreducible component $U_{j}$ in decomposition has the same value of the
central charge $c$ and if $h_{j}$ is the lowest weight of $U_{j}$,
$h_{j}-h_{k}\in\mathbb{Z}$ for every $j,k$. This fact is crucial for our
purpose, which is to extend the conformal symmetry of the net to the larger
group ${\mathcal{D}}^{s}(S^{1})$, $s>3$, in the sense that we want to show
that the conditions in (10) are satisfied for arbitrary $\gamma$ in
${\mathcal{D}}^{s}(S^{1})$ and ${\mathcal{D}}^{s}(I^{\prime})$ respectively.

######  Proposition 4.1.

A conformal net $(\mathcal{A},U,\Omega)$ is
${\mathcal{D}}^{s}(S^{1})$-covariant, $s>3$.

###### Proof.

Let $\\{\gamma_{n}\\}$ be a sequence of diffeomorphisms in ${\rm
Diff}_{+}(S^{1})$ converging to $\gamma\in{\mathcal{D}}^{s}(S^{1})$ in the
topology of ${\mathcal{D}}^{s}(S^{1})$ as in Lemma A.4. For all
$n\in\mathbb{N}$ it holds that

$U(\gamma_{n}){\mathcal{A}}(I)U(\gamma_{n})^{*}={\mathcal{A}}(\gamma_{n}I)\subset{\mathcal{A}}(\textstyle{\bigcup_{k=m}^{n}\gamma_{k}I}),$

where we used isotony of the net ${\mathcal{A}}$. For $x\in{\mathcal{A}}(I)$,
it follows for $m\leq n$ that

$U(\gamma_{n})xU(\gamma_{n})^{*}\in{\mathcal{A}}(\textstyle{\bigcup_{k=m}^{n}\gamma_{k}I})=\bigvee_{k=m}^{\infty}{\mathcal{A}}(\gamma_{k}I),$

by additivity. By Proposition 3.6 it follows that
$U(\gamma)xU(\gamma)^{*}=\lim_{n\rightarrow\infty}U(\gamma_{n})xU(\gamma_{n})^{*}$
(convergence in the strong operator topology) is in
$\bigcup_{k=m}^{\infty}{\mathcal{A}}(\gamma_{k}\cdot I)$ for any $m$, hence we
have by upper semicontinuity that

$U(\gamma){\mathcal{A}}(I)U(\gamma)^{*}\subset\bigcap_{m}{\mathcal{A}}(\textstyle{\bigcup_{k=m}^{\infty}\gamma_{k}I})={\mathcal{A}}(\gamma
I).$

The other inclusion follows by applying ${\hbox{\rm Ad\,}}U(\gamma^{-1})$.

Now consider $\gamma\in{\mathcal{D}}^{s}(I^{\prime})$ and
$x\in{\mathcal{A}}(I)$. We know from lemma A.4 that exists a sequence
$\\{\gamma_{n}\\}\subset{\rm Diff}_{+}(I_{n}^{\prime})$ converging to $\gamma$
in the topology of ${\mathcal{D}}^{s}(S^{1})$ and a decreasing sequence of
intervals $I^{\prime}_{n}\supset{\rm supp\,}(\gamma_{n})\supset I^{\prime}$
such that $\bigcap_{n}I^{\prime}_{n}=I^{\prime}$. For
$x\in{\mathcal{A}}(I_{n})$, $U(\gamma_{m})xU(\gamma_{m})^{*}=x$ if $m\geq n$,
hence by Proposition 3.6 we obtain $U(\gamma)xU(\gamma)^{*}=x$. As $n$ is
arbitrary, this holds for any
$x\in{\mathcal{A}}(\textstyle{\bigcup_{n}I_{n}})={\mathcal{A}}(I)$ by
additivity. ∎

