*subsection[0pt] , [ ][ $\bullet$ ][]

Energy Full Counting Statistics in Return-to-Equilibrium

Jane Panangaden

Department of Mathematics and Statistics

McGill University, Montreal

April 2016

A thesis submitted to McGill University in partial fulfillment of the
requirements of the degree of Master of Science.

©Jane Panangaden 2016

###### Abstract

We consider a finite dimensional quantum system $S$ in an arbitrary initial
state coupled to an infinitely extended quantum thermal reservoir $R$ in
equilibrium at inverse temperature $\beta$. The coupling is given by a bounded
perturbation of the dynamics and the coupling strength is controlled by a
parameter $\lambda$. We assume the system $S+R$ has the property of return to
equilibrium, which means that after sufficiently long time, the joint system
will have reached equilibrium at inverse temperature $\beta$. In this context,
we prove a refinement to the first law of thermodynamics, which states that
the total energy of the system and reservoir is conserved. Specifically, we
define two measures which encode all the information about the fluctuations of
the system and reservoir energy when two measurements are made at time $0$ and
time $t$. These measures are called the full counting statistics (FCS). We
prove weak convergence of the system and reservoir FCS in the double limit
$t\rightarrow\infty$ and $\lambda\rightarrow 0$.
The technical difficulty comes from the fact that the reservoir is infinitely
extended. In order to define the reservoir FCS, we write the finite
dimensional FCS in terms of a relative modular operator: an object which
survives the thermodynamic limit. We then draw on tools from Tomita-Takesaki
modular theory to prove the result.

###### Résumé

On propose dans ce travail d’étudier un système quantique $S$ de dimension
finie couplé à un réservoir de chaleur quantique $R$ infiniement étendu et en
équilibre à température $\beta^{-1}$. Le couplage est causé par une
perturbation bornée de la dynamique et l’intensité du couplage est contrôlée
par un paramètre $\lambda$. Supposons que le système $S+R$ manifeste la
propriété de retour à l’équilibre, c’est-à-dire que, après longtemps, le
système combiné évolue vers un état d’équilibre à temperature $\beta^{-1}$.
Dans ce contexte, on démontre un raffinement du premier principe de la
thermodynamique, qui affirme que l’énérgie totale du systéme est conservée.
Plus précisément, on définit deux mesures qui contiennent les informations
complètes des fluctuations de l’énérgie du système et du réservoir lorsque des
expériences sont effectuées aux temps $0$ et $t$. Ces mesures s’appellent les
statistiques « full counting » (FCS). On démontre la convergence faible des
FCS du système et du rèservoir dans la limite $t\rightarrow\infty$ et
$\lambda\rightarrow 0$.
La difficulté technique se produit du fait que le réservoir est infiniement
étendu. Pour définir son FCS, il faut réécrire le FCS de dimension finie en
utilisant un opérateur modulaire: un objet qui survit la limite
thermodynamique. Ensuite, on fait appel aux outils de la théorie modulaire de
Tomita-Takesaki pour démontrer le théorème.

Dedication

For Emma, forever ago.

Acknowledgments

I would first and foremost like to thank my advisors Prof. Vojkan Jakšić and
Dr. Annalisa Panati. Prof. Jakšić agreed to take me on as a summer student
when I had not yet taken a single course in analysis, and was woefully under-
qualified. His faith in my abilities and expert guidance as I ventured into
research have been an incredibly valuable gift. Dr. Panati has singlehandedly
taught me a huge amount of analysis, starting from the definition of a metric
space all the way to modular theory. She has spent so much time teaching me
and working with me over the past three years. I am extremely grateful for her
investment in my development as a researcher. I would also like to mention Ben
Landon, who was a fantastic mentor when I was first starting in this field,
the entire mathematical physics group at McGill who have made my working
environment enjoyable and friendly, and Renaud Raquépas who fixed all my
French grammar mistakes. I would like to thank the mathematics and statistics
department for giving me such a stimulating environment in which to learn and
study and NSERC for financial support.

Finally, I have to mention my incredible support network: my parents Laurie
and Prakash who have provided me with enriching opportunities throughout my
life and supported me unconditionally, my doctor Dr. Waqqas Afif for keeping
me functioning and my friends who I love deeply. I particularly need to thank
my roommates Adam Bene Watts and Galen Voysey who have become part of my
family during my time at McGill, and who were consistently there for me during
difficult times. Of course, I can’t forget my lovely cat Artemis who sat
beside me during the writing of this thesis, usually on my keyboard or on top
of my notes.

###### Contents

  1. 1 Introduction
  2. 2 Operator Algebra Formalism
    1. 2.1 Quantum Mechanics for Mathematicians
    2. 2.2 Entropy
    3. 2.3 Gibbs State
    4. 2.4 Trace Class Operators
    5. 2.5 $C^{*}$-Algebras
  3. 3 Spectral Theory for $\mathcal{C}^{*}$-algebras
    1. 3.1 Spectral Radius
    2. 3.2 Functional Calculus
    3. 3.3 Spectral Measures
    4. 3.4 Positive Elements and Approximate Identity
  4. 4 States and Measurement
    1. 4.1 Generalized States
    2. 4.2 Geometry of the States
    3. 4.3 Measurement
  5. 5 Dynamics and Equilibrium
    1. 5.1 Generalized Dynamics
    2. 5.2 KMS States
    3. 5.3 Properties of KMS States
  6. 6 Representations and Perturbation Theory
    1. 6.1 The GNS Representation
    2. 6.2 Normal States
    3. 6.3 Araki Perturbation Theory
  7. 7 Modular Theory
    1. 7.1 von Neumann Algebras
    2. 7.2 Modular Operators
    3. 7.3 Natural Cone
    4. 7.4 Relative Modular Operator
    5. 7.5 Modular Dynamics
    6. 7.6 Perturbed Dynamics
  8. 8 Return to Equilibrium Systems
    1. 8.1 Formal Setup and Notation
    2. 8.2 Return to Equilibrium
    3. 8.3 Energy Conservation: Averages
    4. 8.4 Full Counting Statistics
  9. 9 Strong Energy Conservation
    1. 9.1 Main Result
    2. 9.2 Proofs
  10. 10 Conclusion

##  1 Introduction

The $0^{\rm th}$-law of thermodynamics asserts that a large, isolated system
approaches an equilibrium state characterized by a few macroscopic parameters
such as temperature. In particular, a small system coupled to a comparatively
large thermal reservoir at some temperature is expected to eventually reach
its equilibrium state at the reservoir temperature, irrespective of its
initial state. This specific part of the $0^{\rm th}$-law is often called
_return to equilibrium_. From a mechanical point of view, return to
equilibrium holds if the interaction is sufficiently dispersive, which
translates into ergodic properties of the dynamics.

We consider the situation of a small quantum system $S$ coupled to a quantum
thermal reservoir $R$. We assume that the joint system $S+R$ has the property
of return to equilibrium. The precise mathematical formulation of this
property is given by Assumption (1). Although notoriously difficult to prove,
return to equilibrium has been established for several physically relevant
models (spin-boson model, spin-fermion model, electronic black box model,
locally interacting fermionic systems) [AMa, AJPP1, AJPP2, BFS, BM, dRK, DJ,
FMU, FMSU, JOP1, JOP2, JP3, JP2].

$S$$R$$\textrm{temp:}1/\beta$$\lambda$

To study the interaction of the system $S$ with physically relevant
reservoirs, it is mathematically convenient to idealize the reservoir as
infinitely extended. Such an idealization is common in statistical mechanics.
From a mathematical point of view, this idealization is required in order for
our model to exhibit the return to equilibrium behaviour we are interested in
studying. To describe the reservoir, we therefore use the operator algebra
formalism for quantum mechanics.

We consider the exchange of energy between the system and the reservoir as the
joint system relaxes into equilibrium. Let $\lambda$ be a parameter which
controls the coupling strength between the system and reservoir. If two
measurements are made of the energy of $S$ and $R$ at time $0$ and time $t$,
let $\Delta Q_{S}(\lambda,t)$ be the increase in expected value of the energy
of the system and $\Delta Q_{R}(\lambda,t)$ be the decrease in expected value
of the energy of the reservoir. The $1^{\rm st}$ law of thermodynamics is a
conservation of energy law for thermal systems. In this setting it can be
formulated as:

$\Delta Q_{S}:=\lim_{\lambda\rightarrow 0}\lim_{t\rightarrow\infty}\Delta
Q_{S}(\lambda,t)=\lim_{\lambda\rightarrow 0}\lim_{t\rightarrow\infty}\Delta
Q_{R}(\lambda,t)=:\Delta Q_{R}.$

Note that it is crucial that we take the limit $\lambda\rightarrow 0$ after
taking the large-time limit, as we are making measurements of the system
energy and reservoir energy, which do not sum to the total joint system energy
as long as $\lambda\neq 0$. The above statement can be thought of as the
$1^{\rm st}$ law in terms of averages, since it is the change in expected
value of the measurements which converges. This is well-understood and can be
established by using Araki’s perturbation theory. We review this result.

Our goal is to prove a stronger version of the $1^{\rm st}$ law for return to
equilibrium systems. Since we are working with quantum systems, each
measurement is probabilistic, and the measurement order matters. We define
probability measures encoding the full statistics of the fluctuations in
energy of the system and reservoir when two measurements are made at time $0$
and $t$. These measures are called the energy full counting statistics (FCS).
Full counting statistics in various contexts have been studied since their
introduction in [LL]. It is straightforward to define the FCS for the system,
$\mathbb{P}_{S,\lambda,t}$. However, because the reservoir is idealized as
infinitely extended, the definition of the reservoir FCS,
$\mathbb{P}_{R,\lambda,t}$, is more subtle. It requires a reformulation using
Tomita-Takesaki modular theory. We write $\mathbb{P}_{R,\lambda,t}$ in the
case of a confined reservoir in terms of a relative modular operator.
Physically relevant infinitely extended reservoirs can be obtained as a
thermodynamic limit of a sequence of confined reservoirs. The relative modular
operator survives this limit, giving us a way to define the FCS even in the
infinitely extended case. This also gives the relative modular operator a
natural physical interpretation. The objects $\Delta Q_{S/R}(\lambda,t)$ are
shown to be the first moments of the measures $\mathbb{P}_{S/R,\lambda,t}$.
Hence, the full counting statistics contain the information about the average
changes in energy, as well as much more information.

The main result is the following. Under a dynamical assumption which
guarantees return-to-equilibrium behaviour (Assumption 1) and a regularity
assumption on the coupling (Assumption 2), the weak limits

$\mathbb{P}_{S}:=\lim_{\lambda\rightarrow
0}\lim_{t\rightarrow\infty}\mathbb{P}_{S,\lambda,t}\;\;\;\;\;\;\;\;\;\;\mathbb{P}_{R}:=\lim_{\lambda\rightarrow
0}\lim_{t\rightarrow\infty}\mathbb{P}_{R,\lambda,t}$

exist and

$\mathbb{P}_{S}=\mathbb{P}_{R}.$

Since $\Delta Q_{S/R}$ are the first moments of $\mathbb{P}_{S/R}$, this is a
significant strengthening of energy conservation in this setting. This result
has already been announced in [JPPP]. We provide a comprehensive review of the
background and tools required, and more details of the proofs. Furthermore, we
show that under an additional technical assumption the proof simplifies
considerably and we have convergence of all the moments of
$\mathbb{P}_{S/R,\lambda,t}$.

The thesis is structured as follows. Sections (2)-(5) are dedicated to
introducing the operator algebra formalism for quantum mechanics and providing
an overview of some basic but essential tools. In Section (6), the main
results of Araki’s theory of perturbation of the equilibrium states are
reviewed. This is the tool which was used to understand the $1^{\rm st}$ law
in terms of averages, and we will also draw heavily on these results for our
refinement. In Section (7) the modular theory which allows us to define the
reservoir FCS is introduced. In Section (8) a proof of the $1^{\rm st}$ law in
terms of averages is given, the full counting statistics are defined and this
definition is motivated. Section (9) is devoted to the statement and proof of
the main result.

##  2 Operator Algebra Formalism

### 2.1 Quantum Mechanics for Mathematicians

In quantum mechanics, a physical system is represented by a triple
$(\mathcal{H},H,\psi)$ where $\mathcal{H}$ is a Hilbert space, $H$ is a self-
adjoint operator on $\mathcal{H}$ and $\psi$ is a norm-one element of
$\mathcal{H}$. $H$ is called the Hamiltonian and describes the time evolution
of the system via the Schrödinger equation and $\psi$ is a given initial
(pure) state. When $\mathcal{H}$ is finite dimensional we can axiomatize the
theory as follows [Ha].

  1. (i)

In the absence of measurement, the state of the system evolves according to
the Schrödinger equation.

$\displaystyle i\frac{\partial}{\partial t}\psi(t)=H\psi(t)$

This has general solution

$\displaystyle\psi(t)=e^{-itH}\psi.$

Note that for each $t\in\mathbb{R}$, $e^{-itH}$ is a unitary operator on
$\mathcal{H}$, and the map $t\mapsto e^{-itH}$ is a continuous one-parameter
unitary group.

  2. (ii)

A quantity which can be measured is represented by a self-adjoint element
$A\in\mathcal{B}(\mathcal{H})$. By the spectral theorem, $A$ can be decomposed
as

$\displaystyle A=\sum_{i}\lambda_{i}P_{\lambda_{i}}$

where $\lambda_{i}\in\mathbb{R}$ are the eigenvalues of $A$ and
$P_{\lambda_{i}}$ are the projections onto the corresponding eigenspaces. The
possible outcomes of the measurement are $\\{\lambda_{i}\\}$. If the system is
in state $\psi$ the probability of measuring $\lambda_{i}$ is given by
$\langle\psi,P_{\lambda_{i}}\psi\rangle$. It follows that the expected value
of the measurement is $\langle\psi,A\psi\rangle$.

  3. (iii)

After the measurement, if a value of $\lambda_{i}$ is observed, the new state
of the system is given by

$\displaystyle\frac{P_{\lambda_{i}}\psi}{||P_{\lambda_{i}}\psi||}.$

When studying quantum statistical mechanics we wish to work with an ensemble
of pure states. Remaining in the finite dimensional case, we consider a finite
set of pure states, $\\{\psi_{1},...,\psi_{k}\\}$ with probabilities
$p_{1},...,p_{k}$. To this we associate an element of
$\mathcal{B}(\mathcal{H})$ defined by

$\displaystyle\rho=\sum_{i=1}^{k}p_{i}|\psi_{i}\rangle\langle\psi_{i}|.$ (1)

Let $A$ be an observable with spectral decomposition
$A=\sum\lambda_{i}P_{\lambda_{i}}$, and $\phi_{1},...,\phi_{n}$ be an
orthonormal basis for $\mathcal{H}$ of eigenvectors of $A$. The eigenspace of
$\lambda$ is spanned by a subset of these eigenvectors, say $\\{\phi_{1}$, …,
$\phi_{m}\\}$ .

$\displaystyle\textrm{tr}{(\rho P_{\lambda})}$
$\displaystyle=\sum_{i=1}^{n}\langle\phi_{i},\rho
P_{\lambda}\phi_{i}\rangle=\sum_{i=1}^{n}\langle\phi_{i},\rho\sum_{j=1}^{m}\phi_{j}\langle\phi_{j},\phi_{i}\rangle\rangle=\sum_{j=1}^{m}\sum_{i=1}^{n}\langle\phi_{i},\rho\phi_{j}\rangle\delta_{ij}=\sum_{j=1}^{m}\langle\phi_{j},\rho\phi_{j}\rangle$
$\displaystyle=\sum_{j=1}^{m}\langle\phi_{j},\sum_{i=1}^{k}p_{i}\psi_{i}\langle\psi_{i},\phi_{j}\rangle\rangle=\sum_{i=1}^{k}p_{i}\sum_{j=1}^{m}\langle\psi_{i},\phi_{j}\rangle\langle\phi_{j},\psi_{i}\rangle=\sum_{i=1}^{k}p_{i}\langle\psi_{i},P_{\lambda}\psi_{i}\rangle$

Recall that $\langle\psi_{i},P_{\lambda}\psi_{i}\rangle$ is the probability of
measuring $\lambda$ in the state $\psi_{i}$. Hence, $\textrm{tr}(\rho
P_{\lambda})$ is the probability of measuring $\lambda$ for the ensemble of
states. Furthermore, $\textrm{tr}(\rho A)$ is the expected value of the
measurement.

From this computation we can also see, by taking $P_{\lambda}$ to be the
identity, that $\textrm{tr}(\rho)=\sum_{i}p_{i}=1$. In fact, since this
operator is already diagonal, we can see that $p_{i}$ are the eigenvalues, so
all the eigenvalues are contained in $[0,1]$. In the finite dimensional
setting all positive trace-one operators are of the form (1) such that all the
eigenvalues are positive. We therefore define the (mixed) states of the system
to be the elements of $\mathcal{B}(\mathcal{H})$ with positive eigenvalues and
trace one, and modify the axioms to take into account this more general notion
of state.

  1. (i’)

In the absence of measurement, the mixed state of the systems evolves as

$\displaystyle\rho(t)=e^{-itH}\rho e^{itH}.$

It is not hard to check that this is equivalent to evolving each pure state in
the ensemble and then forming a new density matrix.

