Conformally Twisted Spectral Triples

On the Chern–Gauss–Bonnet Theorem
and Conformally Twisted Spectral Triples
for $\boldsymbol{C^{*}}$-Dynamical Systems

Farzad FATHIZADEH † and Olivier GABRIEL ‡

F. Fathizadeh and O. Gabriel

† Department of Mathematics, Mail Code 253-37, California Institute of
Technology,
† 1200 E. California Blvd., Pasadena, CA 91125, USA<EMAIL_ADDRESS>
‡ University of Copenhagen, Universitetsparken 5, 2100 København Ø, Denmark
<EMAIL_ADDRESS>http://oliviergabriel.eu/

Received October 26, 2015, in final form February 04, 2016; Published online
February 10, 2016

The analog of the Chern–Gauss–Bonnet theorem is studied for a
$C^{*}$-dynamical system consisting of a $C^{*}$-algebra $A$ equipped with an
ergodic action of a compact Lie group $G$. The structure of the Lie algebra
$\mathfrak{g}$ of $G$ is used to interpret the Chevalley–Eilenberg complex
with coefficients in the smooth subalgebra $\mathcal{A}\subset A$ as
_noncommutative_ differential forms on the dynamical system. We conformally
perturb the standard metric, which is associated with the unique $G$-invariant
state on $A$, by means of a Weyl conformal factor given by a positive
invertible element of the algebra, and consider the Hermitian structure that
it induces on the complex. A Hodge decomposition theorem is proved, which
allows us to relate the Euler characteristic of the complex to the index
properties of a Hodge–de Rham operator for the perturbed metric. This
operator, which is shown to be selfadjoint, is a key ingredient in our
construction of a spectral triple on $\mathcal{A}$ and a twisted spectral
triple on its opposite algebra. The conformal invariance of the Euler
characteristic is interpreted as an indication of the Chern–Gauss–Bonnet
theorem in this setting. The spectral triples encoding the conformally
perturbed metrics are shown to enjoy the same spectral summability properties
as the unperturbed case.

$C^{*}$-dynamical systems; ergodic action; invariant state; conformal factor;
Hodge–de Rham operator; noncommutative de Rham complex; Euler characteristic;
Chern–Gauss–Bonnet theorem; ordinary and twisted spectral triples; unbounded
selfadjoint operators; spectral dimension

58B34; 47B25; 46L05

##  1 Introduction

In noncommutative geometry [13, 14], $C^{*}$-dynamical systems $(A,G,\alpha)$
have been long studied from a differentiable point of view starting with
extending the basic notions of differential geometry and differential topology
to a differential structure on a $C^{*}$-algebra $A$ endowed with an action
$\alpha\colon G\to\operatorname{Aut}(A)$ of a Lie group $G$. That is, the
notion of a connection, a vector bundle, and Chern classes were introduced for
such a dynamical system, a pseudodifferential calculus was developed and the
analog of the Atiyah–Singer index theorem was proved in [12]. The
noncommutative two torus $\mathbb{T}_{\theta}^{2}$ has been one of the main
motivating examples for these developments. In [55], this line of
investigation has been taken further focusing on general compact Lie groups
and index theory.

Following the seminal work of Connes and Tretkoff on the Gauss–Bonnet theorem
for $\mathbb{T}_{\theta}^{2}$ [18] and its extension in [25] concerning
general translation invariant conformal structures, local differential
geometry of non-flat noncommutative tori has been a subject of increasing
interest in recent years [3, 17, 26, 27, 28, 45]. A Weyl conformal factor may
be used to perturb a flat metric on noncommutative tori, and Connes’
pseudodifferential calculus [12] can be employed along with noncommutative
computational methods to carry out calculation of scalar curvature and to
investigate the related differential geometric statements, see also [20] for
an asymmetric perturbation of the metric.

The idea and the techniques were indeed initiated in a preprint [11], where
with the help of complicated modified logarithmic functions and a modular
automorphism, an expression for the value $\zeta(0)$ of the spectral zeta
function of the Laplacian of a curved metric on $\mathbb{T}^{2}_{\theta}$ was
written. The vanishing of this expression is interpreted as the Gauss–Bonnet
theorem [18], which was suggested by the developments in the following
intimately related theories. In fact, the spectral action principle [9], in
particular the related calculations in the presence of a dilaton [10], and the
theory of twisted spectral triples, which arise naturally in noncommutative
conformal geometry [16, 49], indicate independence of $\zeta(0)$ from the
conformal factor.

Connes’ index formula for Fredholm modules, which involves cyclic cohomology,
is quite broad [13]. It asserts that given a finitely summable Fredholm module
over an algebra, the analytic index, given by pairing a $K$-homology and a
$K$-theory element of the algebra, coincides with the topological index, which
pairs the corresponding elements in periodic cyclic cohomology and homology
obtained by the Chern–Connes characters. The local index formula of Connes and
Moscovici [15] gives a local formula based on residue trace functionals, which
is in the same cyclic cohomology class as the Chern–Connes character, and has
the advantage that one can perform explicit computations with it (see also
[34]). The residue trace functionals are intimately related to the spectral
formulation of Wodzicki’s noncommutative residue [59, 60]. In fact, the
formulation of the noncommutative residue as an integration over the cosphere
bundle of a manifold also is important for explicit computations with
noncommutative geometric spaces, see [22, 28, 29] for a related treatment on
noncommutative tori.

The notion of a twisted spectral triple introduced by Connes and Moscovici
[16] allows to incorporate a variety of new examples, in particular type III
examples in the sense of the Murray–von Neumann classification of operator
algebras. They have shown that the Chern–Connes character of a finitely
summable twisted spectral triple is an ordinary cyclic cocycle and enjoys an
index pairing with $K$-theory. Also, they have constructed a local Hochschild
cocycle, which indicates that the ground is prepared for extending the local
index formula to the twisted case. This was carried out in [47] for a
particular class of twisted spectral triples; the analog of Connes’ character
formula was investigated in [24] for the examples. For treatments using
twisted cyclic theory, in particular for relations of the theory with Cuntz
algebra [19] and quantum groups, we refer to [6, 7, 8], see also [37, 38].
More recent works related to the twisted version of spectral triples reveal
their connections with the Bost–Connes system, Riemann surfaces and graphs
[33], and with the standard model of particle physics [21]. Twisted spectral
triples associated with crossed product algebras are studied in [16, 36, 47],
see also [23] for an algebraic treatment.

Ergodic actions of compact groups on operator algebras are well-studied in the
von Neumann setting (see, e.g., [56, 57, 58]) and in a $C^{*}$-algebraic
context. They were first introduced for $C^{*}$-algebras by E. Størmer [54]
and this initial effort was expanded in various articles. Let us just mention
two of them:

  * •

In their article [1], Albeverio and Høegh-Krohn investigate in particular
ergodic actions on commutative $C^{*}$-algebras $A=C(X)$ and prove that they
correspond to continuous transitive actions on $X$.

  * •

The article [35] by Høegh-Krohn, Landstad and Størmer proves that if $G$ acts
ergodically on a unital $C^{*}$-algebra $A$, its unique $G$-invariant state is
actually a _trace_.

The article [52] was the first to suggest in 1998 that ergodic actions give
rise to interesting spectral triples. This article proceeds with studying the
metric induced on state spaces by ergodic actions. More recently, the article
[30] produced a detailed construction of a so called _Lie–Dirac operator_ on a
$C^{*}$-algebra $A$, based on an ergodic action of a compact Lie group $G$ on
$A$. It also investigated the analytic properties of these Lie–Dirac
operators, proving in particular that they are finitely summable spectral
triples. In the present article, we elaborate on the techniques used in [30]
in order to prove quite different results.

Indeed, in [30] the focus was on a Dirac operator for a “noncommutative spin
manifold”, whereas here the emphasis is on a sort of Hodge–de Rham operator
associated with a conformally perturbed metric, construction of twisted
spectral triples and the analog of the Chern–Gauss–Bonnet theorem. The
Hodge–de Rham operators constructed here are (in general) _not_ Lie–Dirac
operators in the sense of [30]. In this previous article, the algebra
structure of $A$ played only a minor role in the analytical properties of the
spectral triple. Here, the multiplication of $A$ has a central importance.

For a recent approach of Hodge theory using Hilbert modules, we refer to the
recent article [44]. See also [42, 43].

This article is organized as follows. In Section 2, we recall the necessary
statements from representation theory and operator theory, and the notion of
ordinary and twisted spectral triples along with their main properties that
are used in our arguments and concern our constructions. We associate a
complex of noncommutative differential forms to a $C^{*}$-dynamical system
$(A,G,\alpha)$ in Section 3. In the ergodic case, the analog of the Hodge–de
Rham operator is studied when the complex is equipped with a Hermitian
structure determined by a metric in the conformal class of the standard metric
associated with the unique $G$-invariant trace on $A$.

Inspired by a construction in [18], we construct in Section 4 a spectral
triple on $A$ and a twisted spectral triple on the opposite algebra
$A^{\text{op}}$, which encode the geometric information of the conformally
perturbed metric. We study the Dirac operator of the perturbed metric
carefully and prove that it is selfadjoint and enjoys having the same spectral
dimension as the non-perturbed case. It should be stressed that ergodicity
plays a crucial role for the latter to hold.

The existence of an analog of the Chern–Gauss–Bonnet theorem is studied in
Section 5 by proving a Hodge decomposition theorem for our complex and showing
that its Euler characteristic is independent of the conformal factor.
Combining this with the McKean–Singer index formula and small time asymptotic
expansions, which often exist for noncommutative geometric spaces, we explain
how the analog of the Euler class or the Pfaffian of the curvature form can be
computed as local geometric invariants of examples that fit into our setting.
Indeed, such invariants depend on the behavior at infinity of the eigenvalues
of the involved Laplacians and the action of the algebra. Finally, our main
results and conclusions are summarized in Section 6.

##  2 Preliminaries

We start by some reminders about results and notations from various anterior
articles.

######  Definition 2.1.

Given a strongly continuous action $\alpha$ of a compact group $G$ on a unital
$C^{*}$-algebra $A$, we say that it is _ergodic_ if the fixed algebra of
$G$-invariants elements is reduced to the scalars, i.e., if $\forall\,g\in G$,
$\alpha_{g}(a)=a$, then $a\in\mathbb{C}1_{A}$.

Among the important results obtained with this notion of ergodic action, let
us quote the following [35, Theorem 4.1, p. 82]:

######  Theorem 2.2.

Let $A$ be a unital $C^{*}$-algebra, $G$ a compact group and $\alpha$ a
strongly continuous representation of $G$ as an ergodic group of
$*$-automorphisms of $A$, then the unique $G$-invariant state $\varphi_{0}$ on
$A$ is a trace.