### Representations of conformal nets

Let $({\mathcal{A}},U,\Omega)$ a conformal net. A representation $\rho$ of
$({\mathcal{A}},U,\Omega)$ is a family $\rho=\\{\rho_{I}\\}$,
$I\in{\mathcal{I}}$, where $\rho_{I}$ are representations of
${\mathcal{A}}(I)$ on a common Hilbert space ${\mathcal{H}}_{\rho}$ and such
that $\rho_{J}|_{{\mathcal{A}}(I)}=\rho_{I}$ if $I\subset J$. The
representation $\rho$ is said to be locally normal if $\rho_{I}$ is normal for
every $I\in{\mathcal{I}}$ (this is always true if the representation space
${\mathcal{H}}_{\rho}$ is separable [Tak02, Theorem 5.1]). We say that a
representation $\rho$ of a conformal net $({\mathcal{A}},U,\Omega)$ is
diffeomorphism covariant if there exists a positive energy representation
$U^{\rho}$ of $\widetilde{{\rm Diff}_{+}(S^{1})}$ such that

$U^{\rho}(\gamma)\rho_{I}(x)U^{\rho}(\gamma)^{*}=\rho_{\mathring{\gamma}I}(U(\mathring{\gamma})xU(\mathring{\gamma})^{*}),\quad\text{
for }x\in{\mathcal{A}}(I),g\in\widetilde{{\rm Diff}_{+}(S^{1})},$

where $\mathring{\gamma}$ is the image of $\gamma$ in ${\rm Diff}_{+}(S^{1})$
under the covering map.

Now let $\rho$ a locally normal representation of the conformal net
${\mathcal{A}}$ and assume that $e^{i2\pi L^{\rho}_{0}}$ has pure point
spectrum (this is always the case if $\rho$ is a direct sum of irreducibles).
By using [Car04, Proposition 2.2] and arguing as in the proof of [Car04,
Proposition 3.7] it is not hard to see that $\rho$ is diffeomorphism covariant
(this will be directly proved in [Tan18]) and that the corresponding positive
energy projective unitary representation $U_{\rho}$ of $\widetilde{{\rm
Diff}_{+}(S^{1})}$ is a direct sum of irreducibles. By our previous results
$U_{\rho}$ extends to $\widetilde{{\mathcal{D}}^{s}(S^{1})}$, $s>3$, and this
extension makes $\rho$ $\widetilde{{\mathcal{D}}^{s}(S^{1})}$-covariant.
Furthermore, if $\rho$ is a direct sum of irreducible representations, then
the adjoint action ${\hbox{\rm Ad\,}}U_{\rho}(R(2\pi))$ is trivial, and in
this sense $\rho$ is ${\mathcal{D}}^{s}(S^{1})$-covariant. We summarize this
fact in the following proposition.

######  Proposition 4.2.

Let $\rho$ be a locally normal representation of the conformal net
${\mathcal{A}}$ and assume that $e^{i2\pi L^{\rho}_{0}}$ has pure point
spectrum. Then $\rho$ is $\widetilde{{\mathcal{D}}^{s}(S^{1})}$-covariant,
$s>3$. If $\rho$ is a direct sum of irreducible representations, then it is
${\mathcal{D}}^{s}(S^{1})$-covariant.

##  5 Outlook

For integer $n$ and some $h$, the irreducible unitary representation
$U_{(n,h)}$ can be extended to ${\mathcal{D}}^{s}(S^{1}),s>2$ [DIT18]. It
would be interesting to further determine to what point the regularity of the
diffeomorphisms can be weakened in such a way that the representations
$U_{(c,h)}$ may be extended to such a class in a continuous way. The proof of
[DIT18] (based on the strategy of [Vro13]) relies on the better-behaving
$\mathrm{U}(1)$-current, and it appears that such extensions do not act nicely
on the stress-energy tensor $T$, which we are currently able to extend only to
$\mathcal{S}_{\frac{3}{2}}(S^{1})$.

Another interesting question is whether the global multiplier representations
in [FH05] extend to $\widetilde{{\mathcal{D}}^{s}(S^{1})}$. The question is
whether these representations are continuous in the
${\mathcal{D}}^{s}(S^{1})$-topology. Instead, what we used in Proposition 3.12
is the continuity of our extensions as projective representations, and the
existence of local multiplier representations follows. In particular, we do
not know whether there is a multiplier representation of
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ with the Bott-Virasoro cocycle.

#### Acknowledgements.

S.C. would like to thank Gerard Misiołek for inspiring discussions. S.D. and
S.I. would like to thank Stefano Rossi for the valuable discussions. Y.T.
thanks André Henriques and Kathryn Mann for useful information. S.D. and
Y.T. acknowledge the MIUR Excellence Department Project awarded to the
Department of Mathematics, University of Rome Tor Vergata, CUP
E83C18000100006.