  2. (ii’)

If $A$ is an observable with eigenvalue $\lambda$, the probability of
measuring $\lambda$ in state $\rho$ is

$\displaystyle\textrm{tr}(\rho P_{\lambda}).$

  3. (iii’)

After the measurement, the new state of the system is

$\displaystyle\frac{P_{\lambda}\rho P_{\lambda}}{\textrm{tr}(P_{\lambda}\rho
P_{\lambda})}=\frac{P_{\lambda}\rho P_{\lambda}}{\textrm{tr}(\rho
P_{\lambda})}.$

The cyclicity of the trace is used.

Since the only physical (i.e. measurable) quantities are the traces, we may
also choose to view the state as fixed and the observables as dynamical.

$\displaystyle A(t)$ $\displaystyle:=e^{itH}Ae^{-itH}$
$\displaystyle\textrm{tr}(\rho(t)A)$ $\displaystyle=\textrm{tr}(e^{-itH}\rho
e^{itH}A)=\textrm{tr}(\rho e^{itH}Ae^{-itH})=\textrm{tr}(\rho A(t))$

In physics this is called the Heisenberg picture. From a mathematical point of
view, the two descriptions are completely equivalent, but it will be more
convenient for us to work with the dynamics on the observables rather than the
states.

### 2.2 Entropy

In statistical mechanics an important object is the equilibrium state. An
equilibrium state is, in some sense, the maximally chaotic state of a physical
system. To be more precise we will need to introduce the concept of entropy.
The entropy is a function on the space of mixed states of a system which
measures our lack of knowledge about its specific pure state.

We begin in the classical setting. The space of pure states is given by a
finite set $\Omega=\\{\omega_{1},...,\omega_{|\Omega|}\\}$, while the space of
mixed states $\mathcal{P}(\Omega)$ is given by all the probability measures on
$\Omega$. We may identify an element in $P\in\mathcal{P}(\Omega)$ with an
element in $\mathbb{R}^{|\Omega|}$ by associating to $P$ the tuple
$(P(\omega_{1}),...,P(\omega_{|\Omega|}))$. This induces a topology on
$\mathcal{P}(\Omega)$.

######  Definition 2.1.

The entropy is the real-valued function $\mathcal{S}$ defined by

$\displaystyle\mathcal{S}:\bigcup_{\Omega\textrm{ s.t.
}|\Omega|<\infty}\mathcal{P}(\Omega)$ $\displaystyle\rightarrow\mathbb{R}$
$\displaystyle(P(\omega_{1}),...,P(\omega_{|\Omega|}))$
$\displaystyle\mapsto-\sum_{i=1}^{|\Omega|}P(\omega_{i})\log(P(\omega_{i})).$

This is a good way to measure our lack of knowledge about a system because it
scales correctly when two mixed states are combined in a probabilistic way to
form a new mixed state.

######  Proposition 2.2.

Let $\\{\Omega_{k}\\}_{k=1}^{m}$ be a tuple of finite sets and $P_{k}$ be a
probability measure on $\Omega_{k}$. Let $(p_{1},...,p_{m})$ be a probability
measure. Then we can define a new probability measure on
$\bigsqcup_{k=1}^{m}\Omega_{k}$ by $\oplus_{k=1}^{m}p_{k}P_{k}$. The entropy
is the unique function from $\bigcup_{\Omega\textrm{
finite}}\mathcal{P}(\Omega)$ to $\mathbb{R}$ satisfying

  1. 1.

For all $p_{k},P_{k}$,

$\mathcal{S}(\oplus_{k=1}^{m}p_{k}P_{k})=\sum_{k=1}^{m}p_{k}\mathcal{S}(P_{k})+\mathcal{S}((p_{1},...,p_{m})).$

  2. 2.

For each $\Omega$, $\mathcal{S}:\mathcal{P}(\Omega)\rightarrow\mathbb{R}$ is
continuous.

######  Remark 2.1.

An intuitive way to think about condition $1.$ in this proposition is to view
each system $(\Omega_{k},P_{k})$ as a black box from which you can reach in
and pick the “true" state of the system according to some probability
distribution. Then you can think of the larger system created by putting each
one of these boxes into a larger box and assigning them some probabilities
$p_{1},...,p_{m}$. Measuring the pure state of this larger system amounts to
first picking one of the boxes, and then picking an object from that box.

In thermal physics, one often views a pure state as a full list of the
positions and momenta of a large system of particles. However, one can only
measure some macroscopic parameters (such as temperature). It is assumed that
each specific configuration of particles that could correspond to a given set
of observed macroscopic parameters is equally likely. Here, the mixed states
correspond to different values of the macroscopic measurements (such as
temperature). The pure states correspond to a microscopic configuration.
Because of our assumption, the entropy of a given mixed state reduces to
$\log(N)$ where $N$ is the number of microscopic states which could correspond
to the observed macroscopic state. The highest entropy state is the one which
has the greatest number of compatible microstates. However, supposing no
measurements have been made, a priori each microstate is equally likely. The
macrostate most likely to be observed is also the one with the greatest number
of compatible microstates. Hence, the highest entropy state is the one which
is most likely to be observed. This is the statistical formulation of the
second law of thermodynamics.

The definition of entropy can be extended to the quantum case. When working
over a finite dimensional Hilbert space $\mathcal{H}$, the density matrix
plays the role of the system $(\Omega,P)$.

######  Definition 2.3.

The quantum entropy is the functional
$\mathcal{S}:\\{\rho\in\mathcal{B}(\mathcal{H})|\textrm{tr}(\rho)=1,\rho\geq
0\\}\rightarrow\mathbb{R}$ defined by

$\mathcal{S}(\rho)=-\textrm{tr}(\rho\log\rho)=-\sum_{i}\lambda_{i}\log(\lambda_{i}).$

where $\rho=\sum_{i}\lambda_{i}P_{\lambda_{i}}$ is the spectral decomposition.
Note that $\log(\lambda_{i})$ is defined since $\rho$ is positive. If we take
$\rho$ to be a density matrix defined by
$\sum_{i}p_{i}|\psi_{i}\rangle\langle\psi_{i}|$, then the quantum entropy is
the same as the classical entropy of the probability distribution
$(p_{1},...,p_{n})$.

### 2.3 Gibbs State

Now that we have a way to measure the disorder of a state, we can define the
equilibrium states. Recall that in thermodynamics we can interpret the maximal
entropy state (i.e. the most chaotic state) as the most likely macroscopic
state in which to observe the system, supposing that all microscopic states
are equally likely. We can also restrict the microscopic states that are
accessible by, for example, fixing the total energy of the system. We are
going to construct a state that maximizes the entropy functional among states
with a fixed energy.

######  Definition 2.4.

For a finite dimensional quantum system $\mathcal{H}$ with Hamiltonian $H$,
the Gibbs State at inverse temperature $\beta$ is

$\rho_{\beta}:=\frac{e^{-\beta H}}{\textrm{tr}(e^{-\beta H})}.$

Note that this is a positive operator with trace one.

Readers familiar with classical statistical mechanics will recognize this as
the quantum version of the Gibbs canonical ensemble, which describes the
probability distribution of microstates for a system in contact with a thermal
reservoir. A microstate with energy $E$ has probability $e^{-\beta E}/Z$,
where $Z$ is a normalization constant.

If we take $\psi_{i}$ to be a basis of eigenvectors for the Hamiltonian $H$
with eigenvalues $E_{i}$, the corresponding quantum state is

$\sum_{i}\frac{e^{-\beta E_{i}}}{Z}|\psi_{i}\rangle\langle\psi_{i}|=e^{-\beta
H}/Z.$

We will now rigorously state the variational principle satisfied by the Gibbs
state. Fix a Hamiltonian $H$ and consider the map

$\beta\mapsto E(\beta):=\textrm{tr}(H\rho_{\beta}).$

This is a function of real numbers that takes the inverse temperature $\beta$
to the expected energy when the system is in the $\beta$-Gibbs state. It is
continuous and differentiable with derivative

$\displaystyle\frac{d}{d\beta}(\textrm{tr}(H\rho_{\beta}))$
$\displaystyle=\textrm{tr}\left(H\left(\frac{-He^{-\beta
H}}{\textrm{tr}(e^{-\beta H})}-e^{-\beta H}\frac{\textrm{tr}(-He^{-\beta
H})}{\textrm{tr}(e^{-\beta H})^{2}}\right)\right)$
$\displaystyle=\textrm{tr}(\left(-H^{2}+HE(\beta)\right)\rho_{\beta})$
$\displaystyle=-\textrm{tr}((H-E(\beta))^{2}\rho_{\beta})\leq 0$

which shows that $\beta\mapsto\textrm{tr}(H\rho_{\beta})$ is everywhere
decreasing. In fact it is strictly decreasing unless $H$ is a constant
multiple of the identity. We can then compute the limits

$\displaystyle\lim_{\beta\rightarrow\mp\infty}\textrm{tr}(H\rho_{\beta})$
$\displaystyle=\lim_{\beta\rightarrow\mp\infty}\sum_{i}\langle\psi_{i},H\frac{e^{-\beta
H}}{\textrm{tr}(e^{-\beta H})}\psi_{i}\rangle$
$\displaystyle=\lim_{\beta\rightarrow\mp\infty}\sum_{i}\langle\psi_{i},E_{i}\frac{e^{-\beta
E_{i}}}{\textrm{tr}(e^{-\beta H})}\psi_{i}\rangle$
$\displaystyle=\lim_{\beta\rightarrow\mp\infty}\sum_{i}E_{i}\frac{e^{-\beta
E_{i}}}{\sum_{j}e^{-\beta E_{j}}}$
$\displaystyle=E_{\textrm{max}}/E_{\textrm{min}}$

where $E_{\textrm{max}}/E_{\textrm{min}}$ are the largest and smallest
eigenvalues. From this we may conclude that for each
$E\in[E_{\textrm{min}},E_{\textrm{max}}]$ there exists some inverse
temperature $\beta$ such that $E(\beta)=\textrm{tr}(H\rho_{\beta})=E$. Call
this number $\beta_{E}$. We have the following proposition [JOPP].

######  Proposition 2.5.

For $E\in[E_{\min},E_{\max}]$ let $\beta=\beta_{E}\in\mathbb{R}$ such that
$\textrm{tr}(\rho_{\beta}H)=E$, as above. Then

$\max\\{S(\rho):\rho\textrm{ a state,
}\textrm{tr}(H\rho)=E\\}=S(\rho_{\beta}).$

###### Proof.

We begin by computing $S(\rho_{\beta})$. Let $H=\sum_{i}\lambda_{i}\langle
e_{i},\cdot\rangle e_{i}$ be the spectral decomposition of $H$.

$\displaystyle S(\rho_{\beta})$
$\displaystyle=-\sum_{i}\frac{e^{-\beta\lambda_{i}}}{\textrm{tr}(e^{-\beta
H})}\log\left(\frac{e^{-\beta\lambda_{i}}}{\textrm{tr}(e^{-\beta H})}\right)$
$\displaystyle=\beta\sum_{i}\lambda_{i}\frac{e^{-\beta\lambda_{i}}}{\textrm{tr}(e^{-\beta
H})}+\sum_{i}\frac{e^{-\beta\lambda_{i}}}{\textrm{tr}(e^{-\beta
H})}\log(\textrm{tr}(e^{-\beta H}))$ $\displaystyle=\beta
E+\log(\textrm{tr}(e^{-\beta H}))$

Now if $\nu$ is some other state such that $\textrm{tr}(\nu H)=E$, we have the
bound

$\displaystyle S(\nu)$ $\displaystyle=S(\nu)-\beta\textrm{tr}(\nu H)+\beta E$
$\displaystyle\leq\max_{\rho}\left(S(\rho)-\beta\textrm{tr}(\rho
H)\right)+\beta E.$

The last step is to establish the identity

$\log(\textrm{tr}(e^{A}))=\max_{\rho}(\textrm{tr}(\rho A)+S(\rho)).$

From this it follows that $S(\nu)\leq\log(\textrm{tr}(e^{-\beta H}))+\beta
E=S(\rho_{\beta})$. To prove the identity, first note that

$\log(\frac{e^{A}}{\textrm{tr}(e^{A})})=A-\log(\textrm{tr}(e^{A}))\mathbbm{1}.$

Therefore, since $\textrm{tr}(\rho)=1$,

$\displaystyle\textrm{tr}(\rho\log(\frac{e^{A}}{\textrm{tr}(e^{A})}))=\textrm{tr}(\rho
A)-\log(\textrm{tr}(e^{A}))$
$\displaystyle\implies\log(\textrm{tr}(e^{A}))-(\textrm{tr}(\rho A)+S(\rho))$
$\displaystyle=\textrm{tr}(\rho\log(\rho)-\rho\log(\frac{e^{A}}{\textrm{tr}(e^{A})}))$
$\displaystyle\geq\textrm{tr}(\rho-\frac{e^{A}}{\textrm{tr}(e^{A})})=0$

by Klein’s inequality. Equality holds when
$\rho=\frac{e^{A}}{\textrm{tr}(e^{A})}$. ∎

This proposition says that for a given energy $E$, there is a temperature
$\beta^{-1}$ such that the $\beta$-Gibbs state maximizes the entropy among all
states with energy $E$. It is the most chaotic state at that energy. This can
also be interpreted as a stability property due to the second law of
thermodynamics. If a system in the Gibbs state is perturbed slightly, it is
likely to return to the Gibbs state, since the system will evolve in such a
way to increase entropy. This is something stronger than simply saying that
the state is invariant under the dynamics.

### 2.4 Trace Class Operators

The definition of the Gibbs state in the finite dimensional setting relied on
using the trace: $\textrm{tr}(e^{-\beta H})$. When we move to the infinite
dimensional setting, we will see that although we can try to extend the
definition of this quantity, it may diverge. Therefore, the Gibbs state may
not exist. First we need to define the trace in general. To do so we must
introduce the concept of the absolute value of an operator.

For a complex number $z$, we have a unique decomposition into its polar form:
$z=|z|e^{i\theta}$, where $|z|$ is a real number and $\theta\in[0,2\pi)$. We
take $|z|$ to be the definition of the absolute value of $z$. Furthermore we
have the identity $(\overline{z}z)^{1/2}=|z|$. We make use of this identity to
develop a similar decomposition for a bounded operator $A$. $A^{*}$ plays the
role of $\overline{z}$. Just as $\overline{z}z$ is always a positive real
number, $A^{*}A$ is self-adjoint and all its expectation values are positive,
in the sense that for all $\psi\in\mathcal{H}$,

$\langle\psi,A^{*}A\psi\rangle=||A\psi||^{2}\geq 0.$

Such an operator is called a positive operator. Note that this condition is
equivalent to saying that all the eigenvalues are positive in the finite
dimensional case. The next proposition says that for positive operators, there
is a well-defined (unique) positive square root [BR1].

######  Proposition 2.6.

Let $A$ be a positive operator. There is a unique positive operator $B$ such
that $B^{2}=A$. We denote $B$ by $\sqrt{A}$.

In the finite dimensional case, we can construct this square root easily by
using the spectral theorem. If $\sum_{i}\lambda_{i}P_{\lambda_{i}}$ is the
spectral decomposition of a positive operator $A$, then each $\lambda_{i}$ is
positive and we can let $B=\sum_{i}\sqrt{\lambda_{i}}P_{\lambda_{i}}$. $B$ is
positive and it follows from the fact that the projections are mutually
orthogonal that with this definition $B^{2}=A$. In general, the construction
is more involved, but we will see later that it will also follow from a more
general and powerful version of the spectral theorem.

Finally, we define $|A|:=\sqrt{A^{*}A}$. By the last proposition, $|A|$ is a
positive operator, and so all its expectation values are positive. We will use
this in the next definition.

######  Definition 2.7.

Let $A$ be a bounded linear operator over a separable Hilbert space
$\mathcal{H}$ with orthonormal basis $\\{e_{k}\\}$. $A$ is called trace class
if

$\sum_{k}\langle e_{k},|A|e_{k}\rangle<\infty.$

Note that this sum cannot fail to converge in $[0,\infty]$ since for each $k$,
$\langle e_{k},|A|e_{k}\rangle\geq 0$. Furthermore, the sum does not depend on
the choice of basis.

The trace class operators can be thought of as a non-commutative analogue of
$l^{1}$. They are compact operators, and so can be written in the form

$A=\sum_{k}\lambda_{k}\langle\phi_{k},\cdot\rangle\phi_{k}$

where $A\phi_{k}=\lambda_{k}\phi_{k}$ and the set $\\{\phi_{k}\\}$ is
orthonormal. The trace class condition is equivalent to saying that the
sequence of complex numbers $\\{\lambda_{k}\\}$ is $l^{1}$. (See, for example,
[Ma].) Now, if $A$ is trace class we have the following estimate.

$|\sum_{k}\langle\phi_{k},A\phi_{k}\rangle|\leq\sum_{k}|\langle\phi_{k},A\phi_{k}\rangle|=\sum_{k}(\langle\phi_{k},A\phi_{k}\rangle\langle
A\phi_{k},\phi_{k}\rangle)^{1/2}=\sum_{k}(\lambda_{k}\overline{\lambda_{k}})^{1/2}=\sum_{k}\langle\phi_{k},|A|\phi_{k}\rangle<\infty$

In other words, the series which we will define as

$\textrm{tr}(A):=\sum_{k}\langle\phi_{k},A\phi_{k}\rangle$

converges absolutely. From this one can show that the trace is finite and that
the trace does not depend on the choice of basis. The trace class operators
are those operators for which the trace “makes sense". It is important to note
that in the finite-dimensional case, this definition coincides with the usual
definition of trace, and therefore all operators are trace class.