Another result that will play an important role in our article is [35,
Proposition 2.1, p. 76], which we adapt slightly in the following:

######  Proposition 2.3.

Let $A$ be a unital $C^{*}$-algebra, $G$ a compact group and $\alpha$ a
strongly continuous representation of $G$ as an ergodic group of
$*$-automorphisms of $A$. Let $V$ be an irreducible unitary representation of
$G$, $A(V)$ the spectral subspace of $V$ in $A$ and $m(V)$ the multiplicity of
$V$ in $A(V)$. Then we have

$\displaystyle m(V)\leqslant\dim V.$

Among our main results, we prove the finite summability of certain spectral
triples (ordinary and twisted), we therefore define those terms:

######  Definition 2.4.

Let $A$ be a unital $C^{*}$-algebra. An odd (ordinary) _spectral triple_ ,
also called an odd _unbounded Fredholm module_ , is a triple
$(\mathcal{A},\mathscr{H},D)$ where

  * •

$\mathscr{H}$ is a Hilbert space and $\pi\colon A\to B(\mathscr{H})$ a
$*$-representation of $A$ as bounded operators on $\mathscr{H}$,

  * •

$D$ is a selfadjoint unbounded operator – which we will call the _Dirac
operator_ – with domain $\operatorname{Dom}(D)$,

such that

  1. (i)

$(1+D^{2})^{-1}$ is a compact operator,

  2. (ii)

the subalgebra $\mathcal{A}$ of all $a\in A$ such that

$\displaystyle\pi(a)(\operatorname{Dom}(D))\
\subseteq\operatorname{Dom}(D)\qquad\text{and}\qquad[D,\pi(a)]\ \text{ extends
to a bounded map on }\mathscr{H}$

is dense in $A$.

An _even spectral triple_ is given by the same data, but we further require
that a grading $\gamma$ be given on $\mathscr{H}$ such that (i) $A$ acts by
even operators, (ii) $D$ is odd.

######  Remark 2.5.

For a selfadjoint operator $D$, condition (i) of the definition above is
actually equivalent to $\exists\,\lambda\in\mathbb{R}{\setminus}\\{0\\}$ s.t.
$(D+i\lambda)^{-1}$ is a compact operator.

To define _finitely summable_ spectral triples, we now need a brief reminder
regarding _trace ideals_ (also known as _symmetric ideals_), for which we
follow Chapter IV of [14]. For more details concerning symmetrically normed
operator ideals and singular traces we refer the reader to [53] and [46].

######  Definition 2.6.

For $p>1$, the ideal $\mathscr{L}^{p^{+}}$ (also denoted
$\mathscr{L}^{(p,\infty)}$ in [14] and $\mathcal{J}_{p,\omega}$ in [53, p.
21]) consists of all compact operators $T$ on $\mathscr{H}$ such that

$\displaystyle\|T\|_{p^{+}}:=\sup_{k}\frac{\sigma_{k}(T)}{k^{(p-1)/p}}<\infty,$

where $\sigma_{k}$ is defined as the supremum of the trace norms of $TE$, when
$E$ is an orthonormal projection of dimension $k$, i.e.,

$\displaystyle\sigma_{k}(T):=\sup\\{\|TE\|_{1},\dim E=k\\}.$

Equivalently, $\sigma_{k}(T)$ is the sum of the $k$ largest eigenvalues
(counted with their multiplicities) of the positive compact operator
$|T|:=(T^{*}T)^{1/2}$. The definition extends to the case of $p=1$:
$\mathscr{L}^{1^{+}}$ is the ideal of compact operators $T$ s.t.

$\displaystyle\|T\|_{1^{+}}:=\sup_{k}\frac{\sigma_{k}(T)}{\log k}<\infty.$

The elements of $\mathscr{L}^{p^{+}}$ are called _$p^{+}$ -summable_ (or
$(p,\infty)$-summable – see [14, Section IV.2$\alpha$, p. 299 and following]).

A _spectral dimension_ for spectral triples is defined as follows

######  Definition 2.7.

A spectral triple is _$p^{+}$ -summable_ if
$(1+D^{2})^{-1/2}\in\mathscr{L}^{p^{+}}$.

Finally, we will consider _twisted spectral triples_ (also called _$\sigma$
-spectral triples_) as introduced in [16, Definition 3.1]. This is a spectral
triple just like in Definition 2.4, but for a fixed automorphism $\sigma$ of
$\mathcal{A}$, the bounded commutators condition (denoted (ii) above) is
replaced by

  1. (ii)

the subalgebra $\mathcal{A}$ of all $a\in A$ such that

     * •

$\pi(a)(\operatorname{Dom}(D))\subseteq\operatorname{Dom}(D)$,

     * •

$D\pi(a)-\pi(\sigma(a))D$ extends to a bounded map on $\mathscr{H}$

is dense in $A$.

In this paper, we will need the subalgebras $\mathcal{A}^{k}$ for $k\geqslant
0$, corresponding to the $C^{k}$-differentiable class. Following [5, Section
2.2], we introduce the space

$\displaystyle\mathcal{A}^{m}:=\big{\\{}a\in A\colon
g\mapsto\alpha_{g}(a)\text{ is in }C^{m}(G,A)\big{\\}}.$

Let us fix a basis $(\partial_{i})$ of the Lie algebra $\mathfrak{g}$. For
such a choice of basis, the infinitesimal generators $\partial_{i}$ act as
derivations $\mathcal{A}^{m}\to\mathcal{A}^{m-1}$. According to [5, Example
2.2.4, p. 41], $\mathcal{A}^{m}$ equipped with the norm

$\displaystyle\|a\|_{m}:=\|a\|+\sum_{k=1}^{m}\sum_{i_{1}=1}^{n}\cdots\sum_{i_{k}=1}^{n}\frac{\|\partial_{i_{1}}\cdots\partial_{i_{k}}(a)\|}{k!},$

is a Banach algebra with $\|ab\|_{m}\leqslant\|a\|_{m}\|b\|_{m}$. In
particular, if $h\in\mathcal{A}^{1}$, then $e^{\lambda h}\in\mathcal{A}^{1}$,
for all complex number $\lambda$ – see also Lemma 3.3 below for a more precise
estimate.

Following the density properties established in [5] (see, e.g., Definition
2.2.15, p. 47), the intersection
$\mathcal{A}^{\infty}=\bigcap_{j=0}^{\infty}\mathcal{A}^{j}$ is a dense
$*$-subalgebra of the $C^{*}$-algebra $A$, which is stable under the
derivations $\partial_{i}$.

##  3 Hodge–de Rham Dirac operator and $\boldsymbol{C^{*}}$-dynamical systems

In this article, we consider a fixed $A$, a $C^{*}$-algebra with an ergodic
action $\alpha$ of a compact Lie group $G$ of dimension $n$. We write
$\mathcal{A}$ for $\mathcal{A}^{\infty}$, the “smooth subalgebra” of $A$,
which can alternatively be defined as

$\displaystyle\mathcal{A}:=\\{a\in A\colon g\mapsto\alpha_{g}(a)\text{ is in
}C^{\infty}(G,A)\\}.$

The Chevalley–Eilenberg cochain complex with coefficients in $\mathcal{A}$
provides a complex that we interpret as “differential forms” on $\mathcal{A}$.
For the reader’s convenience and to fix notations, we provide a reminder of
this construction. For all $k\in\mathbb{N}$,

$\displaystyle\Omega^{k}:=\mathcal{A}\otimes\bigwedge^{k}\mathfrak{g}^{*},$

where $\mathfrak{g}^{*}$ denotes the linear forms on $\mathfrak{g}$, the Lie
algebra of the Lie group $G$. Given a scalar product on $\mathfrak{g}^{*}$
(e.g., obtained from the Killing form), we can extend it to a scalar product
on $\bigwedge^{k}\mathfrak{g}^{*}$ by setting

$\displaystyle\langle v_{1}\wedge\cdots\wedge v_{k},w_{1}\wedge\cdots\wedge
w_{k}\rangle:=\det(\langle v_{i},w_{j}\rangle),$

i.e., the determinant of the matrix of scalar products. We fix an orthonormal
basis $(\omega_{j})_{j=1,\ldots,n}$ of $\mathfrak{g}^{*}$ for this scalar
product and consider its dual basis $(\partial_{j})_{j=1,\ldots,n}$ in
$\mathfrak{g}$.

Following [41, the model of (4.6), p. 157], we write the exterior derivative
of the complex

$\displaystyle
d(a\otimes\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}})=\sum_{j=1}^{n}\partial_{j}(a)\otimes\omega_{j}\wedge\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}}$
$\displaystyle\qquad{}-\frac{1}{2}\sum_{k=1}^{K}\sum_{\alpha,\beta}(-1)^{k+1}c^{i_{k}}_{\alpha\beta}a\otimes\omega_{\alpha}\wedge\omega_{\beta}\wedge\omega_{i_{1}}\wedge\cdots\wedge\omega_{i_{k-1}}\wedge\omega_{i_{k+1}}\wedge\cdots\wedge\omega_{i_{K}},$
(3.1)

where
$[\partial_{i},\partial_{j}]=\sum\limits_{k=1}^{n}c^{k}_{ij}\partial_{k}$ –
the $c^{k}_{ij}$ are called the _structure constants_ of the Lie algebra
$\mathfrak{g}$. A lengthy but straightforward computation proves that this
exterior derivative satisfies $d^{2}=0$ on $\Omega^{\bullet}$, therefore
$(\Omega^{\bullet},d)$ is a complex.

######  Remark 3.1.

The Chevalley–Eilenberg complex is available even for noncompact groups $G$
and nonergodic actions. In other words, the square $d^{2}$ actually vanishes
even when $G$ is _not_ a  _compact_ Lie group and when the action of $G$ on
$A$ is _not_ ergodic.

The natural product on $\Omega^{\bullet}$ is

$\displaystyle(a\otimes v_{1}\wedge\cdots\wedge v_{k})\cdot(a^{\prime}\otimes
w_{1}\wedge\cdots\wedge w_{k^{\prime}}):=aa^{\prime}\otimes
v_{1}\wedge\cdots\wedge v_{k}\wedge w_{1}\wedge\cdots\wedge w_{k^{\prime}},$
(3.2)

i.e., the product of a $k$-form with a $k^{\prime}$-form is a
$k+k^{\prime}$-form. In particular, for all $k$, $\Omega^{k}$ is an
$\mathcal{A}$-bimodule. The exterior derivative $d$ is compatible with the
right module structure in the following sense

$\displaystyle
d(aa^{\prime}\otimes\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}})=\sum_{j=1}^{n}\partial_{j}(aa^{\prime})\otimes\omega_{j}\wedge\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}}$
$\displaystyle\qquad\quad{}-\frac{1}{2}\sum_{k}\sum_{\alpha,\beta}c^{i_{k}}_{\alpha\beta}aa^{\prime}\otimes\omega_{\alpha}\wedge\omega_{\beta}\wedge\omega_{i_{1}}\wedge\cdots\wedge\omega_{i_{k-1}}\wedge\omega_{i_{k+1}}\wedge\cdots\wedge\omega_{i_{K}}$
$\displaystyle\qquad{}=\left(\sum_{j=1}^{n}\partial_{j}(a)\otimes\omega_{j}\wedge\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}}\right)a^{\prime}$
$\displaystyle\qquad\quad{}+\sum_{j=1}^{n}a\partial_{j}(a^{\prime})\otimes\omega_{j}\wedge\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}}$
$\displaystyle\qquad\quad{}-\frac{1}{2}\left(\sum_{k}\sum_{\alpha,\beta}c^{i_{k}}_{\alpha\beta}a\otimes\omega_{\alpha}\wedge\omega_{\beta}\wedge\omega_{i_{1}}\wedge\cdots\wedge\omega_{i_{k-1}}\wedge\omega_{i_{k+1}}\wedge\cdots\wedge\omega_{i_{K}}\right)a^{\prime}$
$\displaystyle\qquad{}=d(a\otimes\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}})a^{\prime}+(-1)^{K}(a\otimes\omega_{j}\wedge\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}})d(a^{\prime}).$