##  Appendix A Appendix: Various definitions of Sobolev spaces

##### By Fourier coefficients.

In Section 2.3 we introduced the Sobolev space $H^{s}(S^{1})$ through the
Fourier coefficients:

$\displaystyle H^{s}(S^{1})$ $\displaystyle:=\\{f\in
L^{2}(S^{1},{\mathbb{R}}):\sum(1+k^{2})^{s}|\hat{f}_{k}|^{2}<\infty\\},$ (A.1)
$\displaystyle{\mathcal{D}}^{s}(S^{1})$ $\displaystyle:=\\{\gamma\in{\rm
Diff}_{+}^{1}(S^{1}):\tilde{\gamma}-\iota\in H^{s}(S^{1})\\},$ (A.2)

where $\tilde{\gamma}$ is a lift to $\widetilde{{\rm Diff}_{+}^{1}}(S^{1})$
and $\iota$ is the identity map. We can also give the topology first to
$\widetilde{{\mathcal{D}}^{s}(S^{1})}$ as a subspace of $H^{s}(S^{1})+\iota$
with the topology given by $H^{s}(S^{1})$, then to ${\mathcal{D}}^{s}(S^{1})$
by the quotient map. The definition by Fourier coefficients is convenient for
us because it is crucial that ${\mathcal{D}}^{3+\varepsilon}(S^{1})$ acts on
the set of vector fields in $\mathcal{S}_{\frac{3}{2}}(S^{1})$ (to which the
stress-energy tensor can be extended) and the latter is defined again through
Fourier coefficients. However, we also cited Lemmas 2.4, 2.5 from [IKT13,
Lemma B.4, Theorem B.2] where the Sobolev spaces are defined in another way.
Therefore, we have to check that these definitions coincide.

##### By local integral.

Let us first observe that an analogue of [IKT13, Lemma B.1] holds.

######  Lemma A.1.

Let $f\in L^{2}(S^{1},{\mathbb{R}})$ and $s>1$. Then, $f\in H^{s}(S^{1})$ if
and only if $f\in{\mathscr{D}}(i\frac{d}{d\theta})$ and $f^{\prime}\in
H^{s-1}(S^{1})$. Moreover, the norm $\|f\|+\|f^{\prime}\|_{H^{s-1}}$ is
equivalent to $\|f\|_{H^{s}}$, where $\|f\|=\|f\|_{H^{0}}$ is the
$L^{2}$-norm.

With this Lemma, we can consider the second characterization, parallel to
[IKT13, Lemma B.2].

######  Lemma A.2.

Let $s>0,s\notin{\mathbb{Z}}$ and $\lambda=s-\lfloor s\rfloor$, where $\lfloor
s\rfloor$ denotes the largest integer not exceeding $s$. Then $f\in
H^{s}(S^{1})$ if and only if $f\in H^{\lfloor s\rfloor}$ and $[f^{(\lfloor
s\rfloor)}]_{\lambda}<\infty$, where $[f^{(\lfloor s\rfloor)}]_{\lambda}$ is
the $L^{2}$-norm of the following function on $S^{1}\times S^{1}$

$(\theta_{1},\theta_{2})\longmapsto\frac{|f^{(\lfloor
s\rfloor)}(e^{i\theta_{1}})-f^{(\lfloor
s\rfloor)}(e^{i\theta_{2}})|}{|\theta_{1}-\theta_{2}|^{\lambda+\frac{1}{2}}},$

where $|\theta_{1}-\theta_{2}|$ denotes the distance999We may assume that
$-\pi\leq\theta_{1},\theta_{2}<\pi$, and
$|\theta_{2}-\theta_{1}|=\min\\{|\theta_{2}-\theta_{1}|,2\pi-|\theta_{2}-\theta_{1}|\\}$,
hence this depends only on $\theta_{2}-\theta_{2}$. of two points
$\theta_{1},\theta_{2}$ along the circle
$S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$.