In general, it is easy to cook up examples for which the Hamiltonian $H$ is a
bounded self-adjoint operator, and yet $e^{-\beta H}$ is not trace class. A
simple one is to let $H$ be the identity over an infinite dimensional Hilbert
space. Therefore, in the infinite setting, the Gibbs state is not always
defined, and our formalism is not appropriate for describing the equilibrium
behaviour of these systems.

### 2.5 $C^{*}$-Algebras

The solution to this problem is to forget about the underlying Hilbert space,
and view the algebra of bounded operators as the fundamental object. To do
this without referring to a Hilbert space we will need to extract the correct
abstract structure from $\mathcal{B}(\mathcal{H})$. First of all, it is a
vector space with the pointwise operator addition and scalar multiplication.
It has a non-commutative operator multiplication, and an operator adjoint
${}^{*}:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}(\mathcal{H})$, which
satisfies the property $A^{**}=A$. Finally, it has some analytic structure in
the form of the operator norm, $||\cdot||_{op}$, defined by

$||A||_{op}:=\sup_{||\psi||=1}||A\psi||.$

$\mathcal{B}(\mathcal{H})$ is complete in this norm. This property is
inherited from $\mathcal{H}$. If $A_{n}$ is a Cauchy sequence in the operator
norm, it follows that $A_{n}\psi$ is Cauchy in $\mathcal{H}$ for each $\psi$.
Since $\mathcal{H}$ is complete, $A_{n}\psi$ converges. Call this limit
$A\psi$. The map that sends $\psi$ to $A\psi$ is bounded and linear and one
can check that $A_{n}$ converges to $A$ in $\mathcal{B}(\mathcal{H})$.

Furthermore, these structures: multiplication, adjoint and norm, behave well
together.

######  Proposition 2.8.

Let $A,B\in\mathcal{B}(\mathcal{H})$.

  1. 1.

multiplication + adjoint: $(AB)^{*}=B^{*}A^{*}$

  2. 2.

norm + multiplication: $||AB||\leq||A||\cdot||B||$

  3. 3.

norm + adjoint: $||A^{*}A||=||A||^{2}$

###### Proof.

  1. (1)

In $\mathcal{B}(\mathcal{H})$ the adjoint of an operator $A$ is defined as the
unique operator $A^{*}$ such that for all $\phi,\psi\in\mathcal{H}$,

$\langle A^{*}\phi,\psi\rangle=\langle\phi,A\psi\rangle.$

We compute $\langle B^{*}A^{*}\phi,\psi\rangle=\langle
A^{*}\phi,B\phi\rangle=\langle\phi,AB\psi\rangle$. Hence
$(AB)^{*}=B^{*}A^{*}$.

  2. (2)

We use the definition of the operator norm to estimate,

$||AB||=\sup_{||\psi||=1}||AB\psi||\leq\sup_{||\psi||=1}||A||\cdot||B\psi||=||A||\cdot||B||.$

  3. (3)

For this proof we will use the construction of the operator adjoint for a
general Hilbert space. The construction is based on a version of Riesz’s
Hilbert space representation theorem, which states that if
$u:\mathcal{H}\times\mathcal{H}\rightarrow\mathbb{C}$ is a bounded
sesquilinear form, then there is a unique $B\in\mathcal{B}(\mathcal{H})$ such
that

$u(\phi,\psi)=\langle B\phi,\psi\rangle$

for all $\phi,\psi\in\mathcal{H}$. Furthermore, $||B||\leq||u||$ where
$||u||=\inf\\{m:|u(\phi,\psi)|\leq m||\phi||\cdot||\psi||\\}$. To construct
the adjoint of $A$, we take the bounded sesquilinear form

$u_{A}(\phi,\psi)=\langle\phi,A\psi\rangle.$

We then take $A^{*}$ to be the unique operator such that $\langle
A^{*}\phi,\psi\rangle=u_{A}(\phi,\psi)$ from Riesz’s theorem.

Since $||u_{A}||=||A||_{\mathcal{B}(\mathcal{H})}$, we have that
$||A^{*}||\leq||A||$. Also, since $A^{**}=A$, $||A||=||A^{**}||\leq||A^{*}||$
and hence $||A||=||A^{*}||$. Combining this with part (2) gives that
$||A^{*}A||\leq||A||^{2}$. On the other hand, for $\psi\in\mathcal{H}$,

$||A\psi||^{2}=|\langle
A\psi,A\psi\rangle|=\langle\psi,A^{*}A,\psi\rangle\leq||\psi||||A^{*}A\psi||\leq||\psi||^{2}||A^{*}A||.$

Hence

$||A||^{2}=\sup_{\psi\neq 0}\frac{||A\psi||^{2}}{||\psi||^{2}}\leq||A^{*}A||.$

∎

Motivated by this discussion, we define a $\mathcal{C}^{*}$-algebra as
follows.

######  Definition 2.9.

Let $\mathcal{A}$ be a vector space over $\mathbb{C}$.
$\mathcal{A}$ is an algebra if it is equipped with a multiplication
$\times:\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ satisfying
associativity and distributivity, i.e. compatibility with the vector space
structure.

  1. (i)

$A(BC)=(AB)C$

  2. (ii)

$A(B+C)=AB+AC$

  3. (iii)

$\alpha\beta AB=(\alpha A)(\beta B)$

An algebra is called a ∗-algebra if it is equipped with an involution, which
is a map ${}^{*}:\mathcal{A}\rightarrow\mathcal{A}$ satisfying $A^{**}=A$, and
the involution is compatible with the algebra structure.

  1. (iv)

$(AB)^{*}=B^{*}A^{*}$

  2. (v)

$(\alpha A+\beta B)^{*}=\overline{\alpha}A^{*}+\overline{\beta}B^{*}$

An algebra is called a Banach algebra if it is equipped with a norm, is
complete with respect to this norm and the norm is compatible with the algebra
structure.

  1. (vi)

$\|AB\|\leq\|A\|\|B\|$

If $\mathcal{A}$ is a Banach algebra and a ∗-algebra, then it is called a
Banach-∗-algebra if the norm and involution satisfy the following
compatibility condition.

  1. (vii)

$\|A^{*}\|=\|A\|$

Finally, a Banach-∗-algebra is a $\mathcal{C}^{*}$-algebra if the norm and
involution satisfy the $\mathcal{C}^{*}$ condition.

  1. (viii)

$\|A^{*}A\|=\|A\|^{2}$

${\textrm{Algebra:
}(\mathcal{A},+,\cdot,\times)}$${\mathcal{C}^{*}\textrm{-algebra}}$${\textrm{Involution:
}^{*}}$${\textrm{Norm: }||\cdot||}$${{}^{*}-\textrm{algebra}}$Banach algebra
Banach*-algebra${\mathcal{C}^{*}\textrm{ {condition:}
}\mathbf{||A^{*}A||=||A||^{2}}}$

The $\mathcal{C}^{*}$-condition is the linchpin which connects the algebraic
and analytic structures. For one thing, it is needed to prove the following
structure theorem. We have just shown that every space of bounded linear
operators over a Hilbert space is a $\mathcal{C}^{*}$-algebra. The structure
theorem for $\mathcal{C}^{*}$-algebras gives the converse; all
$\mathcal{C}^{*}$-algebras inherit their structure from such a space of
bounded linear operators [BR1].

######  Theorem 2.10.

Every $\mathcal{C}^{*}$-algebra $\mathcal{A}$ is isomorphic to a norm-closed,
self-adjoint (closed under taking adjoints) algebra of bounded operators on a
Hilbert space.

Furthermore we will see later that in a $\mathcal{C}^{*}$-algebra, the norm is
actually completely determined by the algebraic structure. The next section is
devoted to developing an important tool in the analysis of
$\mathcal{C}^{*}$-algebras: the spectral theory, which will allow us to
explore this interplay between algebraic and analytic structure further.

##  3 Spectral Theory for $\mathcal{C}^{*}$-algebras

Having established that a $\mathcal{C}^{*}$-algebra is the correct structure
to generalize $\mathcal{B}(\mathcal{H})$, we now need to reformulate the
axioms of quantum mechanics in this setting. Importantly, we no longer have a
Hilbert Space on which our operators act. Before, we said that the set of
possible outcomes of a measurement was given by the eigenvalues of a bounded
self-adjoint operator. Without a Hilbert space, we do not have a notion of
eigenvalues and eigenvectors. However, we can rewrite the eigenvalue-
eigenvector equation in finite dimensions as a purely algebraic property of
the operator. For $A\in\mathcal{B}(\mathcal{H})$ we have that $\lambda$ is an
eigenvalue of $A$ if

$\exists\psi\in\mathcal{H}\;\;\;\textrm{s.t.}\;\;\;A\psi=\lambda\psi\iff\exists\psi\in\mathcal{H}\;\;\;\textrm{s.t.}\;\;\;(A-\lambda\mathbbm{1})\psi=0\iff(A-\lambda\mathbbm{1})^{-1}\;\;\textrm{does
not exist.}$

Note that the last equivalence only holds for
$\textrm{dim}(\mathcal{H})<\infty$. This motivates the following definition.

######  Definition 3.1.

Let $A$ be an element of a $\mathcal{C}^{*}$-algebra $\mathcal{A}$. The
spectrum of $A$ is

$\sigma(A)=\\{\lambda\in\mathbb{C}:(A-\lambda\mathbbm{1})^{-1}\textrm{does not
exist}\\}.$

We have just argued that in the finite-dimensional case the spectrum of an
operator is exactly the set of its eigenvalues. In general, the spectrum can
have a richer structure. This is related to the fact that a bounded linear
operator on a finite-dimensional vector space is injective if and only if it
is surjective, but that this is not true in infinite dimensions. The
eigenvalue condition can be viewed as the statement that
$(A-\lambda\mathbbm{1})$ fails to be injective. However, even if it is
injective, $(A-\lambda\mathbbm{1})$ could fail to be invertible because
either: $(1)$ it is not bounded below and hence the set-theoretic inverse is
not a bounded operator, or $(2)$ the range is not dense.

Of course, in the $\mathcal{C}^{*}$-algebraic setting we do not refer to the
domain and range of an element, because the algebra is defined abstractly and
not as operators acting on an underlying space. However, keeping in mind the
structure theorem for $\mathcal{C}^{*}$-algebras, the preceeding discussion
should give a feel for what the spectrum will look like. In particular, it
could be larger than the set of eigenvalues and may not be discrete.

### 3.1 Spectral Radius

In a $\mathcal{C}^{*}$-algebra, the spectrum is one tool we can use to explore
the interplay between the algebraic and analytic structure. A priori, it looks
like something that depends only on the algebraic structure, since one just
needs to check whether or not certain elements are invertible. However, we
will see that considering the spectrum of an element as a set in $\mathbb{C}$
and looking at its size will reveal information about the norm of that
element.

######  Definition 3.2.

Let $A\in\mathcal{A}$ be an element of a $\mathcal{C}^{*}$-algebra. The
spectral radius of A is

$R(A)=\sup\\{|\lambda|:\lambda\in\sigma(A)\\}.$

In the finite-dimensional setting the spectral radius of an operator is simply
the norm of its largest eigenvalue. It is also not hard to see from the
definition that the operator norm of a self-adjoint (or more generally normal)
operator is also the largest eigenvalue. One uses the finite dimensional
spectral theorem to choose an orthonormal basis $\\{e_{i}\\}$ of eigenvectors
of $A\in\mathcal{B}(\mathcal{H})$ and then expresses an arbitrary norm one
element of the Hilbert space as a linear combination of these vectors
$\psi=\sum_{i}c_{i}e_{i}$. The coefficients satisfy $\sum_{i}|c_{i}|^{2}=1$.
Then

$||A\psi||^{2}=||\sum_{i}c_{i}\lambda_{i}e_{i}||^{2}=\left(\sum_{i}|c_{i}\lambda_{i}|^{2}\right)^{2}.$

When we sup over all the possible coefficients we see this is maximized when
the coefficient corresponding to the largest eigenvalue is $1$ and all others
are $0$. This shows that the spectral radius of an operator is equal to the
operator norm. In fact, this holds in general as well. We define normal, self-
adjoint and unitary operators abstractly using the involution.

######  Definition 3.3.

An element of a $\mathcal{C}^{*}$-algebra $A\in\mathcal{A}$ is called normal
if $A^{*}A=AA^{*}$, unitary if $A^{*}A=AA^{*}=\mathbbm{1}$ and self-adjoint if
$A^{*}=A$. In particular, unitary and self-adjoint operators are normal.

Then we have the following important result relating the $\mathcal{C}^{*}$
norm to the spectral radius.

######  Proposition 3.4.

For a normal element of a $\mathcal{C}^{*}$-algebra $A\in\mathcal{A}$,

$R(A)=\|A\|.$

###### Proof.

The main difficulty of this proof is to establish the identity
$R(A)=\lim_{n\rightarrow\infty}||A^{n}||^{1/n}$. We begin with the upper
bound. Fix $\lambda$ such that $|\lambda|>||A^{n}||^{1/n}$ for some $n$.
Consider the series

$\lambda^{-1}\sum_{m\geq 0}\left(\frac{A}{\lambda}\right)^{m}.$

Decomposing each $m\in\mathbb{Z}$ as $m=pn+q$ with $0\leq q<n$ and using the
fact that $\left(\|\frac{A}{\lambda}\|\right)^{n}<1$ we can establish that
this series is Cauchy and hence converges since the $\mathcal{C}^{*}$-algebra
is complete. Next we compute,

$(\lambda\mathbbm{1}-A)\lambda^{-1}\sum_{m\geq
0}\left(\frac{A}{\lambda}\right)^{m}=\sum_{m\geq
0}\left(\frac{A}{\lambda}\right)^{m}-\sum_{m\geq
1}\left(\frac{A}{\lambda}\right)^{m}=\mathbbm{1}.$

Hence the limit of the series is the inverse of $(\lambda\mathbbm{1}-A)$ and
$\lambda\nin\sigma(A)$. It follows that $R(A)\leq\|A^{n}\|^{1/n}$ for all $n$
and hence

$R(A)\leq\inf_{n}\|A^{n}\|^{1/n}\leq\liminf_{n}\|A^{n}\|^{1/n}.$

Now we establish the lower bound. Define $r_{A}:=\limsup_{n}\|A^{n}\|^{1/n}$.
Consider the case that $A$ is invertible. Then
$1=\|A^{n}A^{-n}\|\leq\|A^{n}\|\cdot\|A^{-n}\|$. Taking the limsup we get that
$1\leq r_{A}r_{A^{-1}}$. Hence, $r_{A}>0$. The contrapositive of what we have
just proven is that if $r_{A}=0$, then $A$ is not invertible and hence
$0\in\sigma(A)$. Hence $R(A)\geq 0=r_{A}$.

We are left with the case that $r(A)>0$. In the proof of this case, we will
use the fact that for a sequence of operators $A_{n}$ such that
$(\mathbbm{1}-A_{n})^{-1}$ exists,

$\|A_{n}\|\rightarrow 0\iff\|\mathbbm{1}-(\mathbbm{1}-A_{n})^{-1}\|\rightarrow
0.$

To establish this, let $R_{n}=(\mathbbm{1}-A_{n})^{-1}$. Then

$\mathbbm{1}-R_{n}=\mathbbm{1}-(\mathbbm{1}-A_{n})^{-1}=(\mathbbm{1}-A_{n})(\mathbbm{1}-A_{n})^{-1}-(\mathbbm{1}-A_{n})^{-1}=-A_{n}(\mathbbm{1}-A_{n})^{-1}$

and

$A_{n}=\mathbbm{1}-R_{n}^{-1}=R_{n}R_{n}^{-1}-R_{n}^{-1}=(R_{n}-\mathbbm{1})R_{n}^{-1}=-(\mathbbm{1}-R_{n})(\mathbbm{1}-(\mathbbm{1}-R_{n}))^{-1}.$

From these expansions we can see that $\|A_{n}\|\rightarrow 0$ if and only if
$\|\mathbbm{1}-R_{n}\|\rightarrow 0$.