Since we want to treat conformal deformations of the original structure, we
follow [18] and fix a positive invertible element $e^{h}\in\mathcal{A}^{1}$,
where $h$ is a smooth selfadjoint element in $\mathcal{A}^{1}$. Then we define
a scalar product on $\Omega^{k}$ by the formula

$\displaystyle(a\otimes v_{1}\wedge\cdots\wedge v_{k},a^{\prime}\otimes
w_{1}\wedge\cdots\wedge
w_{k})_{\varphi}:=\varphi_{0}\big{(}a^{*}a^{\prime}e^{(n/2-k)h}\big{)}\det(\langle
v_{i},w_{j}\rangle),$ (3.3)

where $\varphi_{0}$ is the unique $G$-invariant state on $A$, which is
actually a trace according to [1, Theorem 3.1, p. 8]. We set the scalar
product of two forms of different degrees to vanish. The scalar product
obtained for $h=0$ is the one we call the _natural scalar product_ on forms.
We define the Hilbert space $\mathscr{H}_{\varphi}$ as the completion of
$\Omega^{\bullet}$ for the scalar product (3.3). In the particular case of
$h=0$, we obtain our reference Hilbert space $\mathscr{H}$. We will also need
the Hilbert spaces
$\mathscr{H}_{0,\varphi}:=\operatorname{GNS}(\mathcal{A},\varphi)$ and
$\mathscr{H}_{0}:=\operatorname{GNS}(\mathcal{A},\varphi_{0})$ as well as the
Hilbert spaces
$\mathscr{H}_{k}:=\mathscr{H}_{0}\otimes\bigwedge^{k}\mathfrak{g}^{*}$ – i.e.,
the completion of $k$-forms – and $\mathscr{H}_{k,\varphi}$.

To understand why we choose the form (3.3) for the conformal deformation, we
compare with the commutative case of a $n$-dimensional compact manifold $M$,
where we have the following property: if the Riemannian metric is transformed
by $g\rightsquigarrow\lambda g$ (for $\lambda>0$), then the (pointwise) norm
of all vectors is multiplied by $\lambda^{1/2}$ and thus the pointwise norm of
$1$-forms is multiplied by $\lambda^{-1/2}$. This in turn implies that the
pointwise norm of $k$-form is multiplied by $\lambda^{-k/2}$. Finally, the
(global) scalar product of $k$-forms is the integral of the pointwise scalar
products. Since under the conformal deformation, the total volume of the
manifold $M$ is multiplied by $\lambda^{n/2}$, the (global) scalar products of
$k$-forms are multiplied by $\lambda^{n/2-k}$. In particular, if $n$ is even
and $k=n/2$, then the scalar product on $n/2$-forms is left invariant under
the conformal deformation.

In order to study $d$ and its adjoint, we introduce the degree $1$ maps
$T_{j}\colon\bigwedge^{\bullet}\mathfrak{g}^{*}\to\bigwedge^{\bullet}\mathfrak{g}^{*}$
defined for all $j\in\\{1,\ldots,n\\}$ by

$\displaystyle T_{j}(v_{1}\wedge\cdots\wedge v_{k})=\omega_{j}\wedge
v_{1}\wedge\cdots\wedge v_{k}.$

Let $\mathcal{R}_{x}$ denote the right multiplication operator for any $x\in
A$: $\mathcal{R}_{x}(a)=ax$, and let $B^{i_{k}}_{\alpha\beta}$ be the bounded
operator on $\bigwedge^{\bullet}\mathfrak{g}^{*}$ defined using the basis
$(\omega_{j})$ by

$\displaystyle
B^{i_{k}}_{\alpha\beta}(\omega_{i_{1}}\wedge\omega_{i_{2}}\wedge\cdots\wedge\omega_{i_{K}})=\omega_{\alpha}\wedge\omega_{\beta}\wedge\omega_{i_{1}}\wedge\cdots\wedge\omega_{i_{k-1}}\wedge\omega_{i_{k+1}}\wedge\cdots\wedge\omega_{i_{K}}.$

We can now give an explicit form to the operator $d$ and its (formal) adjoint
for the unperturbed metric.

######  Lemma 3.2.

With the previous notations, when $h=0$, the operator $d$ can be written

$\displaystyle d=\sum_{j}\partial_{j}\otimes
T_{j}-\frac{1}{2}\sum_{k,\alpha,\beta}c^{i_{k}}_{\alpha\beta}\otimes
B^{i_{k}}_{\alpha\beta}$

and its adjoint $d^{*}$ is

$\displaystyle d^{*}=\sum_{j}\partial_{j}\otimes
T_{j}^{*}-\frac{1}{2}\sum_{k,\alpha,\beta}\overline{c^{i_{k}}_{\alpha\beta}}\otimes\big{(}B^{i_{k}}_{\alpha\beta}\big{)}^{*}.$

###### Proof.

The only point that is not self-explanatory is the behavior of $\partial_{j}$
with respect to the trace $\varphi_{0}$

$\displaystyle\langle[\partial_{j}(a)],[a^{\prime}]\rangle=\varphi_{0}(\partial_{j}(a)^{*}a^{\prime})=\varphi_{0}(-\partial_{j}(a^{*})a^{\prime})=\varphi_{0}(a^{*}\partial_{j}(a^{\prime})),$

where we used the relations $\partial_{j}(a)^{*}=-\partial_{j}(a^{*})$ and
$\varphi_{0}(\partial_{j}(a))=0$. ∎

######  Lemma 3.3.

Let $h$ be an element of $\mathcal{A}^{1}$ and $\partial$ be an infinitesimal
generator of $G$, acting as a derivation on $\mathcal{A}^{1}$, then
$\partial(e^{h})$ is in the $C^{*}$-algebra $A$ and satisfies

$\displaystyle\big{\|}\partial\big{(}e^{h}\big{)}\big{\|}\leqslant\|\partial(h)\|e^{\|h\|}.$

In particular, for a scalar parameter $v\to 0$, $\partial(e^{vh})\to 0$.

###### Proof.

As an operator from $\mathcal{A}^{1}$ to $A$, the derivation $\partial$ is
continuous, therefore we can estimate $\partial(e^{h})$ by using an
$\mathcal{A}^{1}$-converging sequence, like the partial sums of $e^{h}$.

For this sequence, using the derivation property, we get

$\displaystyle\left\|\partial\left(\sum_{k=0}^{N}\frac{h^{k}}{k!}\right)\right\|\leqslant\sum_{k=1}^{N}k\frac{\|\partial
h\|}{k}\frac{\|h\|^{k-1}}{(k-1)!}\leqslant\|\partial h\|e^{\|h\|}.$

The property $\partial(e^{vh})\to 0$ as $v\to 0$ follows immediately. ∎

We call $d_{\varphi}$ the operator defined on $\mathscr{H}_{\varphi}$ by the
formula (3.1). Once the scalar product (3.3) is defined, we want to define an
adjoint $d^{*}_{\varphi}$ to $d_{\varphi}$ for this scalar product. The
Hodge–de Rham operator that we would like to study in fine is
$d_{\varphi}+d_{\varphi}^{*}$. However, the unbounded operator $d_{\varphi}$
is a priori arbitrary, so it is not clear that it admits a densely defined
adjoint. To clarify the relations between $\mathscr{H}$ and
$\mathscr{H}_{\varphi}$, we introduce the following lemma:

######  Lemma 3.4.

For any selfadjoint $h\in\mathcal{A}^{1}$,

  * •

the Hilbert space $\mathscr{H}_{\varphi}$ is equipped with a
$G$-representation defined on degree $k$ forms by $\mathbb{V}_{g}([a\otimes
v_{1}\wedge\cdots\wedge v_{k}]_{\varphi})=[\alpha_{g}(a)\otimes
v_{1}\wedge\cdots\wedge v_{k}]_{\varphi}$, which leads to a $G$-equivariant
left $\mathcal{A}$-module structures on $\mathscr{H}_{\varphi}$;

  * •

the map $L\colon\mathscr{H}_{0}\to\mathscr{H}_{0,\varphi}$ defined by
$L([a]):=[a]_{\varphi}$ is invertible and intertwines the $G$-equivariant left
$\mathcal{A}$-module structures on $\mathscr{H}_{0}$ and
$\mathscr{H}_{0,\varphi}$.

Of course, $L$ extends to
$L\otimes\operatorname{Id}\colon\mathscr{H}\to\mathscr{H}_{\varphi}$ which is
still a continuous and invertible map. We denote its adjoint by
$H\colon\mathscr{H}_{\varphi}\to\mathscr{H}$, whose explicit form is

$\displaystyle H([a\otimes v_{1}\wedge\cdots\wedge
v_{k}]_{\varphi})=\big{[}ae^{(n/2-k)h}\otimes v_{1}\wedge\cdots\wedge
v_{k}\big{]}.$

Finally, $\mathscr{H}$ and $\mathscr{H}_{\varphi}$ are related by the unitary
map $U\colon\mathscr{H}\to\mathscr{H}_{\varphi}$ given on degree $k$ forms by

$\displaystyle U([a\otimes v_{1}\wedge\cdots\wedge
v_{k}])=\big{[}ae^{-(n/2-k)h/2}\otimes v_{1}\wedge\cdots\wedge
v_{k}\big{]}_{\varphi}.$

######  Remark 3.5.

The map $U$ defined above is unitary, but it does not intertwine the
$G$-structures on $\mathscr{H}$ and $\mathscr{H}_{\varphi}$.