###### Proof.

Following [IKT13, Lemma B.2], we prove the Lemma by induction. Let us assume
$\lfloor s\rfloor=0$, hence $s=\lambda$. We have

$\displaystyle\int_{S^{1}}\int_{S^{1}}\frac{|f(e^{i\theta_{1}})-f(e^{i\theta_{2}})|^{2}}{|\theta_{1}-\theta_{2}|^{2\lambda+1}}d\theta_{1}d\theta_{2}$
$\displaystyle=\int_{S^{1}}\int_{S^{1}}\frac{|f(e^{i(\theta_{1}+\theta)})-f(e^{i\theta_{1}})|^{2}}{|\theta|^{2\lambda+1}}d\theta_{1}d\theta$
$\displaystyle=\int_{S^{1}}\frac{1}{|\theta|^{2\lambda+1}}\int_{S^{1}}|f(e^{i(\theta_{1}+\theta)})-f(e^{i\theta_{1}})|^{2}d\theta_{1}d\theta$

By Parseval’s theorem,

$\displaystyle\int_{S^{1}}|f(e^{i(\theta_{1}+\theta)})-f(e^{i\theta_{1}})|^{2}d\theta_{1}$
$\displaystyle=\sum_{k}|\widehat{f(e^{i(\cdot+\theta)})}_{k}-\hat{f}_{k}|^{2}$
$\displaystyle=\sum_{k}|e^{ik\theta}-1|^{2}\cdot|\hat{f}_{k}|^{2},$

therefore,

$\displaystyle\int_{S^{1}}\int_{S^{1}}\frac{|f(e^{i\theta_{1}})-f(e^{i\theta_{2}})|^{2}}{|\theta_{1}-\theta_{2}|^{2\lambda+1}}d\theta_{1}d\theta_{2}$
$\displaystyle=\sum_{k}|\hat{f}_{k}|^{2}\int_{S^{1}}\frac{|e^{ik\theta}-1|^{2}}{|\theta|^{2\lambda+1}}d\theta$
$\displaystyle=\sum_{k}|k|^{2\lambda}|\hat{f}_{k}|^{2}\int_{S^{1}}\frac{|e^{ik\theta}-1|^{2}}{|k|^{2\lambda}|\theta|^{2\lambda+1}}d\theta.$

Note that, for $k\neq 0$, by substitution $\hat{\theta}=k\theta$ we obtain

$\displaystyle\int_{S^{1}}\frac{|e^{ik\theta}-1|^{2}}{|k|^{2\lambda}|\theta|^{2\lambda+1}}d\theta$
$\displaystyle=\int_{-\pi}^{\pi}\frac{|e^{ik\theta}-1|^{2}}{|k|^{2\lambda}|\theta|^{2\lambda+1}}d\theta$
$\displaystyle=\int_{-k\pi}^{k\pi}\frac{|e^{i\hat{\theta}}-1|^{2}}{|\hat{\theta}|^{2\lambda+1}}d\theta$

and this last integral is uniformly bounded both below and above with respect
to $k$. Therefore, for $f\in H^{\lfloor
s\rfloor}(S^{1})=L^{2}(S^{1},{\mathbb{R}})$,
$\|f\|_{H^{s}}=\sum_{k}(1+|k|^{2})^{\lambda}|\hat{f}_{k}|^{2}<\infty$ if and
only if $\sum_{k}|k|^{2\lambda}|\hat{f}_{k}|^{2}<\infty$, if and only if

$\displaystyle\int_{S^{1}}\int_{S^{1}}\frac{|f(e^{i\theta_{1}})-f(e^{i\theta_{2}})|^{2}}{|\theta_{1}-\theta_{2}|^{2\lambda+1}}d\theta_{1}d\theta_{2}<\infty.$

Assuming that the statement holds for $s-1$, we can conclude induction by
applying it to $f^{\prime}$ and using Lemma A.1.

∎

The whole Appendix B of [IKT13] can be adapted to $H^{s}(S^{1})$ and
${\mathcal{D}}^{s}(S^{1})$ using these norms and one obtains Lemma 2.5 for
$s\notin{\mathbb{Z}}$, corresponding to [IKT13, Lemmas B.5,B.6] (for
$s\in{\mathbb{Z}}$, see [IKT13, Section 2]). We believe this is the fastest
way for the reader not familiar with Sobolev spaces.