Now we define $S_{A}=\\{\lambda:|\lambda|\geq r_{A}\\}$. Suppose for
contradiction that $S_{A}\cap\sigma(A)=\phi$. Then for $\omega$ a primitive
$n^{th}$ root of unity the operator

$R_{n}(A,\lambda)=\frac{1}{n}\sum_{k=1}^{n}\left(\mathbbm{1}-\frac{\omega^{k}A}{\lambda}\right)^{-1}$

exists whenever $\lambda\in S_{A}$. We then calculate

$\displaystyle\left(\mathbbm{1}-\frac{A^{n}}{\lambda^{n}}\right)\frac{1}{n}\sum_{k=1}^{n}\left(\mathbbm{1}-\frac{\omega^{k}A}{\lambda}\right)^{-1}$
$\displaystyle=$
$\displaystyle\frac{1}{n}\left[\prod_{j=1}^{n}\left(\mathbbm{1}-\omega^{j}\frac{(\omega
A)}{\lambda}\right)\right]\sum_{k=1}^{n}\left(\mathbbm{1}-\frac{\omega^{k}A}{\lambda}\right)^{-1}$
$\displaystyle=\frac{1}{n}\sum_{k=1}^{n}(\mathbbm{1}-\frac{\omega
A}{\lambda})\cdot\cdot\cdot\widehat{(\mathbbm{1}-\frac{\omega^{k}A}{\lambda})}\cdot\cdot\cdot(\mathbbm{1}-\frac{\omega^{n}A}{\lambda})$
$\displaystyle=\frac{1}{n}(n\mathbbm{1})=\mathbbm{1}$

where we have used the identity $\sum_{i=1}^{n}\omega^{i}=0$ for $n^{th}$
roots of unity. We have established that
$R_{n}(A,\lambda)=(\mathbbm{1}-\frac{A^{n}}{\lambda^{n}})^{-1}$. In particular
this inverse exists. Now we have the estimate

$\displaystyle\left\|\left(\mathbbm{1}-\frac{\omega^{k}A}{r_{A}}\right)^{-1}-\left(\mathbbm{1}-\frac{\omega^{k}A}{\lambda}\right)^{-1}\right\|$
$\displaystyle=\left\|\left(\mathbbm{1}-\frac{\omega^{k}A}{r_{A}}\right)^{-1}\omega^{k}A\left(\frac{1}{\lambda}-\frac{1}{r_{A}}\right)\left(\mathbbm{1}-\frac{\omega^{k}A}{\lambda}\right)^{-1}\right\|$
$\displaystyle\leq\lvert\lambda-r_{A}\rvert\|A\|\sup_{\gamma\in
S_{A}}\|(\gamma\mathbbm{1}-A)^{-1}\|^{2}$

The supremum is finite because we have

$\|(\gamma\mathbbm{1}-A)^{-1}\|\leq\lvert\gamma\rvert^{-1}\sum_{n\geq
0}\frac{\|A^{n}\|}{\lvert\gamma^{n}\rvert}=(\lvert\gamma\rvert-\|A\|)^{-1}$

using the formula for $(\gamma\mathbbm{1}-A)^{-1}$ when $\gamma\geq r_{A}$
established earlier. We have that the map
$\lambda\mapsto(\lvert\lambda\rvert-\|A\|)^{-1}$ is continuous on the
complement of $\sigma(A)$ and since we have assumed that $S_{A}$ has no
intersection with $\sigma(A)$, it is continuous on $S_{A}$.

Note also that the bound is uniform in $k$, so we may use the formula for
$(\mathbbm{1}-\frac{A^{n}}{\lambda^{n}})^{-1}$ to obtain that for all
$\epsilon>0$ there exists $\lambda>r_{A}$ such that

$\left\|\left(\mathbbm{1}-\frac{A^{n}}{r_{A}^{n}}\right)^{-1}-\left(\mathbbm{1}-\frac{A^{n}}{\lambda^{n}}\right)^{-1}\right\|<\epsilon$

for all $n$. For $\lambda>r_{A}$ we have that
$\|A^{n}\|^{1/n}/\lvert\lambda\rvert<1$ and so
$\|A^{n}\|/\lvert\lambda\rvert^{n}\rightarrow 0$ as $n\rightarrow\infty$. It
follows that $\|(\mathbbm{1}-A^{n}/\lambda^{n})^{-1}-\mathbbm{1}\|\rightarrow
0$ and hence by the previous statement that
$\|(\mathbbm{1}-A^{n}/r_{A}^{n})^{-1}-\mathbbm{1}\|\rightarrow 0$. This
implies that $\|A^{n}\|/r_{A}^{n}\rightarrow 0$. But
$r_{A}=\limsup_{n}\|A^{n}\|^{1/n}$ by definition, so this last statement is a
contradiction. Therefore, there is an element $\lambda$ in the spectrum of $A$
which is in $S_{A}$ and hence satisfies $|\lambda|>r_{A}$ and

$R(A)\geq\limsup_{n}\|A^{n}\|^{1/n}.$

The last step is to assume that $A$ is normal and repeatedly apply the
$\mathcal{C}^{*}$ identity.

$\displaystyle\|A^{2^{n}}\|^{2}=\|(A^{*})^{2^{n}}A^{2^{n}}\|=\|(A^{*}A)^{2^{n}}\|=\|(A^{*}A)^{2^{n-1}}\|^{2}=\cdot\cdot\cdot=||A||^{2^{n+1}}$

Then we have
$R(A)=\lim_{n\rightarrow\infty}\|A^{2^{n}}\|^{2^{-n}}=\lim_{n\rightarrow\infty}\left(\|A\|^{2^{n+1}}\right)^{2^{-(n+1)}}=\|A\|$.

∎

A remarkable consequence of this result is that the algebraic structure of a
$\mathcal{C}^{*}$-algebra completely determines the analytic structure.

######  Corollary 3.5.

Let $\mathcal{A}$ be a $*$-algebra. If there is a norm on $\mathcal{A}$ which
makes $\mathcal{A}$ into a $\mathcal{C}^{*}$-algebra, then this norm is the
unique norm which makes $\mathcal{A}$ into a $\mathcal{C}^{*}$-algebra.

###### Proof.

If $A$ is normal in a $*$ -algebra and $\|\cdot\|$ is a
$\mathcal{C}^{*}$-norm, then $\|A\|=R(A)$, which is completely determined by
the algebraic structure since the spectrum of the element does not depend on
the norm. Then if $A$ is an arbitrary element we use the
$\mathcal{C}^{*}$-identity to get

$\|A\|=\|A^{*}A\|^{1/2}=R(A^{*}A)^{1/2}$

since for any $A$, $A^{*}A$ is normal. The norm of every element is determined
by its spectrum. ∎

This illustrates to what extent the analytic and algebraic structures are
interconnected in a $\mathcal{C}^{*}$-algebra. If you have a $\mathcal{C}^{*}$
algebra you never need to ask what norm to put on it, the norm can be deduced
from the algebraic part of the $\mathcal{C}^{*}$-algebra. A similar result
holds for morphisms of $\mathcal{C}^{*}$-algebras. If a map between two
$\mathcal{C}^{*}$ algebras respects all the algebraic structure, then it
automatically respects the analytic structure as well.

######  Corollary 3.6.

Let $\mathcal{A}$ and $\mathcal{B}$ be $\mathcal{C}^{*}$-algebras and
$\pi:\mathcal{A}\rightarrow\mathcal{B}$ be a morphism of ∗-algebras. Then
$||\pi(A)||_{\mathcal{B}}\leq||A||_{\mathcal{A}}$ for all $A\in\mathcal{A}$.
Hence $\pi$ is a morphism of $\mathcal{C}^{*}$-algebras.

###### Proof.

First consider $A$ self-adjoint. If $\lambda\nin\sigma(A)$ then
$(\lambda\mathbb{I}-A)^{-1}$ exists. Then
$\pi((\lambda\mathbb{I}-A)^{-1})=(\pi(\lambda\mathbb{I}-A))^{-1}=(\lambda\mathbb{I}_{\pi(\mathcal{A})}-\pi(A))^{-1}$
exists. Therefore, $\sigma(\pi(A))\subset\sigma(A)$ and so we have that the
spectral radius of $\pi(A)$ is bounded by the spectral radius of $A$ and hence
$||\pi(A)||\leq||A||$.

For general $A$, $A^{*}A$ is self adjoint and so by the previous result,

$||\pi(A)||^{2}=||\pi(A)^{*}\pi(A)||=||\pi(A^{*}A)||\leq||A^{*}A||=||A||^{2}.$

$\pi$ is continuous and hence a morphism of $\mathcal{C}^{*}$-algebras. ∎

Additionally, this result gives us some important information about the
spectra of self-adjoint and unitary operators, which we would like to use to
represent the observables and time evolution of a physical system in quantum
mechanics. Recall that in the finite-dimensional case the observables are time
evolved by

$A\mapsto U(t)AU(t)^{*}$

where $U(t)=e^{itH}$ is a unitary group. Furthermore, we identified the
possible results of measurements with the eigenvalues of the operator
representing the measurement. Since these are self-adjoint, the eigenvalues
are real. This is the reason why we chose the self-adjoint operators to
represent observables: because physical experiments should have real-valued
outcomes. Unitary operators can also be characterized by their eigenvalues.
All the eigenvalues lie on the unit circle. Looking at the spectral
decomposition of an operator we have

$UAU^{*}=U\left(\sum_{i}\lambda_{i}P_{i}\right)U^{*}=\sum_{i}\lambda_{i}UP_{i}U^{*}$

where $UPU^{*}$ is a projection since $U$ is unitary. Time evolution has no
effect on the spectrum, it only moves around the eigenspaces. These spectral
properties of unitary and self-adjoint operators continue to hold in the
general case, which allows us to continue to use our axioms of quantum
mechanics.

######  Corollary 3.7.

If $U$ is a unitary element of a $\mathcal{C}^{*}$-algebra, then the spectrum
of $U$ is contained in the unit circle in $\mathbb{C}$. If $A$ is a self-
adjoint element, then the spectrum of $A$ is contained in
$[-\|A\|,\|A\|]\subset\mathbb{R}$.

###### Proof.

Let $U$ be unitary. Using the $\mathcal{C}^{*}$ condition we get

$\|U\|^{2}=\|U^{*}U\|=\|\mathbbm{1}\|=1.$

Hence, since $U$ is normal, $R(U)=||U||=1$ and $\sigma(U)$ is contained in the
unit disc. Now suppose $\lambda\in\sigma(U)$. It follows that
$\overline{\lambda}\in\sigma(U^{*})=\sigma(U^{-1})$ and hence that
$(\overline{\lambda})^{-1}\in\sigma(U)$. But then
$|(\overline{\lambda})^{-1}|=|\lambda^{-1}|\leq 1$ which means that
$|\lambda|\geq 1$. Hence the spectrum of $U$ is contained in the unit circle.

Let $A$ be self-adjoint. In particular it is normal, so $R(A)=\|A\|$. Fix
$\lambda$ such that $|\lambda^{-1}|>\|A\|$. $\lambda^{-1}$ is not in the
spectrum, and the inverse of
$\mathbbm{1}+i|\lambda|A=-i|\lambda|(i|\lambda^{-1}|\mathbbm{1}-A)$ exists.
Consider the element

$B=(\mathbbm{1}-i|\lambda|A)(\mathbbm{1}+i|\lambda|A)^{-1}.$

We check that $B$ is unitary.

$\displaystyle B^{*}B$
$\displaystyle=(\mathbbm{1}-i|\lambda|A^{*})^{-1}(\mathbbm{1}+i|\lambda|A^{*})(\mathbbm{1}-i|\lambda|A)(\mathbbm{1}+i|\lambda|A)^{-1}$
$\displaystyle=(\mathbbm{1}-i|\lambda|A)^{-1}(\mathbbm{1}-i|\lambda|A)(\mathbbm{1}+i|\lambda|A)(\mathbbm{1}+i|\lambda|A)^{-1}$
$\displaystyle=\mathbbm{1}$

Then we know that the spectrum of $B$ is contained in the unit circle, so for
$\alpha\in\mathbb{C}$, whenever

$\left(\frac{1-i|\lambda|\alpha}{1+i|\lambda|\alpha}\mathbbm{1}-U\right)$

is invertible we must have

$\left\lvert\frac{1-i|\lambda|\alpha}{1+i|\lambda|\alpha}\right\rvert=1$

which is true exactly when $\alpha$ is real. We can factor

$\displaystyle(\frac{1-i|\lambda|\alpha}{1+i|\lambda|\alpha}\mathbbm{1}-U)$
$\displaystyle=(\mathbbm{1}+i|\lambda|\alpha)^{-1}\Big{(}(1-i|\lambda|\alpha)(1+i|\lambda|A)-(1+i|\lambda|\alpha)(1-i|\lambda|A)\Big{)}(\mathbbm{1}+i|\lambda|A)^{-1}$
$\displaystyle=2i|\lambda|(\mathbbm{1}+i|\lambda|\alpha)^{-1}(A-\alpha\mathbbm{1})(\mathbbm{1}+i|\lambda|A)^{-1}$

Hence, if $\alpha\in\sigma(A)$ then $(\mathbbm{1}\alpha-A)$ is invertible and
hence $\alpha$ is real. ∎

We have shown that in a $\mathcal{C}^{*}$-algebra, a self-adjoint element has
real spectrum, so it still makes sense to identify a physical measurement with
a self-adjoint element $A$ and the set of possible outcomes with the set
$\sigma(A)\subset\mathbb{R}$. What we need now is a way to put a probability
measure on this set given some state for the system. Recall that in the
finite-dimensional case we used the spectral theorem to decompose

$A=\sum_{i}\lambda_{i}P_{\lambda_{i}}$

and then for each density matrix $\rho$ we put a probability measure
$p_{\rho}$ on the finite set $\sigma(A)=\\{\lambda_{i}\\}$ by

$p_{\rho}(\lambda_{i})=\textrm{tr}(\rho P_{\lambda_{i}}).$

To extend this to the $\mathcal{C}^{*}$-algebra setting, we will need a more
powerful version of the spectral theorem.

### 3.2 Functional Calculus

The finite-dimensional spectral theorem can be viewed as a way to take
functions of matrices. If $p(x)=\sum_{i=0}^{n}a_{i}x^{i}$ is a polynomial and
$A\in\mathcal{B}(\mathcal{H})$ self-adjoint has spectral decomposition
$A=\sum_{i}\lambda_{i}P_{\lambda_{i}}$ then we can define

$p(A)=\sum_{i=0}^{n}a_{i}\left(\sum_{j}\lambda_{j}P_{\lambda_{j}}\right)^{i}=\sum_{i=0}^{n}a_{i}\left(\sum_{j}\lambda_{j}^{i}P_{\lambda_{j}}\right)=\sum_{j}p(\lambda_{j})P_{\lambda_{j}}.$

We can then avail ourselves of the Stone-Weierstrass theorem, which says that
a continuous function $f:\mathbb{R}\rightarrow\mathbb{C}$ can be approximated
uniformly by a sequence of polynomials $\\{p_{n}\\}$ to define

$f(A)=\lim_{n\rightarrow\infty}p_{n}(A)$

where the limit is taken in the operator norm. This limit exists because

$\|p_{n}(A)-p_{m}(A)\|=\|\sum_{i}(a_{n,i}-a_{m,i})A^{i}\|\leq\sum_{i}|a_{n,i}-a_{m,i}|\|A\|^{i}\leq\|p_{n}-p_{m}\|_{\infty}.$

Or, looking at the spectral form of $p_{n}(A)$ we can see that
$f(A)=\sum_{i}f(\lambda_{i})P_{\lambda_{i}}$. In this section, we develop a
tool which will allow us to do the same thing in the $\mathcal{C}^{*}$-algebra
case. This is known as the functional calculus form of the spectral theorem
[RS1].

######  Theorem 3.8 (Spectral Theorem, Functional Calculus Version).

Let $A$ be a self-adjoint element of a $\mathcal{C}^{*}$-algebra
$\mathcal{A}$, and let $\mathcal{C}^{*}(A)$ be the sub-algebra of
$\mathcal{A}$ generated by $A$. Then there exists a map
$\Phi_{A}:C(\sigma(A))\rightarrow\mathcal{C}^{*}(A)$, where $C(\sigma(A))$ is
the space of continuous functions on $\sigma(A)$ satisfying

  1. 1.

$\Phi_{A}$ is an isomorphism ,

  2. 2.

$\Phi_{A}$ is an isometry (in particular it is continuous) with the sup norm
on $C(\sigma(A))$,

  3. 3.

$\Phi_{A}(1)=\mathbbm{1}$, where $1$ is the map $\mathbb{R}\ni x\mapsto
1\in\mathbb{R}$,

  4. 4.

$\Phi_{A}(id)=A$, where $id$ is the map $\mathbb{R}\ni x\mapsto
x\in\mathbb{R}$.

Furthermore, $\sigma(\Phi_{A}(f))=f(\sigma(A))$. This is sometimes called the
Spectral Mapping Theorem.

The element $\Phi_{A}(f)$ for a continuous function $f$ should be interpreted
as “taking the function" of the element $A$. We denote it by $f(A)$.

To prove the theorem we need to do two things. First, we prove a structure
theorem for commutative $\mathcal{C}^{*}$-algebras, which says that every
unital commutative $\mathcal{C}^{*}$-algebra is isomorphic to a space of
continuous, vanishing at infinity functions on a compact Hausdorff space. This
is called the Gelfand representation. We will then apply the Gelfand
representation to the algebra generated by a self adjoint element, which is
commutative, to prove the spectral theorem. The main technical difficulty is
in proving the Gelfand representation.

We proceed by explicitly constructing the Gelfand representation. Let
$\mathcal{A}$ be a $\mathcal{C}^{*}$-algebra.

######  Definition 3.9.

A character of $\mathcal{A}$ is a surjective (algebraic) homomorphism
$\gamma:\mathcal{A}\rightarrow\mathbb{C}$. Denote by $\widehat{\mathcal{A}}$
the set of all characters of $\mathcal{A}$.

A homomorphism of $\mathcal{C}^{*}$-algebras is automatically continuous, so
$\widehat{\mathcal{A}}\subset\mathcal{A}^{*}$. Our goal will be to identify
$\mathcal{A}$ with $C^{0}(\widehat{\mathcal{A}})$, the space of continuous,
vanishing at infinity functions on $\widehat{\mathcal{A}}$, but we first need
to equip $\widehat{\mathcal{A}}$ with the appropriate topology. As a subspace
of $\mathcal{A}^{*}$ there are many topologies we could put on it, but the one
which turns out to give the desired result is the weak∗-topology.