###### Proof.

Since the sum $\mathscr{H}_{\varphi}=\bigoplus_{k}\mathscr{H}_{k,\varphi}$ is
finite, it suffices to check that $\mathbb{V}_{g}$ is continuous on each
$\mathscr{H}_{k,\varphi}$ separately. Since $\mathbb{V}_{g}$ does not act on
$\bigwedge^{\bullet}\mathfrak{g}^{*}$, it is enough to prove continuity on
$\operatorname{GNS}(A,\tilde{\varphi})$ where
$\tilde{\varphi}(a)=\varphi_{0}(ae^{-h_{k}})$ and
$h_{k}=-(n/2-k)h\in\mathcal{A}^{1}$ (corresponding to forms of degree $k$). We
get

$\displaystyle\|\alpha_{g}(a)\|_{\tilde{\varphi}}^{2}=\varphi_{0}\big{(}\alpha_{g}(a)^{*}\alpha_{g}(a)e^{-h_{k}}\big{)}=\varphi_{0}\big{(}a^{*}a\alpha_{g^{-1}}\big{(}e^{-h_{k}}\big{)}\big{)}$
$\displaystyle\hphantom{\|\alpha_{g}(a)\|_{\tilde{\varphi}}^{2}}{}=\varphi_{0}\big{(}ae^{-h_{k}/2}e^{h_{k}/2}\alpha_{g^{-1}}\big{(}e^{-h_{k}}\big{)}e^{h_{k}/2}e^{-h_{k}/2}a^{*}\big{)}\leqslant
K\varphi_{0}\big{(}ae^{-h_{k}}a^{*}\big{)}=K\|a\|_{\tilde{\varphi}}^{2},$

for a constant $K=\|e^{h_{k}/2}\alpha_{g^{-1}}(e^{-h_{k}})e^{h_{k}/2}\|$. The
above (scalar) inequality follows from the inequality of operators

$\displaystyle
ae^{-h_{k}/2}e^{h_{k}/2}\alpha_{g^{-1}}\big{(}e^{-h_{k}}\big{)}e^{h_{k}/2}e^{-h_{k}/2}a^{*}\leqslant
ae^{-h_{k}/2}Ke^{-h_{k}/2}a^{*}=Kae^{-h_{k}}a^{*},$

which is valid since $e^{h_{k}/2}\alpha_{g^{-1}}(e^{-h_{k}})e^{h_{k}/2}$ is a
positive operator. It then suffices to apply the positive functional
$\varphi_{0}$. Once we know that the map $\mathbb{V}_{g}$ is defined on the
full Hilbert space, proving that it is compatible with the left
$\mathcal{A}$-module structure is a formality.

The process is similar for $L$: it is clear from the definition that if the
map $L$ exists, then it intertwines the $G$-equivariant left
$\mathcal{A}$-module structures on $\mathscr{H}$ and $\mathscr{H}_{\varphi}$.
It remains to prove that $L$ is well-defined and invertible.

We first evaluate

$\displaystyle\|L(a)\|_{\tilde{\varphi}}^{2}=\varphi_{0}\big{(}a^{*}ae^{-h_{k}}\big{)}=\varphi_{0}\big{(}ae^{-h_{k}}a^{*}\big{)}\leqslant\big{\|}e^{-h_{k}}\big{\|}\varphi_{0}(aa^{*})=\big{\|}e^{-h_{k}}\big{\|}\,\|a\|_{\varphi_{0}}^{2},$

by the same argument as above.

To prove that $L$ is invertible, consider the norm of its inverse

$\displaystyle\|a\|_{\varphi_{0}}^{2}=\varphi_{0}\big{(}ae^{-h_{k}/2}e^{h_{k}}e^{-h_{k}/2}a^{*}\big{)}\leqslant\big{\|}e^{h_{k}}\big{\|}\varphi_{0}\big{(}ae^{-h_{k}}a^{*}\big{)}=\big{\|}e^{h_{k}}\big{\|}\,\|a\|_{\tilde{\varphi}}^{2}.$

The evaluation of the adjoint $H$ of $L$ and of the unitary map
$U\colon\mathscr{H}\to\mathscr{H}_{\varphi}$ is an easy exercise. ∎

With the previous notations, we see that
$d_{\varphi}=(L\otimes\operatorname{Id})d(L\otimes\operatorname{Id})^{-1}$ and
thus (at least formally) $d_{\varphi}^{*}=H^{-1}d^{*}H$. However, in order to
facilitate the comparison between $d_{\varphi}+d^{*}_{\varphi}$ acting on
$\mathscr{H}_{\varphi}$ and $d+d^{*}$ acting on $\mathscr{H}$, we “push”
$d_{\varphi}+d_{\varphi}^{*}$ to $\mathscr{H}$ using the unitary $U$. This
leads us to the operators $D_{u}$ studied in the Proposition 3.6 below. But
first, for $h=h^{*}$, we need to introduce the operators
$K_{u}\colon\mathscr{H}\to\mathscr{H}$, where $u\in\mathbb{R}$, defined by

$\displaystyle K_{u}([a\otimes v_{1}\wedge\cdots\wedge
v_{k}])=\big{[}ae^{(n/2-k)hu}\otimes v_{1}\wedge\cdots\wedge v_{k}\big{]}.$
(3.4)

This is a one-parameter group of invertible selfadjoint operators. Moreover,
following our remark on $\mathcal{A}^{1}$ at the end of Section 2, for all
$u\in\mathbb{R}$, the operators $K_{u}$ preserve the space
$\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$.

In the proof below, we consider the orthogonal projections
$\Pi_{k}\colon\mathscr{H}\to\mathscr{H}_{k}$ onto the completion of the
$k$-forms, for all $k\in\\{0,\ldots,n\\}$.

######  Proposition 3.6.

For all $u\in[0,1]$, we consider two unbounded operators defined on the dense
domain
$C=\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}\subseteq\mathscr{H}$,

$\displaystyle d_{u}:=K_{u}dK_{-u},\qquad d_{u}^{*}:=K_{-u}d^{*}K_{u}.$

We have:

  1. $(1)$

if $E_{+,u}$ and $E_{-,u}$ are, respectively, the closures in $\mathscr{H}$ of
the images of the operators $d_{u}$ and $d_{u}^{*}$, then $E_{+,u}$ and
$E_{-,u}$ are orthogonal in $\mathscr{H}$; we denote by $\Pi_{+,u}$ and
$\Pi_{-,u}$, respectively, the orthogonal projections on these spaces;

  2. $(2)$

the operator

$\displaystyle D_{u}=K_{u}dK_{-u}+K_{-u}d^{*}K_{u},$

is essentially selfadjoint on a common core domain
$C=\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$.

The family of operators $D_{u}$ satisfies the estimate

$\displaystyle\|(D_{u+v}-D_{u})(\underline{\omega})\|\leqslant
o_{v}(1)\|D_{u}\underline{\omega}\|+o_{v}(1)\|\underline{\omega}\|,$ (3.5)

where $\underline{\omega}$ is any vector in the common selfadjointness domain
and following Landau’s notations, $o_{v}(1)$ stands for functions of $v$ which
tend to $0$ for $v\to 0$. We can choose these two functions independently of
the parameter $u\in[0,1]$.

###### Proof.

Regarding point (1), we start by proving the property for $u=0$, i.e., for the
untwisted case. There, following Lemma 3.4 the trace $\varphi_{0}$ is
$G$-invariant and therefore the action $\mathbb{V}_{g}$ of $G$ on
$\mathscr{H}_{0}$ is unitary. Consequently, the $G$-representation can be
decomposed into a direct sum of finite-dimensional $G$-representations. Let us
denote by $V$ one of these finite-dimensional spaces.

It is clear from the definition (3.1) that both
$V\otimes\bigwedge^{k}\mathfrak{g}^{*}\subseteq\mathscr{H}$ and its orthogonal
are stable under the action of $d$. Thus the restriction of $d$ to the finite-
dimensional space $V\otimes\bigwedge^{k}\mathfrak{g}^{*}$ is bounded and
admits an adjoint $d^{*}$ whose form is given by Lemma 3.2. Varying the space
$V$, we see that $d^{*}$ is defined on $\mathscr{D}$, the algebraic direct sum
of $V\otimes\bigwedge^{k}\mathfrak{g}^{*}$, which is a dense subset of
$\mathscr{H}$. If we restrict to the case of forms $\underline{\omega}$,
$\underline{\omega}^{\prime}$ in the space
$V\otimes\bigwedge^{k}\mathfrak{g}^{*}\subseteq\mathscr{H}$, we have

$\displaystyle\langle
d\underline{\omega},d^{*}\underline{\omega}^{\prime}\rangle=\langle
d^{2}\underline{\omega},\underline{\omega}^{\prime}\rangle=0,$

there are no considerations of domains for $d$ and $d^{*}$, since we consider
finite-dimensional spaces. The same argument applied to different finite
vector spaces $V$ proves that $E_{+,0}$ (the image of $d$) and $E_{-,0}$ (the
image of $d^{*}$) are orthogonal.

To treat point (1) for a general $u\in[0,1]$, we note that
$E_{+,u}=K_{u}E_{+,0}$: indeed, if $\xi=\lim d\underline{\omega}_{n}$, then
$K_{u}\xi=\lim K_{u}dK_{-u}(K_{u}\underline{\omega}_{n})$ and vice versa.
Similarly, $E_{-,u}=K_{-u}E_{-,0}$. Given $K_{u}e_{+}\in E_{+,u}$ and
$K_{-u}e_{-}\in E_{-,u}$, we have

$\displaystyle\langle K_{u}e_{+},K_{-u}e_{-}\rangle=\langle
e_{+},e_{-}\rangle=0,$

since $K_{u}$ is selfadjoint and $e_{+}\in E_{+,0}$, $e_{-}\in E_{-,0}$ are
orthogonal. This proves the requested orthogonality relation.

Regarding point (2), let us start by giving a sketch of the proof: we first
prove that $D_{0}=D$ is essentially selfadjoint on the requested domain. Using
the estimate (3.5), we then apply Kato–Rellich theorem to show that if $D_{u}$
is essentially selfadjoint for the domain $C$, then so is the operator
$D_{u+v}$ for all $|v|\leqslant\varepsilon$, where $\varepsilon$ is
independent of the point $u\in[0,1]$ chosen. As a consequence, all operators
$D_{u}$ are essentially selfadjoint for the fixed domain.

We first prove that $D_{0}$ is selfadjoint. This is done by using the
Peter–Weyl decomposition of $\mathscr{H}_{0}$ for the unitary action
$\mathbb{V}_{g}$ of $G$ on $\mathscr{H}_{0}$. As mentioned in point (1), the
restriction of $d$ to this finite-dimensional space
$V\otimes\bigwedge^{k}\mathfrak{g}^{*}\subseteq\mathscr{H}$ is well-defined,
as is its adjoint $d^{*}$. Varying the space $V$, we consider $\mathscr{D}$,
the direct sum of $V\otimes\bigwedge^{k}\mathfrak{g}^{*}$, which is a dense
subset of $\mathscr{H}$.

In this situation, we can define $D=d+d^{*}$ on $\mathscr{D}$. If we restrict
$D$ to a component
$V\otimes\bigwedge^{k}\mathfrak{g}^{*}\subseteq\mathscr{H}$, it is formally
selfadjoint by definition. It therefore admits an orthonormal basis of
eigenvectors with real associated eigenvalues. It follows that
$\operatorname{Ran}(D+i)$ and $\operatorname{Ran}(D-i)$ are dense in
$\mathscr{H}$, and this is enough to prove that $D$ is essentially selfadjoint
on the domain $C$ (see [50, Corollary, p. 257]).

Let us now consider an arbitrary $u\in[0,1]$. We want to find $\varepsilon>0$
uniform in $u$ and small enough so that for all $v$ with
$|v|\leqslant\varepsilon$, the operator $D_{u+v}$ is essentially selfadjoint.
By definition,

$\displaystyle D_{u+v}=K_{u+v}dK_{-(u+v)}+K_{-(u+v)}d^{*}K_{u+v}.$

If we introduce $R_{v}=K_{v}-1$, then we can write

$\displaystyle K_{u+v}=K_{u}(1+R_{v}),\qquad K_{-(u+v)}=(1+R_{-v})K_{-u}.$

It is clear that both $R_{v}$ and $R_{-v}$ are bounded with $\|R_{v}\|\to 0$,
$\|R_{-v}\|\to 0$ for $v\to 0$ and by definition, for all $v\in\mathbb{R}$,
$R_{v}$ commutes with $K_{u}$ for $u\in\mathbb{R}$.