##### By local Sobolev spaces.

Alternatively, one may start with the Sobolev spaces on ${\mathbb{R}}$,
following [IKT13]:

$H^{s}({\mathbb{R}},{\mathbb{R}}):=\left\\{f\in
L^{2}({\mathbb{R}},{\mathbb{R}}):\int_{\mathbb{R}}(1+\zeta^{2})^{s}|\hat{f}(\zeta)|^{2}d\zeta<\infty\right\\},$

where $\hat{f}$ denotes the Fourier transform for $f\in
L^{2}({\mathbb{R}},{\mathbb{R}})$ (with a slight abuse of notation: $\hat{f}$
depends on $f\in L^{2}({\mathbb{R}},{\mathbb{R}})$ or $f\in
L^{2}(S^{1},{\mathbb{R}})$). For an open connected set
${\mathcal{U}}\subset{\mathbb{R}}$, we set (see [IKT13, Definition B.1], where
the boundary $\partial{\mathcal{U}}$ is required to be of Lipschitz, but in
${\mathbb{R}}$ it is not necessary):

$H^{s}({\mathcal{U}},{\mathbb{R}}):=\\{f\in
L^{2}({\mathcal{U}},{\mathbb{R}}):\text{ there is }\tilde{f}\in
H^{s}({\mathbb{R}},{\mathbb{R}})\text{ s.t. }f=\tilde{f}|_{\mathcal{U}}\\}$

Let us consider $S^{1}$ as a manifold, namely supplied with an atlas
$\\{{\mathcal{U}}_{\kappa}\\}$.

For $s>\frac{1}{2}$, we may take another definition for
$H^{s}(S^{1})=H^{s}(S^{1},{\mathbb{R}})$:

$\\{f\in C(S^{1},{\mathbb{R}}):\text{ for each }\theta\in S^{1}\text{ there is
}{\mathcal{U}}\ni\theta\text{ s.t.\\! }f|_{{\mathcal{U}}}\in
H^{s}({\mathcal{U}},{\mathbb{R}})\\}.$

Now, from Lemma A.2, it is clear that being in $H^{s}(S^{1})$ is a local
property (note that $S^{1}$ is compact). If $f\in H^{s}(S^{1})$ (in the sense
of (A.1)), for any smooth function $\psi$ with compact support, $\psi f\in
H^{s}(S^{1})$ by Lemma A.2. To a chart ${\mathcal{U}}$ in the atlas, take a
smooth function $\psi$ which is $1$ on $\overline{\mathcal{U}}$ and supported
in a non-dense interval. Then $\psi f$ can be considered as an element of
$H^{s}({\mathbb{R}},{\mathbb{R}})$ by [IKT13, Lemma B.2], hence the definition
(A.1) is stronger. Conversely, if for each $\theta\in S^{1}$ there is
${\mathcal{U}}\ni\theta$ such that $f|_{{\mathcal{U}}}\in
H^{s}({\mathcal{U}},{\mathbb{R}})$, by compactness of $S^{1}$ one can take a
finite cover $\\{{\mathcal{U}}_{k}\\}$ of $S^{1}$ and a smooth partition of
unity $\\{\psi_{k}\\}$ subordinate to it, and it follows that
$f=\sum_{k}\psi_{k}f\in H^{s}(S^{1})$ in the sense of (A.1), therefore, two
definitions are equivalent.

It is also clear that the following definition [Tay11, Section 4.3]

$H^{s}({\mathcal{U}},{\mathbb{R}}):=\\{f\in
L^{2}({\mathcal{U}},{\mathbb{R}}):\psi f\in
H^{s}({\mathbb{R}},{\mathbb{R}})\text{ for any }\psi\in
C^{\infty}({\mathcal{U}},{\mathbb{R}}),{\rm
supp\,}\psi\subset{\mathcal{U}}\\}$

is equivalent to the definition through Fourier coefficients.