######  Definition 3.10.

Let $X$ be a Banach space. There is a natural inclusion
$\iota:X\hookrightarrow X^{**}$ given by

$\iota(A)(\Lambda)=\Lambda(A)$

for $A\in X$ and $\Lambda\in X^{*}$. The weakest topology on $X^{*}$ which
makes every element of $\iota(X)\subset X^{**}$ continuous is called the
weak-*-topology on $X^{*}$.

${X}$${X^{*}}$${X^{**}}$${\iota(X)}$${\iota}$${\subset}$

The weak* topology is the topology of pointwise convergence, in the following
sense.

######  Proposition 3.11 (MacCluer 5.32 ).

A sequence $\Lambda_{n}$ in $X^{*}$ converges in the weak* topology if and
only if $\Lambda_{n}(A)$ converges in $\mathbb{C}$ for all $A\in X$.

It follows that the weak* topology on $X^{*}$ is Hausdorff. Let
$\Lambda\neq\Lambda^{\prime}$ be two elements of $X^{*}$. There is some $A\in
X$ such that $\Lambda(A)\neq\Lambda^{\prime}(A)$. These two points in
$\mathbb{C}$ can be separated by balls of radius $r$. Let

$U=\\{f\in X^{*}:f(A)\in B(r,\Lambda(A))\\}\;\;\;\;\;\;\;\;\;\;\;V=\\{f\in
X^{*}:f(A)\in B(r,\Lambda^{\prime}(A))\\}.$

Since pointwise evaluation is weakly* continuous, $U$ and $V$ are weakly* open
and separate $\Lambda$ and $\Lambda^{\prime}$. Therefore,
$\widehat{\mathcal{A}}$ is Hausdorff with the weak∗ topology. To prove that it
is compact we use a well-known result about the weak∗ topology

######  Theorem 3.12 (Banach-Alaoglu).

Let $X$ be a Banach space. The unit ball of $X^{*}$ is compact in the weak∗
topology.

It follows from a spectral argument that $\widehat{\mathcal{A}}$ is contained
in the unit ball of $\mathcal{A}^{*}$. We first need a lemma which allows us
to characterize the spectrum of $A$ in terms of the characters.

######  Lemma 3.13.

For $A$ in a $\mathcal{C}^{*}$-algebra $\mathcal{A}$,

$\sigma(A)=\\{\gamma(A):\gamma\in\widehat{\mathcal{A}}\\}.$

###### Proof.

The result follows from the fact that there is a correspondence between the
set of characters $\widehat{\mathcal{A}}$ and $M(\mathcal{A})$, the set of
maximal ideals in $\mathcal{A}$, given by

$\widehat{\mathcal{A}}\ni\gamma\leftrightarrow\textrm{ker}(\gamma)\in
M(\mathcal{A}).$

Suppose $\mathcal{I}$ is a maximal ideal and let
$\gamma:\mathcal{A}\rightarrow\mathcal{A}/\mathcal{I}$ be the quotient map.
Since $\mathcal{I}$ is maximal, $\mathcal{A}/\mathcal{I}$ has no proper
ideals. Therefore, for any non-zero element $A\in\mathcal{A}/\mathcal{I}$, we
must have that $A$ is invertible. Otherwise, the ideal generated by $A$ would
be a proper ideal.

Now, fix $A\neq 0$ in $\mathcal{A}/\mathcal{I}$. By the formula for the
spectral radius $R(A)=\lim_{n}\|A^{n}\|^{1/n}$ we know that the spectrum is
non-empty. So for some $\lambda$, $(\lambda\mathbbm{1}-A)$ is not invertible.
But the only non-invertible element in $\mathcal{A}/\mathcal{I}$ is $0$.
Hence, $A=\lambda\mathbbm{1}$. This shows that
$\mathcal{A}/\mathcal{I}=\mathbb{C}$, and hence that
$\gamma:\mathcal{A}\rightarrow\mathbb{C}$ is a character.

Conversely, suppose $\gamma$ is a character. Then
$\mathbb{C}\simeq\mathcal{A}/\textrm{ker}(\gamma)$. Since $\mathbb{C}$ is a
field, $\textrm{ker}(\gamma)$ is a maximal ideal. We have established a
bijection between $\widehat{\mathcal{A}}$ and $M(\mathcal{A})$.

We now fix $A\in\mathcal{A}$. Suppose $\lambda\in\sigma(A)$. Then
$\lambda\mathbbm{1}-A$ is not invertible, and hence it generates a proper
ideal of $\mathcal{A}$, because $\mathbbm{1}$ is not contained in this ideal.
We can then take the family of all ideals containing $A$ but not $\mathbbm{1}$
and order them by inclusion. Using Zorn’s lemma, we find that $A$ is contained
in a proper maximal ideal. Then using the correspondence, there is a character
$\gamma$ such that $\lambda\mathbbm{1}-A\in\textrm{ker}(\gamma)$. It follows
that $\gamma(A)=\lambda$.

Conversely, suppose for some character $\gamma$, $\gamma(A)=\lambda$. Then
$\gamma(\lambda\mathbbm{1}-A)=0$, so $\lambda\mathbbm{1}-A\in\ker(\gamma)$,
which is a proper maximal ideal by the correspondence, and hence
$\lambda\mathbbm{1}-A$ is not invertible, i.e. $\lambda\in\sigma(A)$. ∎

A consequence of the preceding lemma is that for any character $\gamma$,

$|\gamma(A)|\leq R(A)\leq\|A\|$

which implies that $\|\gamma\|_{\mathcal{A}^{*}}\leq 1$, and is contained in a
weak∗ compact set. Hence, to show that $\widehat{\mathcal{A}}$ is weak∗
compact, it suffices to show that it is weak∗ closed. Let $\gamma_{\alpha}$ be
a net in $\widehat{\mathcal{A}}$ which converges to $\gamma$ in
$\mathcal{A}^{*}$. We need to show that $\gamma$ is a character. First,
$\gamma$ is a homomorphism.

$\displaystyle\gamma(AB)$
$\displaystyle=\lim_{\alpha}\gamma_{\alpha}(AB)=\lim_{\alpha}\gamma_{\alpha}(A)\gamma_{\alpha}(B)=\gamma(A)\gamma(B)$
$\displaystyle\gamma(A^{*})$
$\displaystyle=\lim_{\alpha}\gamma_{\alpha}(A^{*})=\lim_{\alpha}\overline{\gamma_{\alpha}(A)}=\overline{(\gamma(A))}$

Since $\gamma$ is a homomorphism, it must be surjective. If
$\gamma(\mathbbm{1})=0$ then $\gamma$ is trivial since
$\gamma(A)=\gamma(\mathbbm{1})\gamma(A)=0$. But this contradicts the fact that
$\gamma_{\alpha}$ converges to $\gamma$ in the weak∗ topology. Otherwise if
$\gamma(\mathbbm{1})=z\neq 0$, then for all $\lambda\in\mathbb{C}$,
$\gamma(\lambda z^{-1}\mathbbm{1})=\lambda$ and hence $\gamma$ is surjective.

We have shown that $\widehat{\mathcal{A}}$ with the weak∗ topology is a
compact Hausdorff space. We will use this topological space to define the
Gelfand transform.

######  Definition 3.14.

Let $\mathcal{A}$ be a unital, commutative $\mathcal{C}^{*}$-algebra. The
Gelfand transform is the map $\Gamma$ defined by:

$\displaystyle\Gamma:\mathcal{A}$ $\displaystyle\rightarrow
C(\widehat{\mathcal{A}})$ $\displaystyle A$
$\displaystyle\mapsto\widehat{A}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\textrm{where}$
$\displaystyle\widehat{A}:\widehat{\mathcal{A}}$
$\displaystyle\rightarrow\mathbb{C}$ $\displaystyle\gamma$
$\displaystyle\mapsto\gamma(A)$

$\widehat{A}$ is the “evaluation at $A$" map on the characters, which is
continuous by the definition of the weak∗ topology. Lemma 3.13 can be
reformulated in terms of $\Gamma$ as

$\Gamma(A)(\widehat{\mathcal{A}})=\sigma(A)$

from which it follows that

$\|\Gamma(A)\|_{\infty}=R(A).$

We can also rewrite the bound $|\gamma(A)|\leq\|A\|$ as the fact that
$\|\Gamma(A)\|\leq\|A\|$, so $\Gamma$ is norm decreasing. $\Gamma$ is a
homomorphism when we equip $C(\widehat{\mathcal{A}})$ with pointwise addition
and multiplication and with an involution given by complex conjugation.
Finally, $\Gamma(\mathcal{A})$ clearly separates points in
$\widehat{\mathcal{A}}$ since if $\gamma_{1}\neq\gamma_{2}$ in
$\widehat{\mathcal{A}}$, then for some $A\in\mathcal{A}$,
$\gamma_{1}(A)\neq\gamma_{2}(A)$. Hence
$\widehat{A}(\gamma_{1})\neq\widehat{\mathcal{A}}(\gamma_{2})$. The last
ingredient we need to prove the structure theorem for commutative unital
$\mathcal{C}^{*}$-algebras is the general Stone-Weierstrass theorem.

######  Theorem 3.15 (Stone-Weierstrass ).

Let $X$ be a compact Hausdorff space and $\mathcal{A}$ be a self-adjoint
($f\in\mathcal{A}\implies\overline{f}\in\mathcal{A}$) sub-algebra of $C(X)$
which contains the constant functions and separates points in $X$. Then
$\mathcal{A}$ is uniformly dense in $C(X)$.

######  Theorem 3.16 (Structure Theorem for Commutative Unital
$\mathcal{C}^{*}$-algebras).

Every commutative unital $\mathcal{C}^{*}$-algebra $\mathcal{A}$ is
isometrically isomorphic to $C(\widehat{\mathcal{A}})$.

###### Proof.

Since $\mathcal{A}$ is commutative, every element is normal. For any
$A\in\mathcal{A}$,

$\|\Gamma(A)\|_{\infty}=R(A)=\|A\|.$

$\Gamma$ is an isometry, and hence it is also injective. $\Gamma(\mathcal{A})$
is a sub-algebra of $C(\widehat{\mathcal{A}})$ which separates points in
$\widehat{\mathcal{A}}$ and contains the constant functions. We just need to
show that $\Gamma(\mathcal{A})$ is self-adjoint. Suppose $A\in\mathcal{A}$ is
self-adjoint. Then for all $\gamma\in\widehat{\mathcal{A}}$

$\Gamma(A)(\gamma)=\widehat{A}(\gamma)=\gamma(A)\in\sigma(A)\subset\mathbb{R}.$

Hence, as a function on $\widehat{\mathcal{A}}$,
$\Gamma(A)=\overline{\Gamma(A)}$. Now let $A\in\mathcal{A}$ be arbitrary. It
can be decomposed into a linear combination of self-adjoint elements, $A=B+iC$
where $B,C$ are self-adjoint. Then,

$\Gamma(A^{*})=\Gamma(B-iC)=\Gamma(B)-i\Gamma(C)=\overline{\Gamma(B)+i\Gamma(C)}=\overline{\Gamma(A)}.$

Hence $\Gamma(\mathcal{A})$ is self-adjoint. It follows from Stone-Weierstrass
that $\Gamma(\mathcal{A})$ is dense in $C(\widehat{A})$, and since $\Gamma$ is
an isometry in fact $\Gamma(\mathcal{A})=C(\widehat{\mathcal{A}})$. ∎

We will now use the structure theorem to prove the spectral theorem.

###### Proof. (of the Spectral Theorem).

Fix $A\in\mathcal{A}$ a normal element of a unital $\mathcal{C}^{*}$-algebra.
Let $Pol(\mathbbm{1},A,A^{*})$ be the set of all polynomials in the variables
$\mathbbm{1}$, $A$, and $A^{*}$. These three variables commute because of the
normality assumption, so $Pol(\mathbbm{1},A,A^{*})$ is commutative. The
closure of $Pol(\mathbbm{1},A,A^{*})$ is the smallest sub-algebra containing
$A$ and $\mathbbm{1}$, so
$\mathcal{C}^{*}(A)=\overline{Pol(\mathbbm{1},A,A^{*})}$, and in particular
$\mathcal{C}^{*}(A)$ is a commutative unital $\mathcal{C}^{*}$-algebra.
Applying the Gelfand’s theorem to $\mathcal{C}^{*}(A)$ we get that

$\Gamma:\mathcal{C}^{*}(A)\rightarrow C(\widehat{\mathcal{C}^{*}(A)})$

is an isometric isomorphism.

Now let $\sigma(A)$ be the spectrum of $A$ in $\mathcal{A}$ and
$\sigma^{*}(A)$ be the spectrum of $A$ in $\mathcal{C}^{*}(A)$. We have that
$\sigma^{*}(A)\subset\sigma(A)$. By Lemma 3.13 we have that the map

$\widehat{\mathcal{C}^{*}(A)}\ni\gamma\mapsto\gamma(A)\in\sigma^{*}(A)$

is a surjection. Furthermore it is continuous because
$\widehat{\mathcal{C}^{*}(A)}$ has the weak∗ topology induced by the dual
space of $\mathcal{C}^{*}(A)$. We will now show that it is injective. Suppose
that $\gamma_{1}(A)=\gamma_{2}(A)$. Then,

$\gamma_{1}(A^{*})=\overline{\gamma_{1}(A)}=\overline{\gamma_{2}(A)}=\gamma_{2}(A^{*})$

and also $\gamma_{1}(\mathbbm{1})=1=\gamma_{2}(\mathbbm{1})$. Therefore,
$\gamma_{1}$ and $\gamma_{2}$ agree on $Pol(\mathbbm{1},A,A^{*})$ and since
they are continuous, they also agree on $\mathcal{C}^{*}(A)$.
$\gamma\mapsto\gamma(A)$ is a homeomorphism. Using this we define a map

$\displaystyle\Psi:C(\sigma^{*}(A))$ $\displaystyle\rightarrow
C(\widehat{\mathcal{C}^{*}(A)})$ $\displaystyle\Psi(f)(\gamma)$
$\displaystyle=f(\gamma(A))\;\;\;\;\;\;\;\;\;\;f\in
C(\sigma^{*}(A)),\gamma\in\widehat{\mathcal{C}^{*}(A)}$

which is an isometric isomorphism. Then $\Phi:=\Gamma^{-1}\circ\Psi$ is an
isometric isomorphism from $C(\sigma^{*}(A))$ to $\mathcal{C}^{*}(A)$. We
compute:

$\Gamma(A)(\gamma)=\gamma(A)=id(\gamma(A))=\Psi(id)(\gamma)$

hence $\Gamma(A)=\Psi(id)$ which implies that $\Phi(id)=A$. Similarly,

$\Gamma(\mathbbm{1})(\gamma)=\gamma(\mathbbm{1})=1=1(\gamma(\mathbbm{1}))=\Psi(1)(\gamma)$

hence $\Gamma(\mathbbm{1})=\Psi(1)$ which implies that $\Phi(1)=\mathbbm{1}$.

The last thing to do to get the desired mapping is to show that
$\sigma^{*}(A)=\sigma(A)$. Let $\lambda\in\sigma^{*}(A)$ and fix $\epsilon>0$.
We can find $f\in C(\sigma^{*}(A))$ such that $\|f\|_{\infty}=1$,
$0\leq|f(x)|\leq 1$, $f(\lambda)=1$ and $f(x)=0$ whenever
$|x-\lambda|>\epsilon$. Then $\Phi(f)$ is an element of $\mathcal{C}^{*}(A)$
and we can compute

$\|(A-\lambda\mathbbm{1})\Phi(f)\|=\|\Phi^{-1}((A-\lambda\mathbbm{1})\Phi(f))\|_{\infty}=\|(id-\lambda)f\|_{\infty}\leq\epsilon.$

We also have that $\|\Phi(f)\|=\|f\|_{\infty}=1$, so
$\|A-\lambda\mathbbm{1}\|\leq\epsilon$. Hence, $A-\lambda\mathbbm{1}$ cannot
be invertible in $\mathcal{A}$ (if it has a set-theoretic inverse it must be
unbounded) and $\lambda\in\sigma(A)$. ∎

###### Proof. (of the spectral mapping theorem).

First let $p(x,y)=\sum_{i,j}c_{ij}x^{i}y^{j}$ be a polynomial in two variables
with coefficients in $\mathbb{C}$ and $\phi$ be a character. Since $\phi$ is a
homomorphism,

$\phi(p(A,A^{*}))=\phi(\sum_{i,j}c_{ij}A^{i}(A^{*})^{j})=\sum_{i,j}^{n}c_{ij}\phi(A)^{i}\overline{\phi(A)}^{j}=p(\phi(A),\overline{\phi(A)}).$

Now let $f$ be a continuous function on $\sigma(A)$. By Stone-Weierstrass,
there is a sequence of elements $p_{n}(A,A^{*})\in Pol(\mathbbm{1},A,A^{*})$
converging to $f(A)$. Then since any character $\phi$ is continuous, we have
that

$\phi(p_{n}(A,A^{*}))\rightarrow\phi(f(A)).$

Consider the sequence, $\Phi^{-1}(p_{n}(A,A^{*}))\in C(\sigma(A))$, which
exists since $\Phi$ is an isomorphism. We can compute
$\Phi^{-1}(p_{n}(A,A^{*}))$ concretely using the definition of $\Phi$. First,

$\Gamma(p_{n}(A,A^{*}))=\widehat{p_{n}(A,A^{*})}:\gamma\mapsto\gamma(p_{n}(A,A^{*}))=p_{n}(\gamma(A),\overline{\gamma(A)}).$

But also, letting $p_{n}^{\prime}(x)=p_{n}(x,\overline{x})$ we have

$\Psi(p_{n}^{\prime}):\gamma\mapsto
p_{n}^{\prime}(\gamma(A))=p_{n}(\gamma(A),\overline{\gamma(A)}).$

Hence $\Psi(p_{n}^{\prime})=\Gamma(p_{n}(A,A^{*}))$ which implies that
$p_{n}^{\prime}=\Phi^{-1}(p_{n}(A,A^{*}))$. Since $\Phi^{-1}$ is continuous

$p_{n}^{\prime}=\Phi^{-1}(p_{n}(A))\rightarrow\Phi^{-1}(f(A))=f$

uniformly, and so for all $\gamma$ characters,
$p_{n}^{\prime}(\gamma(A))\rightarrow f(\gamma(A))$. Hence
$f(\gamma(A))=\gamma(f(A))$. The spectral mapping theorem follows by applying
Lemma 3.13 to $A$ and $f(A)$. ∎

We have developed a tool that allows us to take functions of elements of a
$\mathcal{C}^{*}$ algebra in a way that is compatible with the structure of
the algebra of continuous functions, and that is also compatible with the
spectral characterization of elements. We now see how this can be used to give
the rules for states and measurement in general quantum systems.