We write

$\displaystyle
D_{u+v}=K_{u}(1+R_{v})d(1+R_{-v})K_{-u}+K_{-u}(1+R_{-v})d^{*}(1+R_{v})K_{u}$
$\displaystyle\hphantom{D_{u+v}}{}=K_{u}dK_{-u}+K_{u}dR_{-v}K_{-u}+K_{u}R_{v}dK_{-u}+K_{u}R_{v}dR_{-v}K_{-u}$
$\displaystyle\hphantom{D_{u+v}=}{}+K_{-u}d^{*}K_{u}+K_{-u}d^{*}R_{v}K_{u}+K_{-u}R_{-v}d^{*}K_{u}+K_{-u}R_{-v}d^{*}R_{v}K_{u}.$

The sum of the terms $K_{u}dK_{-u}$ and $K_{-u}d^{*}K_{u}$ gives back $D_{u}$.
Since $E_{+,u}$ and $E_{-,u}$ are orthogonal, we have

$\displaystyle\|D_{u}\underline{\omega}\|^{2}=\|\Pi_{+,u}(D_{u}\underline{\omega})\|^{2}+\|\Pi_{-,u}(D_{u}\underline{\omega})\|^{2}=\|d_{u}\underline{\omega}\|^{2}+\|d_{u}^{*}\underline{\omega}\|^{2}.$
(3.6)

Both $D_{u+v}$ and $D_{u}$ are symmetric operators, so their difference
(namely the sum $\Sigma$ of the six remaining terms) is a symmetric operator.
By hypothesis, $D_{u}$ is selfadjoint. According to Kato–Rellich theorem as
stated in [51, Theorem X.12, p. 162], it therefore only remains to prove that
$C$ is also a domain for $\Sigma$ and that for all $\underline{\omega}\in C$,

$\displaystyle\|\Sigma(\underline{\omega})\|\leqslant
a\|D_{u}\underline{\omega}\|+b\|\underline{\omega}\|,$

where both real numbers $a$, $b$ are positive and $a<1$. It is clear from the
definition of $K_{\pm u}$ and $R_{v}$ that their actions preserve the core $C$
of $C^{1}$-functions on $G$ and thus $\Sigma(\underline{\omega})$ has a well-
defined meaning for all $\underline{\omega}\in C$. We decompose
$\underline{\omega}\in C$ into a sum
$\underline{\omega}=\sum_{k}\underline{\omega}_{k}$ of $C^{1}$-forms of degree
$k$ and start by an estimate of the different terms
$\|\Sigma\underline{\omega}_{k}\|$ for any fixed $k$.

Remember from Lemma 3.2 that $d$ can be written

$\displaystyle d=\sum_{j}\partial_{j}\otimes
T_{j}-\frac{1}{2}\sum_{k,\alpha,\beta}c^{i_{k}}_{\alpha\beta}\otimes
B^{i_{k}}_{\alpha\beta},$

where the different $B^{i_{k}}_{\alpha\beta}$ are bounded operators. We remark
that the $B^{i_{k}}_{\alpha\beta}$ commute with right multiplications, like
the one appearing in the definition of $K_{u}$ acting on an element of given
degree. For $\underline{\omega}=a\otimes\underline{v}$ of degree $k$, we write

$\displaystyle
K_{u}dR_{-v}K_{-u}a\otimes\underline{v}=\sum\partial_{j}(ae^{-(n/2-k)hu}(e^{-(n/2-k)hv}-1))e^{(n/2-(k+1))hu}\otimes
T_{j}(\underline{v})$
$\displaystyle\hphantom{K_{u}dR_{-v}K_{-u}a\otimes\underline{v}=}{}-\frac{1}{2}\sum_{k}\sum_{\alpha,\beta}c^{i_{k}}_{\alpha\beta}ae^{-(n/2-k)hu}(e^{-(n/2-k)hv}-1)e^{(n/2-(k+1))hu}\otimes
B^{i_{k}}_{\alpha\beta}\underline{v}$
$\displaystyle\hphantom{K_{u}dR_{-v}K_{-u}a\otimes\underline{v}}{}=\sum\partial_{j}(ae^{-(n/2-k)hu})e^{(n/2-(k+1))hu}(e^{-(n/2-k)hv}-1)\otimes
T_{j}(\underline{v})$
$\displaystyle\hphantom{K_{u}dR_{-v}K_{-u}a\otimes\underline{v}=}{}-\frac{1}{2}\sum_{k}\sum_{\alpha,\beta}c^{i_{k}}_{\alpha\beta}ae^{-hu}(e^{-(n/2-k)hv}-1)\otimes
B^{i_{k}}_{\alpha\beta}\underline{v}$
$\displaystyle\hphantom{K_{u}dR_{-v}K_{-u}a\otimes\underline{v}=}{}+\sum
ae^{-(n/2-k)hu}\partial_{j}(e^{-(n/2-k)hv}-1))e^{(n/2-(k+1))hu}\otimes
T_{j}(\underline{v})$
$\displaystyle\hphantom{K_{u}dR_{-v}K_{-u}a\otimes\underline{v}}{}=R_{-v}(K_{u}dK_{-u})(a\otimes\underline{\omega})\\!+K_{u}\circ\\!\sum_{j}(\mathcal{R}_{\partial_{j}(e^{-(n/2-k)hv})}\\!\otimes
T_{j})\circ K_{-u}(a\otimes\underline{v}).$

Taking a linear combination to treat the case of a sum
$\underline{\omega}=\sum\underline{\omega}_{k}$, we get

$\displaystyle
K_{u}dR_{-v}K_{-u}=R_{-v}(K_{u}dK_{-u})+K_{u}\circ\bigg{(}\sum_{j,k}(\mathcal{R}_{\partial_{j}(e^{-(n/2-k)hv})}\otimes
T_{j})\circ\Pi_{k}\bigg{)}\circ K_{-u}.$

In this equality,
$\sum_{j,k}(\mathcal{R}_{\partial_{j}(e^{-(n/2-k)hv})}\otimes
T_{j})\circ\Pi_{k}$ is a finite sum of bounded operators. As a consequence of
Lemma 3.3, the norm of these operators tend to $0$ for $v\to 0$. We already
know that $R_{-v}$ tends to $0$ in norm for $v\to 0$, we therefore get the
estimate

$\displaystyle\|K_{u}dR_{-v}K_{-u}(\underline{\omega})\|\leqslant
o_{v}(1)\|K_{u}dK_{-u}(\underline{\omega})\|+o_{v}(1)\|\underline{\omega}\|.$
(3.7)

The two functions $o_{v}(1)$ can be taken uniform in $u\in[0,1]$, since
$[0,1]$ is a compact.

The term $K_{u}R_{v}dK_{-u}$ is easily treated:
$K_{u}R_{v}dK_{-u}=R_{v}K_{u}dK_{-u}$. The term $K_{u}R_{v}dR_{-v}K_{-u}$ is
processed similarly: $K_{u}R_{v}dR_{-v}K_{-u}=R_{v}K_{u}dR_{-v}K_{-u}$ and
then the estimate (3.7) enables us to write

$\displaystyle\|K_{u}R_{v}dR_{-v}K_{-u}(\underline{\omega})\|\leqslant
o_{v}(1)\big{\|}\big{(}K_{u}dK_{u}^{-1}\big{)}(\underline{\omega})\big{\|}+o_{v}(1)\|\underline{\omega}\|.$

As a result, we get

$\displaystyle\|(d_{u+v}-d_{u})\underline{\omega}\|\leqslant
o_{v}(1)\|d_{u}\underline{\omega}\|+o_{v}(1)\|\underline{\omega}\|.$ (3.8)

Lemma 3.2 affords a similar treatment of the term $K_{-u}d^{*}R_{v}K_{u}$,
just replacing $T_{j}$ by $T^{*}_{j}$, $B^{i_{k}}_{\alpha\beta}$ by
$(B^{i_{k}}_{\alpha\beta})^{*}$ and $c^{i_{k}}_{\alpha\beta}$ by
$\overline{c^{i_{k}}_{\alpha\beta}}$. We get an estimate

$\displaystyle\|(d_{u+v}^{*}-d_{u}^{*})\underline{\omega}\|\leqslant
o_{v}(1)\|d_{u}^{*}\underline{\omega}\|+o_{v}(1)\|\underline{\omega}\|.$ (3.9)

Using the equation (3.6), which ensures that
$\|d_{u}\underline{\omega}\|\leqslant\|D_{u}\underline{\omega}\|$ and
$\|d_{u}^{*}\underline{\omega}\|\leqslant\|D_{u}\underline{\omega}\|$, we can
combine (3.8) and (3.9) to show that the relation (3.5) is satisfied.

We can therefore apply the Kato–Rellich theorem for all $u\in[0,1]$ and this
proves that all $D_{u}$ (including $D_{1}$) have selfadjoint extensions with
the same core $C=\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$. ∎

######  Remark 3.7.

It appears from the proof of point (2) that we could also take
$\mathcal{A}^{\infty}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$ as core for
the operator $D_{0}$ (using the Peter–Weyl decomposition). If we further
assume $h\in\mathcal{A}^{\infty}$, the rest of the proof applies verbatim and
shows that all $D_{u}$ have a common core, namely
$\mathcal{A}^{\infty}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$.

######  Corollary 3.8.

For all selfadjoint elements $h\in\mathcal{A}^{1}$ and all parameters
$u\in[0,1]$, the operators $D_{u}$ are $n^{+}$-summable.

###### Proof.

In the untwisted case, i.e., for $D_{0}$, we can follow the argument of
Theorem 5.5 of [30] to prove that $d+d^{*}$ is $n^{+}$-summable, where $n$ is
the dimension of $G$. Indeed, according to Proposition 2.3 from [35], as
$G$-vector spaces, we have
$\mathscr{H}\hookrightarrow\mathscr{H}_{\text{ref}}$ where
$\mathscr{H}_{\text{ref}}:=L^{2}(G)\otimes\bigwedge^{k}\mathfrak{g}^{*}$.
Moreover, the operator $d+d^{*}:=D_{\text{ref}}$ on this space is just the
Hodge–de Rham operator on $G$ and therefore it is $n^{+}$-summable. Since
$D_{\text{ref}}$ also preserves the finite-dimensional spaces
$V\otimes\bigwedge^{k}\mathfrak{g}^{*}$ obtained by Peter–Weyl decomposition,
the eigenvalues of $|D|$ coincide with those of $|D_{\text{ref}}|$ except that
they may have lower (and possibly zero) multiplicities. Consequently, the same
computation as in [30] proves that $D$ is $n^{+}$-summable.

To extend this property to all $D_{u}$ for $u\in[0,1]$, we first note that to
prove $D_{u}$ is $n^{+}$-summable, it suffices to show that the operator
$(D_{u}+i)^{-1}$ is in the symmetric ideal $\mathscr{L}^{n^{+}}$ – as
mentioned in Remark 2.5. The existence of the operator $(D_{u}+i)^{-1}$ is a
consequence of Proposition 3.6. The discussion above proves that
$(D_{0}+i)^{-1}$ is in this ideal.