Now, we define $H^{s}(S^{1},S^{1})$ to be the maps $f:S^{1}\to S^{1}$ such
that there are two atlases
$\\{{\mathcal{U}}_{k}\\},\\{{\mathcal{V}}_{\kappa}\\}$ of $S^{1}$ such that
$f|_{{\mathcal{U}}_{k}}\in H^{s}({\mathcal{U}}_{k},{\mathbb{R}})$ where we
identified $f({\mathcal{U}}_{k})\subset{\mathcal{V}}_{k}$ as a subset of
${\mathbb{R}}$ by the chart (see [IKT13, Section 3.1]) and

${\mathcal{D}}^{s}(S^{1})=\\{\gamma\in{\rm Diff}_{+}^{1}(S^{1}):\gamma\in
H^{s}(S^{1},S^{1})\\}.$

Recall that the definition (A.2) is local, and the identity map $\iota$ is a
smooth function, hence it is equivalent to the definition above which is
manifestly local.

Now that we have the equivalence of definitions, we can use [IKT13, Theorem
B.2], which we cited and specialized as Lemma 2.5.

##### Local approximation of diffeomorphisms.

We also need that elements in ${\mathcal{D}}^{s}(S^{1})$ with compact support
can be approximated by elements ${\mathcal{D}}^{s}(S^{1})$ with slightly
larger support.

######  Lemma A.3.

For a fixed $f\in H^{s}(S^{1})$, the rotation ${\mathbb{R}}\ni t\mapsto
f_{t}=f(e^{i(\cdot-t)})\in H^{s}(S^{1})$ is continuous.

###### Proof.

We have $\hat{f}_{t,k}=e^{ikt}\hat{f}_{k}$, and hence
$|\hat{f}_{t,k}|=|\hat{f}_{k}|$ and $\hat{f}_{t,k}\to\hat{f}_{k}$ as $t\to 0$.
By Lebesgue’s dominated convergence theorem (applied to the measure space
${\mathbb{Z}}$ with the counting measure, with the dominating function
$k\mapsto 4|(1+k^{2})^{s}\hat{f}_{k}|^{2}$)

$\sum_{k}(1+k^{2})^{s}|\hat{f}_{t,k}-\hat{f}_{k}|^{2}\to 0.$

This means $\|f-f_{t}\|_{H^{s}}\to 0$. ∎

######  Lemma A.4.

For every $\gamma\in{\mathcal{D}}^{s}(S^{1})$, there exists a sequence
$\\{\gamma_{n}\\}$ converging to $\gamma$ in the topology of
${\mathcal{D}}^{s}(S^{1})$. Furthermore, if $\gamma$ is supported in $I$, we
can take $\gamma_{n}$ such that ${\rm supp\,}\gamma_{n}\supset\gamma_{n+1}$
and $\bigcap_{n}{\rm supp\,}\gamma_{n}=I$.

###### Proof.

Let $\gamma\in{\mathcal{D}}^{s}(S^{1})$ and
$\varphi\in\widetilde{{\mathcal{D}}^{s}(S^{1})}$ such that
$\varphi(\theta+2\pi)=\varphi(\theta)+2\pi$ and
$\gamma(e^{i\theta})=e^{i\varphi(\theta)}$. If $\gamma$ is supported in a
proper interval we may assume without loss of generality that
$\varphi(\theta)=\theta$ if $\theta\in[-\pi,a)\cup(b,\pi]$. The function
$\psi\coloneqq\varphi^{\prime}-1$ is $2\pi$-periodic and has compact support
$[a,b]$ as a function on $[-\pi,\pi]$.

We now choose a set of $C^{\infty}$-functions $\\{g_{n}\\}$ with compact
support strictly contained in $[-\pi,\pi]$ such that for all $n\in\mathbb{N}$
$g_{n}\geq 0$, $\int g_{n}=1$, ${\rm supp\,}(g_{n})\supset{\rm
supp\,}(g_{n+1})$, ${\rm supp\,}(g_{n})\rightarrow\\{0\\}$. In addition, if
$\gamma$ is supported in $[a,b]$, we may assume that $[a,b]+{\rm
supp\,}(g_{n})\supset{\rm supp\,}(\psi*g_{n})$, where the convolution is
defined on $S^{1}={\mathbb{R}}/2\pi{\mathbb{Z}}$ as an abelian group.

To obtain the claim, it is enough to show that $\|\psi-\psi*g_{n}\|_{H^{s}}\to
0$ as $n\to 0$. This follows from

$\|\psi-\psi*g_{n}\|_{H^{s}}\leq\int_{S^{1}}g_{n}(t)\|\psi-\psi_{t}\|_{H^{s}}dt$

and Lemma A.3.

∎

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