### 3.3 Spectral Measures

We are looking for a way to construct a measure on the spectrum of an element
of our $\mathcal{C}^{*}$-algebra. One important application of this is to give
a probability distribution on the set of outcomes of a measurement in quantum
mechanics. However, we will also use this result in other contexts, such as to
define the full counting statistics measure which describes the energy
fluctuations of the system and the reservoir. The main tool we will use is the
Riesz-Markov theorem, which allows us to associate measures to continuous
linear functionals on the space of continuous functions.

######  Theorem 3.17 (Riesz-Markov).

Let $X$ be a locally compact Hausdorff space and $\Lambda$ be a continuous
linear functional on $C^{0}(X)$, the space of complex-valued continuous
functions on $X$ which vanish at infinity. There is a unique regular complex
Borel measure $\mu$ on $X$ satisfying

$\Lambda(f)=\int_{X}f(x)d\mu(x)$

for each $f\in C^{0}(X)$.

Now consider a normal element in a $\mathcal{C}^{*}$-algebra
$A\in\mathcal{A}$. By using the functional calculus, we may define
$f(A)\in\mathcal{A}$ for each $f\in C^{0}(\sigma(A))$. Furthermore the map
$f\mapsto f(A)$ is continuous with the uniform topology on $C^{0}(\sigma(A))$.
Therefore, a continuous linear functional $\omega$ on $\mathcal{A}$ induces a
continuous linear functional on $C^{0}(\sigma(A))$ by

$\Lambda_{A,\omega}(f):=\omega(f(A)).$

We can then apply the Riesz-Markov theorem to obtain a measure
$\mu_{A,\omega}$ on $\sigma(A)$ satisfying

$\omega(f(A))=\int_{\sigma(A)}f(x)d\mu_{A,\omega}(x).$

for all $f\in C^{0}(\sigma(A))$. $\mu_{A,\omega}$ is called the spectral
measure of the pair $(A,\omega)$. In the special case that
$\mathcal{A}=\mathcal{B}(\mathcal{H})$ for some Hilbert space $\mathcal{H}$,
we may consider a linear functional induced by a vector $\psi\in\mathcal{H}$

$\omega_{\psi}(B)=\langle\psi,B\psi\rangle$

for all $B\in\mathcal{A}$. The spectral measure of the pair
$(A,\omega_{\psi})$ is then denoted by $\mu_{A,\psi}$.

In this setting, the spectral measure allows us to extend the functional
calculus to all bounded Borel functions on the spectrum. For $f$ bounded and
Borel on $\sigma(A)$ we define $f(A)$ as a map from $\mathcal{H}$ to
$\mathcal{H}$ such that

$\langle\psi,f(A)\psi\rangle=\int_{\sigma(A)}f(x)d\mu_{A,\psi}(x)$

for all $\psi\in\mathcal{H}$. We may then use the polarization identity to
recover $\langle\psi,f(A)\phi\rangle$ for all $\psi,\phi\in\mathcal{H}$. Then
using the Riesz lemma on the sesquilinear form
$(\psi,\phi)\mapsto\langle\psi,f(A)\phi\rangle$ we can reconstruct the bounded
operator $f(A)$. All the desired properties follow by applying the continuous
functional calculus.

### 3.4 Positive Elements and Approximate Identity

To conclude our discussion of the spectral theory, we will introduce two tools
which are useful in practice: the positive elements and approximate
identities.

The positive elements are a special class of self-adjoint elements. They form
a cone inside the $\mathcal{C}^{*}$ algebra and can be used to induce an order
relation.

######  Definition 3.18.

Let $A\in\mathcal{A}$ be an element of a $\mathcal{C}^{*}$-algebra. $A$ is
called positive if $\sigma(A)\subset[0,\infty)$. We denote the set of all
positive elements of $\mathcal{A}$ by $\mathcal{A}^{+}$.

Since self-adjoint elements have real spectrum, a positive element is
automatically self-adjoint. This definition is equivalent to several other
conditions. First, we can reformulate it in terms of an analytic property.

######  Lemma 3.19.

A self-adjoint element $A\in\mathcal{A}$ is positive if and only if
$\|\mathbbm{1}-A\|/\|A\|\leq 1$.

###### Proof.

Suppose $A$ is positive. Then by the spectral radius,
$\sigma(A)\subset[0,\|A\|]$ and hence by the spectral mapping theorem
$\sigma((\mathbbm{1}-A)/\|A\|)\subset[0,1]$. This implies that
$\|\mathbbm{1}-A\|/\|A\|\leq 1$, again by the spectral radius.

Conversely, suppose $\|\mathbbm{1}-A\|/\|A\|\leq 1$. Then by the spectral
radius $\sigma((\mathbbm{1}-A)/\|A\|)\subset[-1,1]$. Then
$\sigma(A)\subset[0,2\|A\|]$ which implies that $A$ is positive. ∎

We also have some equivalent algebraic characterizations [BR1]. The last one
in particular is useful in practice.

######  Lemma 3.20.

TFAE:

  1. 1.

$A$ is positive,

  2. 2.

$A=B^{2}$ for some $B\in\mathcal{A}$ positive (this $B$ is also unique and we
denote it by $\sqrt{A}$),

  3. 3.

$A=B^{*}B$ for some $B\in\mathcal{A}$.

We can use the positive elements to put an order relation on the
$\mathcal{C}^{*}$-algebra by

$A>B\iff A-B\in\mathcal{A}^{+}.$

The fact that this is an order relation follows from the cone structure of the
positive elements: $\mathcal{A}^{+}$ is closed under addition and
multiplication by positive scalars, and it satisfies
$\mathcal{A}^{+}\cap-\mathcal{A}^{+}=\\{0\\}$. In addition to being an order
relation, because of the $\mathcal{C}^{*}$-algebra structure $>$ has some
additional properties.

######  Lemma 3.21.

Let $A\geq B\geq 0$.

  1. 1.

$\|A\|\geq\|B\|$

  2. 2.

$C^{*}AC\geq C^{*}BC\geq 0$ for any $C\in\mathcal{A}$.

  3. 3.

If $\mathcal{A}$ has an identity and $\lambda>0$ then
$(B+\lambda\mathbbm{1})^{-1}\geq(A+\lambda\mathbbm{1})^{-1}.$

We will use this order relation to define an approximate identity.

######  Definition 3.22.

An approximate identity of a $\mathcal{C}^{*}$-algebra $\mathcal{A}$ is a net
of elements $\\{E_{\alpha}\\}_{\alpha}$ satisfying

  1. 1.

$\|E_{\alpha}\|\leq 1$,

  2. 2.

$\alpha\leq\beta\implies E_{\alpha}\leq E_{\beta}$,

  3. 3.

$\lim_{\alpha}\|E_{\alpha}A-A\|=\lim_{\alpha}\|AE_{\alpha}-A\|=0$ for all
$A\in\mathcal{A}$.

######  Proposition 3.23.

Every $\mathcal{C}^{*}$-algebra has an approximate identity.

###### Proof.

For a $\mathcal{C}^{*}$-algebra $\mathcal{A}$, let $\Lambda$ be the collection
of all finite subsets of $\mathcal{A}$. Order $\Lambda$ by inclusion. Define
the sequence of functions $f_{n}:[0,\infty)\rightarrow[0,1)$ by

$f_{n}(t)=\frac{nt}{1+nt}.$

Suppose $\alpha=\\{A_{1},...,A_{n}\\}\in\Lambda$. Then since
$A_{1}^{2}+...+A_{n}^{2}$ is a positive element we can define

$E_{\alpha}:=f_{n}(A_{1}^{2}+...+A_{n}^{2}).$

Since $f_{n}$ is non-negative, by spectral mapping $E_{\alpha}$ is a positive
element. Also since $\|f_{n}\|_{\infty}\leq 1$ we have that
$\|E_{\alpha}\|\leq 1$. Now let $\alpha\leq\beta$ in $\Lambda$ so that
$\alpha=\\{A_{1},...,A_{m}\\}$ and $\beta=\\{A_{1},...,A_{n}\\}$ with $m\leq
n$. We have that

$\mathbbm{1}+m(A_{1}^{2}+...+A_{m}^{2})\leq\mathbbm{1}+n(A_{1}^{2}+...+A_{n}^{2})$

and by the properties of our order relation it follows that

$(\mathbbm{1}+n(A_{1}^{2}+...+A_{n}^{2}))^{-1}\leq(\mathbbm{1}+m(A_{1}^{2}+...+A_{m}^{2}))^{-1}.$

Since $1-f_{n}(t)=1-\frac{nt}{1+nt}=\frac{1}{1+nt}$ it follows that
$\mathbbm{1}-E_{\beta}\leq\mathbbm{1}-E_{\alpha}$ and hence $E_{\alpha}\leq
E_{\beta}$.
To prove the last property, fix $A\in\mathcal{A}$. Then $\\{A\\}\in\Lambda$.
Fix $\alpha_{0}>\\{A\\}$ so that $\alpha_{0}=\\{A_{1},...,A_{m}\\}$ with
$A_{i}=A$ for some $i$. Let $\alpha=\\{A_{1},...A_{n}\\}$ be an arbitrary
element such that $\alpha>\alpha_{0}$, i.e. $n>m$. We have that $A^{2}\leq
A_{1}^{2}+...+A_{n}^{2}$ and hence by Lemma 3.21

$(\mathbbm{1}-E_{\alpha})A^{2}(\mathbbm{1}-E_{\alpha})\leq(\mathbbm{1}-E_{\alpha})(A_{1}^{2}+...+A_{n}^{2})(\mathbbm{1}-E_{\alpha}).$

Define $g_{n}(t)=(1-f_{n}(t))t(1-f_{n}(t))$. The right hand side of this
inequality is $g_{n}(A_{1}^{2}+...+A_{n}^{2})$ so we can rewrite is as

$(A-AE_{\alpha})^{*}(A-AE_{\alpha})\leq g_{n}(A_{1}^{2}+...+A_{n}^{2}).$

We can now compute $g_{n}(t)=\frac{t}{(1+nt)^{2}}$, which has a maximum at
$t=1/n$. Therefore, $\|g_{n}\|_{\infty}=1/4n$ and hence

$\|A-AE_{\alpha}\|^{2}\leq\frac{1}{4n}\leq\frac{1}{4m}.$

It follows that $\|A-AE_{\alpha}\|\rightarrow 0$. ∎

##  4 States and Measurement

### 4.1 Generalized States

In the finite dimensional setting, the states of a system are given by density
matrices. In the more general $\mathcal{C}^{*}$-algebraic setting, this needs
to be modified as we again run into issues with the convergence of the trace.
Given a density matrix $\rho$ over a (finite dimensional) Hilbert space
$\mathcal{H}$, we can construct $\omega_{\rho}$, a linear functional on
$\mathcal{B}(\mathcal{H})$. For $A\in\mathcal{B}(\mathcal{H})$,

$\omega_{\rho}(A):=\textrm{tr}(\rho A).$

We have that $\omega_{\rho}$ is positive, in the sense that it maps positive
operators to positive numbers. This follows by considering the eigenvalues of
$\rho A$, which are all positive since both $\rho$ and $A$ are positive.
Furthermore, we may put a norm on the space of linear functionals defined by

$||\omega||=\sup_{||A||=1}|\omega(A)|.$

This norm can equally well be defined on the space of linear functionals over
an abstract $\mathcal{C}^{*}$-algebra. Since $\rho$ is trace-one, it follows
that $||\omega_{\rho}||=1$.

Importantly, these positive linear functionals which are induced by the
density matrices are objects which still make sense in the
$\mathcal{C}^{*}$-algebra setting. Motivated by this discussion, we give the
following definition.

######  Definition 4.1.

A linear functional $\omega$ over a $\mathcal{C}^{*}$-algebra $\mathcal{A}$ is
positive if $\omega(A^{*}A)\geq 0$ for all $A\in\mathcal{A}$.
A positive linear functional $\omega$ with $||\omega||=1$ is called a state.

Conveniently, the positivity condition automatically gives continuity.

######  Proposition 4.2.

Let $\omega$ be a linear functional over a $\mathcal{C}^{*}$-algebra
$\mathcal{A}$. Then if $\omega$ is positive it is continuous (or equivalently
bounded).

###### Proof.

Let $\\{A_{i}\\}$ be an arbitrary sequence of positive elements with
$||A_{i}||\leq 1$. Let $\lambda_{i}$ be an arbitrary sequence of of positive
real numbers such that $\sum_{i}\lambda_{i}<\infty$.

$||\sum_{i=1}^{m}\lambda_{i}A_{i}-\sum_{i=1}^{n}\lambda_{i}A_{i}||=||\sum_{i=n}^{m}\lambda_{i}A_{i}||\leq\sum_{i=n}^{m}\lambda_{i}||A_{i}||\leq\sum_{i=n}^{m}\lambda_{i}$

Since $\sum_{i}\lambda_{i}$ converges, $\sum_{i}\lambda_{i}A_{i}$ converges in
norm to a positive element $A$, and since each term is positive this
convergence is monotone. It follows that

$\sum_{i}\lambda_{i}\omega(A_{i})=\omega(\sum_{i}\lambda_{i}A_{i})\leq\omega(A)<\infty$

where we have used the positivity of $\omega$. Since $\lambda_{i}$ was an
arbitrary sequence, $\omega(A_{i})$ is uniformly bounded and

$M=\sup\\{\omega(A):A\geq 0,||A||\leq 1\\}<\infty.$

But any operator can be written as a linear combination of four positive
operators using the decomposition

$A=\frac{A+A^{*}}{2}+i\frac{A-A^{*}}{2i}$

where $A_{re}:=\frac{A+A^{*}}{2}$ and $A_{im}:=\frac{A-A^{*}}{2i}$ are self-
adjoint and then the further decomposition of each of those terms into

$A_{re/im}=\frac{|A_{re/im}|+A_{re/im}}{2}-\frac{|A_{re/im}|-A_{re/im}}{2}.$

Hence $||\omega||\leq 4M<\infty$ and $\omega$ is bounded and hence continuous.

∎

A natural question to ask is whether all states defined in this was can be
recovered by tracing against an appropriate density matrix. We will see later
that this is not the case, and that the states which can be written in such a
form, called normal states, have various special properties. However, for a
fixed state we can construct something similar by using a specific
representation.

### 4.2 Geometry of the States

It is interesting to look at the geometry of the set of states. First we need
a technical lemma.

######  Lemma 4.3.

For $A,B\in\mathcal{A}$, and $\omega$ a positive linear functional,

$|\omega(AB^{*})|^{2}\leq\omega(A^{*}A)\omega(B^{*}B).$

This is a Cauchy-Schwartz type inequality. Furthermore, if we take an
approximate identity $\\{E_{\alpha}\\}$,

$||\omega||=\sup_{\alpha}\omega(E_{\alpha}^{2}).$

###### Proof.

To prove the first bound, we consider $\omega((\lambda A+B)^{*}(\lambda
A+B))$, which must be a positive real number for any choice of $\lambda$, by
the positivity of $\omega$. It follows by linearity that

$|\lambda|^{2}\omega(A^{*}A)+\overline{\lambda}\omega(A^{*}B)+\lambda\omega(B^{*}A)+\omega(B^{*}B)\geq
0.$

In particular the above quantity must be real, from which it follows that
$\omega(A^{*}B)=\overline{\omega(B^{*}A)}$. Writing $\lambda=a+ib$ where
$a,b\in\mathbb{R}$ we have that for any choice of $a$ and $b$ the following
must hold.

$a^{2}\omega(A^{*}A)+2a\textrm{Re}(\omega(B^{*}A))-2b\textrm{Im}(\omega(B^{*}A))+b^{2}\omega(A^{*}A)+\omega(B^{*}B)\geq
0$

Calculating the first and second order partial derivatives, we see that the
above quantity is positive definite everywhere and has one local minimum at
$a=-\textrm{Re}(\omega(B^{*}A))/\omega(A^{*}A)$ and
$b=\textrm{Im}(\omega(B^{*}A))/\omega(A^{*}A)$. Therefore, our inequality
being satisfied for all $a,b$ is equivalent to the inequality being satisfied
at this one point. Substituting these values, the desired bound follows.