We then use [39, Theorem 1.16, p. 196] to prove that if
$(D_{u}+i)^{-1}\in\mathscr{L}^{n^{+}}$ then for some $\varepsilon>0$ small
enough but independent of $u\in[0,1]$, and for any $v$ in
$|v|\leqslant\varepsilon$, then $(D_{u+v}+i)^{-1}\in\mathscr{L}^{n^{+}}$. For
all $u$, $v$,

$\displaystyle(D_{u+v}+i)-(D_{u}+i)=D_{u+v}-D_{u},$

and to apply Kato’s stability property, we need to give a relative bound on
$D_{u+v}-D_{u}$, expressed in terms of $D_{u}+i$. We are going to obtain this
using the relation (3.5). Indeed, since we know that $D$ is selfadjoint,
$\langle D\xi,\xi\rangle=\langle\xi,D\xi\rangle$ and thus

$\displaystyle\|(D+i)\xi\|^{2}=\|D\xi\|^{2}+\|\xi\|^{2}.$

which shows that $\|D\xi\|\leqslant\|(D+i)\xi\|$. From this fact and (3.5), we
deduce

$\displaystyle\|(D_{u+v}-D_{u})(\underline{\omega})\|\leqslant
o_{v}(1)\|(D_{u}+i)\underline{\omega}\|+o_{v}(1)\|\underline{\omega}\|,$

which let us apply [39, Theorem 1.16, p. 196] to $D_{u}+i$ and
$D_{u+v}-D_{u}$, leading to the expression

$\displaystyle(D_{u+v}+i)^{-1}=(D_{u}+i)^{-1}\big{(}1+(D_{u+v}-D_{u})(D_{u}+i)^{-1}\big{)}^{-1},$

where both $(D_{u+v}-D_{u})(D_{u}+i)^{-1}$ and
$(1+(D_{u+v}-D_{u})(D_{u}+i)^{-1})^{-1}$ are bounded operators. This
expression shows that $(D_{u+v}+i)^{-1}$ is a product of $(D_{u}+i)^{-1}$ in
the ideal $\mathscr{L}^{n^{+}}$ and a bounded operator. It is therefore itself
in the ideal $\mathscr{L}^{n^{+}}$ and this completes the proof. ∎

The operator $d_{u}$ of Proposition 3.6 induces a cochain complex:

######  Proposition 3.9.

The operator $d_{u}:=K_{u}dK_{-u}$, defined from $d=\Pi_{+}D$ on the domain of
selfadjointness of $D_{u}$ is closable. Taking its closure, there is a cochain
complex $(d_{u},\mathscr{H}_{k})$

$\displaystyle
0\to\mathscr{H}_{0}\xrightarrow{d_{u,0}}\mathscr{H}_{1}\to\cdots\to\mathscr{H}_{n-1}\xrightarrow{d_{u,n-1}}\mathscr{H}_{n}\to
0.$ (3.10)

######  Remark 3.10.

In the complex (3.10), the map
$d_{u,k}\colon\mathscr{H}_{k}\to\mathscr{H}_{k+1}$ is of course (the closure
of) the restriction of $d_{u}$ to
$\mathscr{H}_{k}\cap\operatorname{Dom}(D_{u})$, where
$\operatorname{Dom}(D_{u})$ is the domain of selfadjointness of $D_{u}$.

###### Proof.

We first treat the case of $d$ (for $h=0$). In this case, if $x_{n}\to x$ and
$y_{n}\to x$ while both $dx_{n}$ and $dy_{n}$ converge, we want to prove that
$\lim dx_{n}=\lim dy_{n}$. Consider any $z\in\mathscr{H}$ which lives in a
finite-dimensional vector space $V\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$
obtained from the Peter–Weyl decomposition. This ensures that $\Pi_{+}z$ is in
$V\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$ and thus in the domain of $D$.
We then have

$\displaystyle\langle z,\Pi_{+}Dx_{n}\rangle=\langle
D\Pi_{+}z,x_{n}\rangle\to\langle D\Pi_{+}z,x\rangle\leftarrow\langle
z,\Pi_{+}Dy_{n}\rangle.$

Since we know that both $dx_{n}$ and $dy_{n}$ converge in $\mathscr{H}$ and
that $\mathscr{D}$, the algebraic direct sum of all
$V\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$ is dense, it is necessary that
$\lim dx_{n}=\lim dy_{n}$ and this proves that $d$ is closable.

It follows that the kernel $\ker(\overline{d})$ is closed. Since
$\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$ is a core for $D$,
any $x$ in the domain $\operatorname{Dom}(\overline{d})$ can be approximated
by $x_{n}\in\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$ such
that $x_{n}\to x$ and $dx_{n}\to dx$. The density of $\mathcal{A}^{\infty}$
inside $\mathcal{A}^{1}$ (as discussed at the end of Section 2) then provides
an approximation of the original $x\in\dim(\overline{d})$ by
$y_{n}\in\mathcal{A}^{\infty}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}=\Omega^{\bullet}$.
For this sequence $y_{n}$, we know from Section 3 that $d^{2}y_{n}=0$. By
density, we obtain that (3.10) is a cochain complex.

Similarly, for $d_{u}=K_{u}dK_{-u}$ if $x_{n}\to x$ and $y_{n}\to x$ while
both $d_{u}x_{n}$ and $d_{u}y_{n}$ converge, we have $K_{-u}x_{n}\to
K_{-u}x\leftarrow K_{-u}y_{n}$ and $K_{-u}d_{u}x_{n}=dK_{-u}x_{n}$,
$K_{-u}d_{u}y_{n}=dK_{-u}y_{n}$. Since $d$ is closable, we get $\lim
K_{-u}d_{u}x_{n}=\lim K_{-u}d_{u}y_{n}$, which suffices to prove that $d_{u}$
is also closable. The cochain property then follows from
$d_{u}^{2}=K_{u}d^{2}K_{-u}=0$. ∎

In the rest of this section, we will be interested in the _reduced cohomology_
of the complex (3.10), namely the cohomology groups

$\displaystyle
H^{k}(d_{u},\mathscr{H}_{k}):=\ker(d_{u,k})/\overline{\operatorname{Ran}(d_{u,k-1})}.$
(3.11)

For any $u\in[0,1]$, let us write $E_{0,u}$ for the kernel of $D_{u}$. We have
the following Hodge decomposition theorem for the conformally perturbed
metric:

######  Theorem 3.11.

Let $G$ be a _compact_ Lie group of dimension $n$ acting _ergodically_ on a
unital $C^{*}$-algebra $A$. With the notations introduced previously, for any
parameter $u\in[0,1]$, there is a decomposition of $\mathscr{H}$ into a direct
sum of orthogonal Hilbert spaces

$\displaystyle\mathscr{H}=E_{-,u}\oplus E_{0,u}\oplus E_{+,u}.$

###### Proof.

The operator $D_{u}$ is selfadjoint with compact resolvent, as a consequence
of Proposition 3.6 and Corollary 3.8. Thus, we have an orthogonal sum
$\mathscr{H}=E_{0,u}\oplus\overline{\operatorname{Ran}(D_{u})}$. Following
Proposition 3.6, $\overline{\operatorname{Ran}(D_{u})}=E_{-,u}\oplus E_{+,u}$
and the sum is orthogonal, which proves the result. ∎

We call the restriction of $D_{u}^{2}$ to $\mathscr{H}_{k}$ the _Laplacian on
$\mathscr{H}_{k}$_ and denote it by $\Delta_{k}$, which is thus an unbounded
operator on $\mathscr{H}_{k}$, defined on the domain
$\mathcal{A}^{\infty}\otimes\bigwedge^{k}\mathfrak{g}^{*}$. Note that
$\Delta_{k}$ actually depends on our choice of conformal perturbation
$h\in\mathcal{A}^{1}$.

######  Corollary 3.12.

Let $H^{k}(d_{u},\mathscr{H}_{k})$ be the cohomology groups introduced in
(3.11), they identify naturally with the kernel of $\Delta_{k}$, i.e.,

$\displaystyle\ker(\Delta_{k})\simeq H^{k}(d_{u},\mathscr{H}_{k}).$

######  Remark 3.13.

This Corollary implies in particular that these cohomology groups are finite-
dimensional, since $\ker(\Delta_{k})=\ker(D_{u})$ and $D_{u}$ has compact
resolvent by Corollary 3.8.

###### Proof.

The cohomology group $H^{k}(d_{u},\mathscr{H}_{k})$ is defined as
$\ker(d_{u,k})/\overline{\operatorname{Ran}(d_{u,k-1})}$. The Hodge
decomposition Theorem 3.11 can be combined with the projections $\Pi_{\pm}$
and $\Pi_{k}$ on $\mathscr{H}_{k}$ to prove that
$\mathscr{H}_{k}=\ker(\Delta_{k})\oplus\overline{\operatorname{Ran}(d_{u,k-1})}\oplus\overline{\operatorname{Ran}(d_{u,k}^{*})}$.
We know that $\ker(d_{u,k})=\operatorname{Ran}(d_{u,k}^{*})^{\perp}$.
Therefore
$\ker(d_{u,k})=\overline{\operatorname{Ran}(d_{u,k-1})}\oplus\ker(\Delta_{k})$
from which it follows immediately that
$H^{k}(d_{u},\mathscr{H}_{k})=\ker(d_{u,k})/\overline{\operatorname{Ran}(d_{u,k-1})}\simeq\ker(\Delta_{k})$.
∎

######  Proposition 3.14.

The cohomology groups $H^{k}(d_{u},\mathscr{H}_{k})$ are abstractly isomorphic
to the nonperturbed $(h=0)$ cohomology groups $H^{k}(d,\mathscr{H}_{k})$.

###### Proof.

It is easy to check that $\ker(d_{u})=K_{u}\ker(d)$ and
$E_{+,u}=K_{u}E_{+,0}$. Thus, as abstract vector space,
$\ker(d_{u})/E_{+,u}=K_{u}\ker(d_{0})/K_{u}E_{+,0}$ is finite-dimensional,
with the same dimension as $\ker(d_{0})/E_{+,0}$. ∎

######  Remark 3.15.

The dimensions of $\ker(d_{u})/E_{+,u}$ and $\ker(d_{0})/E_{+,0}$ are the
same, but there are not “concretely isomorphic” for the scalar product we
consider. The _concrete realisation_ of $\ker(d_{u})/E_{+,u}$ is
$\\{\underline{\omega}\in\ker(d_{u})\colon\forall\,\underline{\omega}^{\prime}\in
E_{+,u},\langle\underline{\omega},\underline{\omega}^{\prime}\rangle=0\\}$.
However, $K_{u}$ does not preserve scalar products and therefore,
$\ker(d_{u})/E_{+,u}$ is not realised concretely by $K_{u}E_{0,0}$. In other
words, $K_{u}E_{0,0}$ is _not_ the space of harmonic forms for $D_{u}$.

##  4 Conformally twisted spectral triples
for $\boldsymbol{C^{*}}$-dynamical systems

In the following theorem, we use the selfadjoint operator $D_{u}$ to construct
spectral triples for the natural actions of the algebra $A$ (with its left
action on $\mathscr{H}$) and the algebra $A^{\text{op}}$ (acting on the right
of $\mathscr{H}$).

######  Theorem 4.1.

Let $G$ be a _compact_ Lie group of dimension $n$ acting ergodically on a
unital $C^{*}$-algebra $A$, then using the unique $G$-invariant trace
$\varphi_{0}$ of Theorem 2.2, we write
$\mathscr{H}_{0}:=\operatorname{GNS}(A,\varphi_{0})$.

For any fixed $h\in\mathcal{A}^{1}$ and any $u\in[0,1]$, the data
$(A,\mathscr{H}_{0}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*},D_{u})$ with
grading $\gamma$ defines an even $n^{+}$-summable spectral triple, where

  * •

the representation $\pi$ of $A$ on
$\mathscr{H}=\mathscr{H}_{0}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$ is
given by restriction of the left multiplication (3.2);

  * •

the unbounded operator $D_{u}$ is the unique selfadjoint extension of

$\displaystyle D_{u}=K_{u}dK_{-u}+K_{-u}d^{*}K_{u},$

defined on the core
$C=\mathcal{A}^{1}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*}$, the operator
$K_{u}$ being defined by (3.4);

  * •

the grading operator $\gamma$ is defined on degree $k$ forms by

$\displaystyle\gamma(a\otimes v_{1}\wedge\cdots\wedge v_{k})=(-1)^{k}(a\otimes
v_{1}\wedge\cdots\wedge v_{k}).$

For any fixed $h\in\mathcal{A}^{1}$ and any $u\in[0,1]$, the data
$(A^{\text{op}},\mathscr{H}_{0}\otimes\bigwedge^{\bullet}\mathfrak{g}^{*},D_{u})$
with grading $\gamma$ defines an even $n^{+}$-summable _twisted_ spectral
triple, with the automorphism $\beta$ on $A$ given by
$\beta(a)=e^{hu}ae^{-hu}$ – we use this $\beta$ to define an automorphism on
$A^{\text{op}}$.

######  Remark 4.2.

The morphism $\beta$ defined above preserves the multiplication of
$A^{\text{op}}$. It also satisfies the relation _unitarity condition_ (see
[16, equation (3.4)]) that is
$\beta\big{(}(a^{\text{op}})^{*}\big{)}=(\beta^{-1}(a^{\text{op}}))^{*}$.

###### Proof.

It is clear from the definition of $\pi$ that $A$ is represented on
$\mathscr{H}$ by bounded operators. The existence and uniqueness of the
selfadjoint extension of $D_{u}$ is proved in Proposition 3.6, while the
compact resolvent and finite summability properties are shown in Corollary
3.8.

We now prove that the commutator of $D_{u}$ with $a\in\mathcal{A}^{1}$ is
bounded. To this end, we use the notations of Lemma 3.2 to decompose the
operator $D_{u}$. We call

  * •

Part (0) is the “bounded part” of $D_{u}$, that is the terms

$\displaystyle-
K_{u}\bigg{(}\frac{1}{2}\sum_{k,\alpha,\beta}c^{i_{k}}_{\alpha\beta}\otimes
B^{i_{k}}_{\alpha,\beta}\bigg{)}K_{-u}\qquad\text{and}\qquad-
K_{-u}\bigg{(}\frac{1}{2}\sum_{k,\alpha,\beta}\overline{c^{i_{k}}_{\alpha,\beta}}\otimes(B^{i_{k}}_{\alpha,\beta})^{*}\bigg{)}K_{u}.$

  * •

Part (I) consists of the terms

$\displaystyle K_{u}\bigg{(}\sum_{j}\partial_{j}\otimes T_{j}\bigg{)}K_{-u}.$

  * •

Part (II) consists of the terms

$\displaystyle K_{-u}\bigg{(}\sum_{j}\partial_{j}\otimes
T_{j}^{*}\bigg{)}K_{u}.$

Part (0) commutes with the left multiplication by $a\in\mathcal{A}^{1}$, and
thus it does not contribute to the commutator. We therefore only need to
estimate Parts (I) and (II) of $D_{u}(a^{\prime}\underline{\omega})$ for
$\underline{\omega}=a\otimes v_{1}\wedge\cdots\wedge v_{k}$, that is

$\displaystyle\sum_{j}\partial_{j}\big{(}a^{\prime}ae^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad\quad{}+\sum_{j}\partial_{j}\big{(}a^{\prime}ae^{(n/2-k)hu}\big{)}e^{-(n/2-(k-1))hu}\otimes
T_{j}^{*}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad{}=\sum_{j}\big{(}\partial_{j}(a^{\prime})ae^{-(n/2-k)hu}+a^{\prime}\partial_{j}\big{(}ae^{-(n/2-k)hu}\big{)}\big{)}e^{(n/2-(k+1))hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad\quad{}+\sum_{j}\big{(}\partial_{j}(a^{\prime})ae^{(n/2-k)hu}+a^{\prime}\partial_{j}\big{(}ae^{(n/2-k)hu}\big{)}\big{)}e^{-(n/2-(k-1))hu}\otimes
T_{j}^{*}(v_{1}\wedge\cdots\wedge v_{k}).$

It follows from these considerations that

$\displaystyle[D_{u},a^{\prime}]\underline{\omega}=\sum_{j}\partial_{j}(a^{\prime})ae^{-hu}\otimes(T_{j}+T_{j}^{*})(v_{1}\wedge\cdots\wedge
v_{k}),$

which is clearly a bounded function of $\underline{\omega}$ for any
$a^{\prime}\in\mathcal{A}^{1}$. Moreover, such $a^{\prime}\in\mathcal{A}^{1}$
sends the core $C$ of our selfadjoint operator $D_{u}$ to itself and following
[48, Proposition A.1, p. 293], this suffices to ensure that
$a^{\prime}\in\mathcal{A}^{1}$ sends the domain of $D_{u}$ to itself. The
algebra $\mathcal{A}$ of Definition 2.4 thus contains $\mathcal{A}^{1}$ and is
dense in the $C^{*}$-algebra $A$. This completes the proof that
$(A,\mathscr{H},D_{u})$ is a $n^{+}$-summable spectral triple.

It remains to study its parity: it is clear from the definition that $\gamma$
sends the core $C$ to itself and thus it leaves the full domain of the
selfadjoint operator $D_{u}$ stable. Clearly, $\gamma$ distinguishes only
between
$\mathscr{H}_{\text{even}}:=A\otimes\bigwedge^{\text{even}}\mathfrak{g}^{*}$
and $\mathscr{H}_{\text{odd}}:=A\otimes\bigwedge^{\text{odd}}\mathfrak{g}^{*}$
and $\pi(a)$ leaves both spaces invariant, while $D_{u}$ is an odd operator.
This proves that $(A,\mathscr{H},D_{u})$ with $\gamma$ is an even spectral
triple.

The parity paragraph above applies verbatim to the spectral triple constructed
from the right action of $A^{\text{op}}$. The summability property is also
conserved. It remains to investigate the bounded twisted commutators. Notice
first that if $a^{\prime},h\in\mathcal{A}^{1}$ then both right multiplications
by $a^{\prime}$ and by $\beta(a^{\prime})$ leave the core $C$ of $D_{u}$
invariant and therefore the domain of $D_{u}$ is also stable under these right
multiplication.

Using the decomposition of $D_{u}$ into Parts (0), (I) and (II), it appears
that Part (0) commutes with the right action of $A^{\text{op}}$ and therefore
does not contribute to the commutator. We treat Parts (I) and (II) separately.
Keeping only Part (I) in the expression $D_{u}(\underline{\omega}\cdot
a^{\prime})$ for $\underline{\omega}=a\otimes v_{1}\wedge\cdots\wedge v_{k}$,
we get

$\displaystyle\sum_{j}\partial_{j}\big{(}aa^{\prime}e^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad{}=\sum_{j}\big{(}\partial_{j}(a)a^{\prime}e^{-(n/2-k)hu}+a\partial_{j}(a^{\prime})e^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}\otimes
T_{j}(v_{1}\wedge\ldots\wedge v_{k})$
$\displaystyle\qquad\quad{}+\sum_{j}aa^{\prime}\partial_{j}\big{(}e^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad{}=\sum_{j}\big{(}\partial_{j}(a)a^{\prime}e^{-hu}+a\partial_{j}(a^{\prime})e^{-hu}+aa^{\prime}\partial_{j}\big{(}e^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}\big{)}$
$\displaystyle\qquad\quad{}\otimes T_{j}(v_{1}\wedge\cdots\wedge v_{k}).$

We compare this expression to $D_{u}(\underline{\omega})\beta(a^{\prime})$,
i.e.,

$\displaystyle\sum_{j}\partial_{j}\big{(}ae^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}e^{hu}a^{\prime}e^{-hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad{}=\sum_{j}\partial_{j}(a)a^{\prime}e^{-hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k})$
$\displaystyle\qquad\quad{}+\sum_{j}a\partial_{j}\big{(}e^{-(n/2-k)hu}\big{)}e^{(n/2-(k+1))hu}e^{hu}a^{\prime}e^{-hu}\otimes
T_{j}(v_{1}\wedge\cdots\wedge v_{k}).$

In these two sums, the only terms that could lead to an unbounded contribution
are those containing $\partial_{j}(a)$, but these two terms cancel. At this
point, we must perform the same computation on Part (II) to make sure that the
automorphism $\beta$ is also suitable for this case. A very similar
computation proves that this it is indeed the case – the key property is that
$e^{-(n/2-k)hu}e^{(n/2-(k+1))hu}=e^{-hu}=e^{(n/2-k)hu}e^{-(n/2-(k-1))hu}$ –
and thus the operator (defined a priori only on $C$)

$\displaystyle
D_{u}\pi^{\text{op}}\big{(}(a^{\prime})^{\text{op}}\big{)}-\pi^{\text{op}}(\beta(a^{\prime})^{\text{op}})D_{u},$

where
$\pi^{\text{op}}\big{(}(a^{\prime})^{\text{op}}\big{)}=\mathcal{R}_{a^{\prime}}\otimes\operatorname{Id}_{\bigwedge^{\bullet}\mathfrak{g}^{*}}$,
extends to a bounded operator on $\mathscr{H}$. ∎

Since the operator $D_{u}$ is odd with respect to the grading operator
$\gamma$, we can write $D_{u}$ as combination of
$D_{u}^{+}\colon\mathscr{H}_{\text{even}}\to\mathscr{H}_{\text{odd}}$ and
$D_{u}^{-}\colon\mathscr{H}_{\text{odd}}\to\mathscr{H}_{\text{even}}$. The odd
Fredholm operator admits a (possibly) nontrivial index defined as

$\displaystyle\operatorname{Index}_{\text{odd}}(D_{u})=\dim\ker(D_{u}^{+})-\dim\ker(D_{u}^{-})$
(4.1)

(see, e.g., [32, equation (9.36), p. 397]).