Now we apply the previous inequality to the approximate identity.

$|\omega(AE_{\alpha})|^{2}\leq\omega(A^{*}A)\omega(E_{\alpha}^{2})\leq
M||A||^{2}\omega(E_{\alpha}^{2})$

where as before $M:=\sup\\{\omega(A):A\geq 0,||A||\leq 1\\}$. Taking the limit
in $\alpha$ and using the continuity of $\omega$ yields

$|\omega(A)|^{2}\leq
M||A||^{2}\sup_{\alpha}\omega(E_{\alpha}^{2})\implies||\omega||^{2}\leq
M\sup_{\alpha}\omega(E_{\alpha}^{2}).$

We also have that for each $\alpha$, $E_{\alpha}^{2}<1$, and hence
$\sup_{\alpha}\omega(E_{\alpha}^{2})\leq||\omega||$. From the definition,
$M\leq||\omega||$. It follows that
$||\omega||=\lim_{\alpha}\omega(E_{\alpha}^{2})$.

∎

An important corollary of this lemma is that for two linear functionals
$\omega_{1}$ and $\omega_{2}$,
$||\omega_{1}+\omega_{2}||=||\omega_{1}||+||\omega_{2}||$. In particular if
$||\omega_{1/2}||=1$,

$||\lambda\omega_{1}+(1-\lambda)\omega_{2}||=\lambda||\omega_{1}||+(1-\lambda)||\omega_{2}||=1.$

In other words, the set of states is convex. It is then natural to define the
pure states as the extremal points of the set of states, i.e. those which
cannot be written as convex combinations of distinct states. This mirrors the
classical case when we view the set of mixed states, which are probability
measures, as a subset of the hyperplane in $\mathbb{R}^{n}$. The set is convex
in the euclidean geometry with the extremal points being the pure probability
measures. The difference is that in the classical setting there is a unique
way to write each mixed state as a linear combination of pure states: the set
of states is a simplex. However, in the quantum case we do not have this
unique decomposition.

For example, in a two-level quantum system, given by the Hilbert space
$\mathbb{C}^{2}$, the states are given by $2\times 2$ matrices over
$\mathbb{C}$ with trace-one and positive eigenvalues. Any such density matrix
$\rho$ may be decomposed as a linear combination of Pauli matrices and the
identity.

$\rho=\mu\mathbbm{1}+\vec{\lambda}\cdot\vec{\sigma}$

where $\vec{\lambda}=(\lambda_{1},\lambda_{2},\lambda_{3})$ is a vector of
coefficients and $\vec{\sigma}=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the
Pauli matrices. Since all the Pauli matrices are traceless, we must have that
if $\rho$ is trace-one, $\mu=1/2$. We then compute the eigenvalues of $\rho$
in terms of $\vec{\lambda}$ to be $1/2(1\pm\|\vec{\lambda}\|)$. Since both
eigenvalues must be positive, $\|\vec{\lambda}\|\leq 1$. Therefore the map
which associates the trace-one matrix $\rho$ to the vector $\vec{\lambda}$
sends the density matrices to the unit ball. Note also that this map is linear
and so preserves the convexity structure. Recall that a state is pure if
$\textrm{tr}(\rho^{2})=1$. Using the eigenvalues we computed, we have that
$\textrm{tr}(\rho^{2})=1/2(1+\|\vec{\lambda}\|^{2})$. Hence, if $\rho$ is
pure, $\|\vec{\lambda}\|=1$ and is mapped to the unit sphere.

In both the classical and quantum case the set of states is convex with the
extremal points being the pure states. However, in the classical case each
state is a unique linear combination of the pure states, while in the quantum
case we have lost the uniqueness. There is no canonical way to build the mixed
states out of pure states.

Using again an approximate identity technique and Lemma 4.3 we can also obtain
some bounds which hold in general for states, which will be useful later.

######  Proposition 4.4.

Let $\omega$ be a positive linear functional over a $\mathcal{C}^{*}$-algebra
$\mathcal{A}$. Then for all $A,B\in\mathcal{A}$.

  1. 1.

$|\omega(A)|^{2}\leq\omega(A^{*}A)||\omega||$

  2. 2.

$|\omega(A^{*}BA)|\leq\omega(A^{*}A)||B||$

### 4.3 Measurement

In the finite-dimensional case we had a finite set of outcomes for a
measurement: the eigenvalues of a particular observable
$A=\sum_{i}e_{i}P_{i}$. If we were in a pure state $\psi$ we obtained a
probability measure on this finite set given by
$prob(e_{i})=\langle\psi,P_{i}\psi\rangle$. In a mixed state $\rho$ the
probability distribution was given by $prob(e_{i})=\textrm{tr}(\rho P_{i})$.

In the general setting we have that our set of possible outcomes for an
observable $A\in\mathcal{A}$ is $\sigma(A)\subset\mathbb{R}$, and our mixed
state is given by a positive linear functional $\omega\in\mathcal{A}^{*}$. We
will use the spectral measure formulation of the spectral theorem to obtain a
probability distribution on the set of outcomes. Recall that we have a measure
$\mu_{A,\omega}$ which satisfies

$\omega(f(A))=\int_{\sigma(A)}fd\mu_{A,\omega}.$

In analogy with the finite case, we want to consider our state evaluated on
projections associated with certain subsets of $\sigma(A)$. Let
$E\subset\sigma(A)$. We can use the Borel functional calculus to define an
operator $1_{E}(A)$ where $1_{E}$ is the characteristic function of $E$. This
generalizes the finite case since if we take $E=\\{e_{i}\\}$ then
$1_{E}(A)=P_{i}$. This gives that

$\omega(1_{E}(A))=\int_{E}d\mu_{A,\omega}=\mu_{A,\omega}(E)$

which allows us to interpret the spectral measure $\mu_{A,\omega}(E)$ as
giving the probability of the measured outcome lying in $E$. In particular,
the expected value of a measurement is given by

$\langle A\rangle=\omega(A)=\int_{\sigma(A)}xd\mu_{A,\omega}(x).$

##  5 Dynamics and Equilibrium

### 5.1 Generalized Dynamics

Recall that in the finite dimensional setting, our dynamics on the observables
was given by a Hamiltonian $H$

$\tau^{t}(A)=e^{itH}Ae^{-itH}.$

We can compute the derivative to obtain the differential equation

$\frac{d}{dt}\tau^{t}(A)=iH\tau^{t}(A)-\tau^{t}(A)(iH)=i[H,\tau^{t}(A)]$

with initial condition $\tau^{0}(A)=A$. A solution to this is

$\tau^{t}(A)=e^{it[H,\cdot]}A$

where we view $\delta_{H}:=i[H,\cdot]:A\mapsto i[H,A]$ as a bounded linear
operator on the space $\mathcal{B}(\mathcal{H})$, noting that in this setting
$\mathcal{B}(\mathcal{H})$ is a finite dimensional Banach space. $\delta_{H}$
is self-adjoint and is called the generator of $\tau_{t}$. Importantly,
$\delta_{H}$ is in the $\mathcal{C}^{*}$-algebra of operators rather than the
underlying Hilbert space, so this is the formulation we will use when moving
to the general setting.

First let us consider what happens if the Hilbert space is infinite
dimensional. We define dynamics using unitary operators, which preserve the
inner product structure.

######  Definition 5.1.

Let $\mathcal{H}$ be an infinite dimensional Hilbert space. A dynamics on
$\mathcal{B}(\mathcal{H})$ is a map $\tau^{t}(A)=U(t)AU(t)^{*}$ where $U(t)$
is a one-paramter strongly-continuous unitary group. In other words,

  1. 1.

for each $t$, $U(t)$ is unitary,

  2. 2.

for $s,t>0$, $U(t+s)=U(t)U(s)$,

  3. 3.

$U(0)=\mathbbm{1}$,

  4. 4.

for each $A$, $t\mapsto U(t)AU(t)^{*}$ is continuous.

We can apply Stone’s theorem to our dynamics, which gives a correspondence
between one parameter-strongly continuous unitary groups $U(t)$ and self-
adjoint (but not necessarily bounded) operators $A$.

$U(t)\leftrightarrow e^{itA}$

The exponential is well-defined using a power series if $A$ is bounded or
using spectral theory if $A$ is unbounded [RS1]. Therefore, while we can still
write our dynamics in the form $\tau^{t}(A)=e^{itH}Ae^{-itH}$, $H$ may be
unbounded and hence not an element of the algebra $\mathcal{B}(\mathcal{H})$.
From a physical point of view, this corresponds to an infinitely extended
system having infinite energy, but it will pose problems when we move to the
$\mathcal{C}^{*}$-algebraic formalism, where we do not have access to the
Hilbert space.

We can rewrite the definition of a dynamics in terms of the operator algebra
by using the fact that unitary operators preserve inner products, and hence
conjugation by unitaries preserves operator norm.

######  Definition 5.2.

A dynamics on a $\mathcal{C}^{*}$-algebra $\mathcal{A}$ is a map

$\mathbb{R}\ni t\mapsto\tau^{t}\in Aut(\mathcal{A})$

which satisfies

  1. 1.

$\tau^{t+s}=\tau^{t}\circ\tau^{s}$,

  2. 2.

$t\mapsto\tau^{t}(A)$ is continuous for all $A\in\mathcal{A}$ (strong
continuity).

For a $\mathcal{C}^{*}$-dynamics, which is a one-parameter strongly continuous
automorphism group on a Banach space, we define the generator as the possibly
unbounded operator $\delta:\mathcal{D}(\delta)\rightarrow\mathcal{A}$:

$\displaystyle\mathcal{D}(\delta)$
$\displaystyle=\\{A\in\mathcal{A}:\lim_{t\rightarrow
0}\frac{1}{t}(\tau^{t}(A)-A)\textrm{ exists}\\},$ $\displaystyle\delta(A)$
$\displaystyle=\lim_{t\rightarrow 0}\frac{1}{t}(\tau^{t}(A)-A)\textrm{ for
}A\in\mathcal{D}(\delta).$

The following theorem says that the generator does indeed generate the
dynamics [Ge].

######  Theorem 5.3 (Hille-Yosida).

Let $t\mapsto\tau^{t}$ be a one-parameter strongly continuous group of
automorphisms on a Banach space $X$ with generator $\delta$. Then the domain
of $\delta$, $\mathcal{D}(\delta)$, is dense and $\delta$ is closed.
Furthermore if $A\in\mathcal{D}(\delta)$ then

$\left(t\mapsto\tau^{t}(A)\right)\in C^{0}(\mathbb{R},\mathcal{D}(\delta))\cap
C^{1}(\mathbb{R},X)$

and $\delta$ satisfies the differential equation

$\frac{d}{dt}\tau^{t}(A)=\delta\tau^{t}(A)=\tau^{t}\delta(A).$

For this reason we use the formal notation $\tau^{t}=e^{t\delta}$. Note that
if we have a Hamiltonian dynamics on a finite dimensional Hilbert space given
by Hamiltonian $H$, then we may compute the generator $\delta$ in this last
more general sense and find that $\delta=\delta_{H}=i[H,\cdot]$.

### 5.2 KMS States

Now that we know how to define states and dynamics in general, we need to find
a better way of defining the equilibrium states for a given dynamics. First of
all, if the Gibbs state exists, it should be an equilibrium state. We also
want our equilibrium states to be invariant and to have certain stability
properties under small perturbations of the dynamics. First we derive a
condition which is equivalent to the Gibbs state in finite dimensions, but
which is formulated in terms of linear functionals rather than density
matrices.

We have a Hamiltonian dynamics acting on $\mathcal{B}(\mathcal{H})$ given by
$\tau^{t}(A)=e^{itH}Ae^{-itH}$. For fixed $A,B$ we can extend the function
$\mathbb{R}\ni t\mapsto\omega(A\tau^{t}(B))$ to a strip in the complex plane,
just by applying the functional calculus to the Hamiltonian. Then the Gibbs
state is equivalent to an “approximate commutation" property [JOPP].

######  Proposition 5.4.

Let $\mathcal{H}$ be a finite dimensional Hilbert space. The state
$\omega\in\mathcal{B}(\mathcal{H})^{*}$ is a Gibbs state at inverse
temperature $\beta$ if and only if

$\omega(A\tau^{i\beta}(B))=\omega(BA)$

for all $A,B\in\mathcal{B}(\mathcal{H})$. This condition is called the KMS
condition.

###### Proof.

Suppose $\omega_{\beta}$ is the Gibbs state.

$\omega_{\beta}(A\tau^{i\beta}(B))=\textrm{tr}(\rho_{\beta}Ae^{-\beta
H}Be^{\beta H})=\textrm{tr}(\frac{e^{-\beta H}}{\textrm{tr}(e^{-\beta
H})}Ae^{-\beta H}Be^{\beta H})=\textrm{tr}(\rho_{\beta}BA)=\omega_{\beta}(BA)$

The KMS condition follows from the cyclicity of the trace. This implication
actually holds as long as the Gibbs state exists, even if $H$ is not finite-
dimensional. The traceclass operators are an ideal in the space of bounded
operators, and so the above calculation still makes sense.

Now suppose $\omega$ is a state satisfying the KMS condition. Since we are in
the finite dimensional case, we have some density matrix $\rho$ such that
$\omega(\cdot)=\textrm{tr}(\rho\;\cdot)$. The KMS condition can then be
written as

$\textrm{tr}(\rho BA)=\textrm{tr}(\rho Ae^{-\beta H}Be^{\beta
H})\;\;\;\;\;\;\forall A,B\in\mathcal{B}(\mathcal{H}).$

Let $X=e^{\beta H}\rho$ and $Y=Ae^{-\beta H}$. Then we have

$\textrm{tr}(XBY)=\textrm{tr}(XYB)\;\;\;\;\;\forall
B,Y\in\mathcal{B}(\mathcal{H}).$

It follows that $[X,B]=0$ for all $B$ and hence $X=\alpha\mathbbm{1}$ for some
scalar $\alpha$. Hence $\rho=\alpha e^{-\beta H}$. Normalizing so that
$\textrm{tr}(\rho)=1$ gives the Gibbs state. ∎

######  Remark 5.1.

The KMS condition can be rewritten as

$\omega(\tau^{t}(B)A)=\omega(A\tau^{t+i\beta}(B))\;\;\;\;\;\forall
t\in\mathbb{R}.$

In this form we can see it as a boundary condition of the function

$F_{A,B}(z):=\omega(A\tau^{z}(B))$

on the strip $0\leq\textrm{Im}(z)\leq\beta$:
$F_{A,B}(t+i\beta)=\omega(\tau^{t}(B)A)$.

Our goal now is to generalize the KMS condition to the operator algebra
setting. First we need to do some work to see if, for a general
$\mathcal{C}^{*}$ dynamics, we can extend the function
$t\mapsto\omega(A\tau^{t}(B))$ analytically to a strip in the complex plane.

######  Definition 5.5.

Let $t\mapsto\tau^{t}$ be a strongly continuous, one-parameter group of
$*$-automorphisms of a $\mathcal{C}^{*}$-algebra $\mathcal{A}$. An element
$A\in\mathcal{A}$ is analytic for $\tau^{t}$ if there exists a strip
$I_{\lambda}=\\{z\in\mathbb{C}:|\textrm{Im}(z)|<\lambda\\}$ and a function
$f:I_{\lambda}\rightarrow\mathcal{A}$ such that

  1. 1.

for $t\in\mathbb{R}$, $f(t)=\tau^{t}(A)$ and

  2. 2.

for each $\omega\in\mathcal{A}^{*}$, the function $z\mapsto\omega(f(z))$ is
analytic.

This “pointwise analyticity" condition is actually equivalent to a stronger
notion of analyticity in the space $\mathcal{A}$ itself.

######  Proposition 5.6.

If $A\in\mathcal{A}$ is analytic for $\tau^{t}$ on the strip $I_{\lambda}$,
then for any $z\in I_{\lambda}$ the limit

$\lim_{h\rightarrow 0}\frac{f(z+h)-f(z)}{h}$

exists in norm.

###### Proof.

Let $z\in I_{\lambda}$ and let $B(z,r)$ be a ball contained in $I_{\lambda}$.
Let $C$ be the boundary of this ball. Define $K=B(z,r/2)\subset B(z,r)$. For
$x\in K$ and $\omega\in\mathcal{A}^{*}$ we have by Cauchy’s integral formula

$\omega(f(x))=\frac{1}{2\pi i}\int_{C}\frac{\omega(f(y))}{y-x}dy.$

Then for $z+h,z+g\in K$

$\displaystyle\frac{1}{(h-g)}\left(\frac{1}{h}(\omega(f(z+h))-\omega(f(z)))-\frac{1}{g}(\omega(f(z+g))-\omega(f(z)))\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i(h-g)}\int_{C}\omega(f(y))\left(\frac{1}{h(y-z-h)}-\frac{1}{g(y-z-g)}-\frac{1}{h(y-z)}+\frac{1}{g(y-z)}\right)dy$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{C}\omega(f(y))(y-z-g)^{-1}(y-z-h)^{-1}(y-z)^{-1}dy$

$|y-z-g|$ is bounded below uniformly in $g$ by $r/2$. Hence for fixed $\omega$
the quantity above is uniformly bounded in $g$ and $h$. Consider the family of
bounded linear maps indexed on $(h,g)$

$\displaystyle\Lambda_{h,g}:\mathcal{A}^{*}$
$\displaystyle\rightarrow\mathbb{C}$ $\displaystyle\omega$
$\displaystyle\mapsto\frac{1}{(h-g)}\left(\frac{1}{h}(\omega(f(z+h))-\omega(f(z)))-\frac{1}{g}(\omega(f(z+g))-\omega(f(z)))\right)$

We have shown that for each fixed $\omega$,
$\sup_{h,g}|\Lambda_{h,g}(\omega)|<\infty$. Therefore by uniform boundedness,
$\sup_{h,g}||\Lambda_{h,g}||<\infty$. It follows that for some $\gamma<\infty$
the following inequality holds for all $h,g$ such that $|z-g|<r/2$ and
$|z-h|<r/2$.