##  5 Existence of a Chern–Gauss–Bonnet theorem
for conformal perturbations of $\boldsymbol{C^{*}}$-dynamical systems

In this section we show that the Hodge decomposition theorem proved in Section
3 indicates the existence of an analog of the Chern–Gauss–Bonnet theorem for
the $C^{*}$-dynamical systems studied in the present article. Let us explain
the classical case before stating the statement for our setting. Indeed,
because of the natural isomorphism between the space of harmonic differential
forms and the de Rham cohomology groups, for a classical closed manifold $M$,
the index of the operator
$d+d^{*}\colon\Omega^{\text{even}}M\to\Omega^{\text{odd}}M$ is equal to the
Euler characteristic of $M$. On the other hand the McKean–Singer index theorem
asserts that the index is given by

$\displaystyle\operatorname{Index}\big{(}d+d^{*}\colon\Omega^{\text{even}}\to\Omega^{\text{odd}}\big{)}=\sum_{i=0}^{\dim
M}(-1)^{i}\operatorname{Tr}\big{(}e^{-t\triangle_{i}}\big{)},$

where $\triangle_{i}=d^{*}d+dd^{*}$ is the Laplacian on the space of
$i$-differential forms on $M$, and $t$ is any positive number. This formula,
furthermore, contains local geometric information as $t\to 0^{+}$, since there
is a small time asymptotic expansion of the form

$\displaystyle\operatorname{Tr}\big{(}e^{-t\triangle_{i}}\big{)}\sim t^{-\dim
M/2}\sum_{j=0}^{\infty}a_{2j}(\triangle_{i})t^{j}.$

The coefficients $a_{2j}(\triangle_{i})$ are local geometric invariants, which
depend on the high frequency behaviour of the eigenvalues of the Laplacian and
are the integrals of some invariantly defined local functions
$a_{2j}(x,\triangle_{i})$ against the volume form of $M$. Independence of the
index from $t$ implies that the alternating sum of the constant terms in the
above asymptotic expansions for $\triangle_{i}$ gives the index. Hence, using
the Hodge decomposition theorem,

$\displaystyle\chi(M)=\operatorname{Index}\big{(}d+d^{*}\colon\Omega^{\text{even}}\to\Omega^{\text{odd}}\big{)}=\int_{M}\sum_{i=0}^{\dim
M}(-1)^{i}a_{\dim M}(x,\triangle_{i})\,d\text{vol}_{g}.$

In fact, the integrand in the latter coincides with the Pfaffian of the
curvature form, which is a remarkable and difficult identification [2].

With notations and assumptions as in Section 3, we obtain the following result
which indicates the existence of an analog of the Chern–Gauss–Bonnet theorem
in the setting of $C^{*}$-dynamical systems studied in this article.

######  Theorem 5.1.

The Euler characteristic $\chi$ of the complex $(d_{u},\mathscr{H}_{k})$ is
related to the odd index defined in (4.1)

$\displaystyle\chi=\sum_{k=0}^{n}(-1)^{k}\dim
H^{k}(d_{u},\mathscr{H}_{k})=\sum_{k=0}^{n}(-1)^{k}\ker(\Delta_{k})=\operatorname{Index}_{\textnormal{odd}}(D_{u}),$

and is independent of the conformal factor $e^{-h}$.

###### Proof.

The first equality is actually the definition of the Euler characteristic
$\chi$. The second equality is an immediate consequence of Corollary 3.12. The
third equality and the last statement can be justified by using Remark 3.13
and Proposition 3.14. That is, $\underline{\omega}\in\ker(\Delta_{k})$ means
in particular that $\underline{\omega}$ is in the domain of $\Delta_{k}$,
which is included in the domain of $D_{u}$ (by definition). We then have

$\displaystyle
0=\langle\underline{\omega},\Delta_{k}\underline{\omega}\rangle=\langle
D_{u}\underline{\omega},D_{u}\underline{\omega}\rangle,$

which proves that $\underline{\omega}\in\ker(D_{u})$. For a $k$-form
$\underline{\omega}$, the converse is obvious. It follows that
$\ker(D_{u}^{+})=\bigoplus_{k\geqslant 0}\ker(\Delta_{2k})$ and
$\ker(D_{u}^{-})=\bigoplus_{k\geqslant 0}\ker(\Delta_{2k+1})$, which yields

$\displaystyle\operatorname{Index}_{\text{odd}}(D_{u})=\dim\ker(D_{u}^{+})-\dim\ker(D_{u}^{-})=\bigoplus_{k\geqslant
0}\dim\ker(\Delta_{2k})-\bigoplus_{k\geqslant 0}\ker(\Delta_{2k+1})=\chi.$

The dimension of these groups are independent of the conformal factor $e^{-h}$
as a consequence of Proposition 3.14. ∎

######  Remark 5.2.

An alternative proof of the index property using only bounded operators can be
obtained using Sobolev spaces. For a clear account of these spaces and their
analytic properties in our setting, we refer the reader to the paper [55].
Also, in order to have a complete analog of the Chern–Gauss–Bonnet theorem,
one needs to find a local geometric formula for the index, which is proved
above to be a conformal invariant.

The heat kernels of Laplacians of conformally perturbed metrics on certain
noncommutative spaces such as the noncommutative $n$-tori
$\mathbb{T}_{\Theta}^{n}$ admit asymptotic expansions of the form

$\displaystyle\operatorname{Tr}(e^{-t\Delta_{k}})\sim\sum_{j=0}^{\infty}a_{j}(\Delta_{k})t^{(j-n)/2},\qquad
t\to 0^{+}.$ (5.1)

In fact, for noncommutative tori, each Laplacian $\Delta_{k}$ is an elliptic
selfadjoint differential operator of order 2, and asymptotic expansions of
this form can be derived by using the heat kernel method explained in [31]
while employing Connes’ pseudodifferential calculus [12]. This method was
indeed used in [17, 18, 25, 26, 28], for calculating and studying the term in
the expansion that is related to the scalar curvature of noncommutative two
and four tori. Going through this process for noncommutative tori
$\mathbb{T}_{\Theta}^{n}$, one can see that the odd coefficients in the latter
asymptotic expansion will vanish, since in their explicit formula in terms of
the pseudodifferential symbol of $\Delta_{k}$, there is an integration over
the Euclidean space $\mathbb{R}^{n}$ of an odd function involved (see [31, p.
54 and Theorem 1.7.6, p. 58]). Thus in the case of the noncommutative torus
$\mathbb{T}_{\Theta}^{n}$ we can write (5.1) as

$\displaystyle\operatorname{Tr}\big{(}e^{-t\Delta_{k}}\big{)}\sim
t^{-n/2}\sum_{j=0}^{\infty}a_{2j}(\Delta_{k})t^{j},\qquad t\to 0^{+}.$ (5.2)

Now, using the McKean–Singer index formula [31, Lemma 1.6.5, p. 47] and our
analog of Hodge decomposition theorem, for any $t>0$ we have

$\displaystyle\chi=\sum_{k=0}^{n}(-1)^{k}\operatorname{Tr}\big{(}e^{-t\Delta_{k}}\big{)}.$
(5.3)

Thus from equations (5.2) and (5.3), and using the independence of the Euler
characteristic from $t$ which implies that only the constant term from (5.2)
contributes to the calculation of the Euler characteristic, we can write

$\displaystyle\chi=\sum_{k=0}^{n}(-1)^{k}a_{n}(\Delta_{k})=\sum_{k=0}^{n}(-1)^{k}\varphi_{0}(\mathcal{R}_{k}),$

where the local geometric invariants $\mathcal{R}_{k}$ are derived from the
pseudodifferential symbols of the Laplacian $\Delta_{k}$, by a heat kernel
method. This method was used for example in [17, 26, 28] for computation of
scalar curvature for noncommutative two and four tori. The alternating sum of
the $\mathcal{R}_{k}$ gives a noncommutative analog of the local expression
for the Euler class.

##  6 Summary and conclusions

The Chern–Gauss–Bonnet theorem is an important generalization of the
Gauss–Bonnet theorem for surfaces, which states that the Euler characteristic
of an even-dimensional Riemannian manifold can be computed as the integral of
a characteristic class, namely the Pfaffian of the curvature form, which is a
local invariant of the geometry. In particular, it shows that the integral of
this geometric invariant is independent of the metric and depends only on the
topology of the manifold. The results obtained in this paper show that the
analog of this theorem holds for a general ergodic $C^{*}$-dynamical system,
whose algebra and Lie group are not necessarily commutative. To be more
precise, the family of metrics considered for a dynamical system is obtained
by using an invertible positive element of the $C^{*}$-algebra to conformally
perturb a fixed metric defined via the unique invariant trace, and our result
is about the invariance of a quantity, which is a natural analog of the Euler
characteristic, from the conformal factor.

This type of results were previously proved for the noncommutative two torus
$\mathbb{T}_{\theta}^{2}$. That is, the analog of the Gauss–Bonnet theorem was
proved in [18] and extended to general translation invariant complex
structures on these very important but particular $C^{*}$-algebras in [25],
where a conformal factor varies the metric. The differential geometry of
$C^{*}$-dynamical systems were developed and studied in [12], where the
noncommutative two torus $\mathbb{T}_{\theta}^{2}$ played a crucial role.
However the investigation of the analog of the Gauss–Bonnet theorem for
$\mathbb{T}_{\theta}^{2}$, when the flat metric is conformally perturbed was
pioneered in [11], where after heavy calculations, some noncommutative
features seemingly indicated that the theorem does not hold. However, studying
the spectral action in the presence of a dilaton [10], the development of the
theory of twisted spectral triples [16], and further studies of examples of
complex structures on noncommutative manifolds [40], led to convincing
observations that the Gauss–Bonnet theorem holds for the noncommutative two
torus. Then, by further analysis of the expressions and functions of a modular
automorphism obtained in [11], Connes and Tretkoff proved the desired result
in [18] for the simplest translation invariant conformal structure, and the
generalization of their result was established in [25] (where the use of a
computer for the heavy computations was inevitable).

It is remarkable that, a non-computational proof of the Gauss–Bonnet theorem
for the noncommutative two torus is given in [17], which is based on the work
[4], where the conformal index of a Riemannian manifold is defined using
properties of conformally covariant operators and the variational properties
of their spectral zeta functions. Therefore, since computations are enormously
more involved in dimensions higher than two, it is of great importance to use
spectral methods to show the existence of the analog of the Chern–Gauss–Bonnet
theorem, which is presented in this article, not only for nocommutative tori,
but for general $C^{*}$-dynamical systems. We have also paid special attention
to the spectral properties of the analog of the Hodge–de Rham operator of the
perturbed metric: we have proved its selfadjointness and shown that the
spectral dimension is preserved. We have then shown that this operator gives
rise to a spectral triple with the unitary left action of the algebra, and
gives a twisted spectral triple with the unitary action of the opposite
algebra on the right, generalizing the construction in [18] on the
noncommutative two torus and providing abstractly a large family of twisted
spectral triples.

### Acknowledgements

The authors thank the Hausdorff Research Institute for Mathematics (HIM) for
their hospitality and support during the trimester program on Noncommutative
Geometry and its Applications in 2014, where the present work was partially
carried out. They also thank the anonymous referees for their constructive
feedback. Parts of this article were obtained and written while the second
author was working as a postdoc at the University of Glasgow. He would like to
thank C. Voigt for enabling his stay in Scotland.

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