$||\frac{1}{h}(f(z+h)-f(z))-\frac{1}{g}(f(z+g)-f(z))||\leq\gamma|h-g|$

Since $\mathcal{A}$ is complete it follows that the derivative at $z$ exists.
∎

Now that we have seen how the analytic elements behave, we will see that we
can in fact approximate any element by (entire) analytic elements. To do this,
we use the fact that we can define a “pointwise" integral on $\mathcal{A}$.

######  Proposition 5.7.

Let $A\in\mathcal{A}$ and $\mu$ be a Borel measure on $\mathbb{R}$. There
exists $B\in\mathcal{A}$ such that for each $\omega\in\mathcal{A}^{*}$

$\omega(B)=\int\omega(\tau^{t}(A))d\mu(t).$

We denote

$B=\int\tau^{t}(A)d\mu(t).$

Using this construction we may define a sequence

$A_{n}=\sqrt{\frac{n}{\pi}}\int\tau^{t}(A)e^{-nt^{2}}dt.$

To show that $A_{n}$ is entire analytic we must construct an analytic
extension of the function $t\mapsto\tau^{t}(A_{n})$ to the whole complex
plane. Define for $z\in\mathbb{C}$

$f_{n}(z)=\sqrt{\frac{n}{\pi}}\int\tau^{t}(A)e^{-n(t-z)^{2}}dt.$

For $z=s\in\mathbb{R}$ we have

$f_{n}(s)=\sqrt{\frac{n}{\pi}}\int\tau^{t}(A)e^{-n(t-s)^{2}}dt=\sqrt{\frac{n}{\pi}}\int\tau^{t+s}(A)e^{-nt^{2}}dt=\tau^{s}(\sqrt{\frac{n}{\pi}}\int\tau^{t}(A)e^{-nt^{2}}dt)=\tau^{s}(A_{n}).$

From here, one can show that $f_{n}(z)$ is an analytic function for each $n$
and hence that each $A_{n}$ is analytic for $\tau^{t}$. Finally, one checks
that $A_{n}$ converges to $A$ in norm.

The norm dense sub-algebra of entire analytic elements (for a given dynamics
$\tau^{t}$) is denoted by $\mathcal{A}_{\tau}$. To show a set is dense in
$\mathcal{A}$ it is sufficient to show it is dense in $\mathcal{A}_{\tau}$. We
are finally ready to give the general definition of our equilibrium states.

######  Definition 5.8.

Let $(\mathcal{A},\tau)$ be a $\mathcal{C}^{*}$-dynamical system. A state
$\omega$ is a $(\tau,\beta)$-KMS state if

$\omega(A\tau^{i\beta}(B))=\omega(BA)$

for all $A,B$ in some norm-dense, $\tau$-invariant sub-algebra
$\mathcal{B}_{\tau}\subset\mathcal{A}_{\tau}$.

### 5.3 Properties of KMS States

It is not as simple to rewrite the KMS condition as a boundary condition as we
have done in the finite-dimensional case. This is because it is not clear that
the map $t\mapsto\omega(A\tau^{t}(B))$ has an analytic extension for arbitrary
$A,B\in\mathcal{A}$, since they may not be analytic elements. However, the
next proposition says that this is indeed the case.

######  Proposition 5.9.

Let $(\mathcal{A},\tau)$ be a $\mathcal{C}^{*}$-dynamical system and $\omega$
a state. Define

$D_{\beta}=\\{z\in\mathbb{C}:0<\textrm{Im}(z)<\beta\\}.$

If $\omega$ is a $(\beta,\tau)$-KMS state, then for any $A,B\in\mathcal{A}$
there exists a complex-valued function $F_{A,B}$ which is bounded and
continuous on $\overline{D_{\beta}}$ and analytic on $D_{\beta}$ satisfying
the following boundary conditions:

$F_{A,B}(t)=\omega(A\tau^{t}(B))$ $F_{A,B}(t+i\beta)=\omega(\tau^{t}(B)A)$

for all $t\in\mathbb{R}$.

This result follows from the density of the analytic elements, and the fact
that we already have an entire analytic extension $z\mapsto\tau^{z}(A)$
satisfying these properties for $A\in\mathcal{A}_{\tau}$. ([BR 5.3.7] for
details) This characterization of KMS states will be useful in practice.

The next thing to do is check that the KMS states satisfy the most basic
property that we desire from an equilibrium state: invariance under the
dynamics.

######  Lemma 5.10.

Let $(\mathcal{A},\tau)$ be a $\mathcal{C}^{*}$-dynamical system. The set of
entire analytic elements $\mathcal{A}_{\tau}$ is invariant under the dynamics.

###### Proof.

Suppose $A\in\mathcal{A}_{\tau}$. There is a function
$f:\mathbb{C}\rightarrow\mathcal{A}$ such that $f(t)=\tau^{t}(A)$ and for each
$\omega\in\mathcal{A}^{*}$, $z\mapsto\omega(f(z))$ is analytic. Define
$g(z)=f(z+s)$ for fixed $s\in\mathbb{R}$. Then

$g(t)=f(t+s)=\tau^{t+s}(A)=\tau^{t}(\tau^{s}(A))$

for all $t\in\mathbb{R}$. Also for $\omega\in\mathcal{A}^{*}$,
$z\mapsto\omega(g(z))=\omega(f(z+s))$ is analytic since it is just a shifted
analytic function. Hence $\tau^{s}(A)\in\mathcal{A}_{\tau}$ and the analytic
elements are invariant under the dynamics. ∎

######  Proposition 5.11.

Let $\omega$ be a $(\beta,\tau)$-KMS state over $\mathcal{A}$. Then for all
$A\in\mathcal{A}$,

$\omega(\tau^{t}(A))=\omega(A).$

###### Proof.

Wlog, we assume $\beta=-1$. Let $B\in\mathcal{B}_{\tau}$, the dense sub-
algebra for which the KMS condition holds. Define

$F(z)=\omega(\tau^{z}(B)).$

This is an analytic function. We will bound this function on the strip
$I=\\{z\in\mathbb{C}:-1\leq Im(z)\leq 0\\}$.

$|F(z)|\leq\|\tau^{z}(B)\|=\|\tau^{Rez}(\tau^{Imz}(B))\|=\|\tau^{Imz}(B)\|$

Hence

$\sup_{z\in I}|F(z)|\leq\sup_{\lambda\in[-1,0]}\|\tau^{i\lambda}(B)\|.$

Since $\lambda\mapsto\|\tau^{i\lambda}(B)\|$ is continuous, this supremum is
attained and $F$ is bounded on $I$ by some constant $M$. We now use the KMS
condition to show that $F$ is periodic in $i$.

$F(z-i)=\omega(\mathbbm{1}\tau^{-i}(\tau^{z}(B)))=\omega(\tau^{z}(B)\mathbbm{1})=F(z)$

(If $\mathcal{A}$ does not have an identity, use the approximate identity.)
This implies that $F$ is bounded by $M$ everywhere and hence is constant by
Liouville’s theorem. ∎

Unlike in the finite dimensional case, where there is exactly one equilibrium
state (the Gibbs state) for a given dynamics and temperature, in the general
setting a KMS state may not exist, and if it does it may not be unique.
However, we can study the geometry of the set of KMS states at a fixed
temperature.

######  Proposition 5.12.

Let $(\mathcal{A},\tau^{t})$ be a $\mathcal{C}^{*}$-dynamical system, and
$\beta\in\mathbb{R}$. The set of $(\beta,\tau)$-KMS states is convex, and in
fact it is a simplex.

A physical interpretation for this result is that the extremal points of the
set of KMS states represent different pure phases, with each KMS state having
a unique decomposition as a convex combination of these pure phases. To
motivate this further and to explore the stability properties of the KMS
states under perturbations of the dynamics, we will need to introduce several
more tools.

##  6 Representations and Perturbation Theory

### 6.1 The GNS Representation

Although we have formulated our theory without reference to an underlying
Hilbert space in order to extend it to the infinite setting, it is sometimes
useful to exploit the structure theorem for $\mathcal{C}^{*}$-algebras to work
in the more concrete setting of an algebra of bounded operators over a Hilbert
space. We refer to this process as working in a representation.

######  Definition 6.1.

Let $\mathcal{A}$ be a $\mathcal{C}^{*}$-algebra and $\mathcal{H}$ be a
Hilbert space. We say a map
$\Pi:\mathcal{A}\rightarrow\mathcal{B}(\mathcal{H})$ is a representation of
$\mathcal{A}$ in $\mathcal{H}$ if $\pi$ is a morphism of $\mathcal{C}^{*}$\-
algebras.

By Corollary 3.6 we know that a ∗-algebra morphism between
$\mathcal{C}^{*}$-algebras is automatically a $\mathcal{C}^{*}$-algebra
morphism, so we only need to check that the algebraic structure is preserved
to find a representation. Furthermore, we know from the structure theorem that
a representation always exists. However, not every rep gives an isomorphism
between $\mathcal{A}$ and $\pi(\mathcal{A})$.

######  Definition 6.2.

A representation $\pi$ of $\mathcal{A}$ over $\mathcal{H}$ is called faithful
if $\ker(\pi)=0$.

Faithful reps are particularly nice because they preserve all the structure.
The structure theorem tells us that every $\mathcal{C}^{*}$-algebra has a
faithful rep. The power of using a representation comes into play when we are
also considering a state on the $\mathcal{C}^{*}$-algebra, and we want to
represent the state in the Hilbert space as well.

######  Definition 6.3.

Given a $\mathcal{C}^{*}$-algebra $\mathcal{A}$ and a representation
$(\mathcal{H},\pi)$, a state $\omega$ is a vector state if

$\omega(A)=(\Omega,\pi(A)\Omega)$

for some $\Omega\in\mathcal{H}$. $\Omega$ is a vector representative of
$\omega$.

The GNS representation is a powerful result that allows any
$\mathcal{C}^{*}$-algebra with a state to be represented in the bounded
operators of some Hilbert space with the state being described by a cyclic
vector in this Hilbert space. It says that all states can be realised as
vector states by constructing the appropriate representation.

######  Proposition 6.4 (GNS Representation).

Given a $\mathcal{C}^{*}$-algebra $\mathcal{A}$ and a state $\omega$, there
exists a representation of $\mathcal{A}$ such that $\omega$ is a vector state
and its vector representative is a cyclic vector. This representation is
unique up to unitary equivalence, and is called the GNS representation.

###### Proof.

(Existence)

Begin with a $\mathcal{C}^{*}$-algebra $\mathcal{A}$ and a state $\omega$. Let
$I_{\omega}=\\{A\in\mathcal{A}:\omega(A^{*}A)=0\\}$. This is a left ideal of
$\mathcal{A}$. (This is not easy to show, so omit for now.) Now let
$\mathcal{H}=\mathcal{A}/I_{\omega}$. Define an inner product on
$\mathcal{H}_{\omega}$ by $([A],[B])_{\omega}=\omega(A^{*}B)$. It is not hard
to check that this is well-defined using the fact that $I_{\omega}$ is an
ideal. Finally let
$\mathcal{H}_{\omega}=\overline{\left(\mathcal{A}/I_{\omega}\right)}$ where
the closure is with respect to this inner product.
$(\mathcal{H}_{\omega},()_{\omega})$ is a Hilbert space.

We construct a map

$\displaystyle\pi_{\omega}:\mathcal{A}$
$\displaystyle\rightarrow\mathcal{B}(\mathcal{H}_{\omega})$

where $\pi_{\omega}(A)$ is defined as follows. First, for all
$B\in\mathcal{A}$ let

$\pi_{\omega}(A)_{0}:[B]\mapsto[AB]$

This is densely defined on $\mathcal{H}_{\omega}$. We now use Useful Bound 2
to see that $\forall B\in\mathcal{A}$

$||\pi_{\omega}(A)_{0}[B]||_{\mathcal{H}_{\omega}}^{2}=([AB],[AB])=\omega(B^{*}A^{*}AB)\leq||A||^{2}\omega(B^{*}B)=||A||^{2}||[B]||_{\mathcal{H}_{\omega}}^{2}$

Hence the closure of $\pi_{\omega}(A)_{0}$ is in
$\mathcal{B}(\mathcal{H}_{\omega})$ and we let
$\pi_{\omega}(A)=\overline{\pi_{\omega}(A)_{0}}$. With this definition the map
$\pi_{\omega}$ is a morphism of $\mathcal{C}^{*}$-algebras.

Finally let $\Omega_{\omega}=\mathbb{I}$. For any element $[A]$ of
$\mathcal{H}_{\omega}$, $\pi(A)\Omega_{\omega}$ = $[A\mathbb{I}]=[A]$, so
$\Omega_{\omega}$ is cyclic. Then

$(\Omega_{\omega},\pi_{\omega}(A)\Omega_{\omega})_{\omega}=([\mathbb{I}],\pi_{\omega}(A)[\mathbb{I}])_{\omega}=([\mathbb{I}],[A])_{\omega}=\omega(\mathbb{I}^{*}A)=\omega(A)$

This construction satisfies the requirements of Proposition 6.4. It is called
the canonical GNS representation.

(Uniqueness)
Let $(\mathcal{H}_{\omega},\pi_{\omega},\Omega_{\omega})$ be the cannonical
GNS representation, and
$(\mathcal{H}_{\omega}^{\prime},\pi_{\omega}^{\prime},\Omega_{\omega}^{\prime})$
be another representation satisfying the conditions of proposition 6.4.

Construct an operator $\mathcal{U}$ as follows.

$\displaystyle\mathcal{\tilde{U}}:\textrm{Dom}(\mathcal{\tilde{U}})\subset\mathcal{H}_{\omega}$
$\displaystyle\rightarrow\mathcal{H}_{\omega}^{\prime}$
$\displaystyle\Omega_{\omega}$ $\displaystyle\mapsto\Omega_{\omega}^{\prime}$
$\displaystyle\pi_{\omega}(A)\Omega_{\omega}$
$\displaystyle\mapsto\pi_{\omega}^{\prime}(A)\Omega_{\omega}^{\prime}\qquad\forall
A\in\mathcal{A}$

$\mathcal{\tilde{U}}$ is bounded on its domain.

$\displaystyle||\mathcal{\tilde{U}}(\pi_{\omega}(A)\Omega_{\omega})||_{\mathcal{H}_{\omega}^{\prime}}$
$\displaystyle=||\pi_{\omega}^{\prime}(A)\Omega_{\omega}^{\prime}||_{\mathcal{H}_{\omega}^{\prime}}$
$\displaystyle=(\pi_{\omega}^{\prime}(A)\Omega_{\omega}^{\prime},\pi_{\omega}^{\prime}(A)\Omega_{\omega}^{\prime})_{\mathcal{H}_{\omega}^{\prime}}^{2}$
$\displaystyle=(\Omega_{\omega}^{\prime},\pi_{\omega}^{\prime}(A^{*}A)\Omega_{\omega}^{\prime})^{2}$
$\displaystyle=\omega(A^{*}A)^{2}$
$\displaystyle=||\pi_{\omega}(A)\Omega_{\omega}||_{\mathcal{H}_{\omega}}$

Since
$\textrm{Dom}(\mathcal{\tilde{U}})=\\{\pi_{\omega}(A)\Omega_{\omega}:A\in\mathcal{A}\\}$
is dense in $\mathcal{H}_{\omega}$ we can extend $\mathcal{\tilde{U}}$ to a
bounded linear operator $\mathcal{U}$ on all of $\mathcal{H}_{\omega}$ by the
B.L.T. theorem. Note that since the range of $\mathcal{\tilde{U}}$ is dense in
$\mathcal{H}_{\omega}^{\prime}$ the range of $\mathcal{U}$ is also dense.
Finally we verify that $\mathcal{\tilde{U}}$ and hence $\mathcal{U}$ preserve
the inner product. Let $\xi_{A}=\pi_{\omega}(A)\Omega_{\omega}$ and
$\xi_{B}=\pi_{\omega}(B)\Omega_{\omega}$ .

$\displaystyle(\mathcal{U}\xi_{A},\mathcal{U}\xi_{B})$
$\displaystyle=(\pi_{\omega}^{\prime}(A)\Omega_{\omega}^{\prime},\pi_{\omega}^{\prime}(B)\Omega_{\omega}^{\prime})$
$\displaystyle=\omega(A^{*}B)$ $\displaystyle=(\xi_{A},\xi_{B})$

Let $\xi,\xi^{\prime}\in\mathcal{H}_{\omega}$. Then there are sequences
$A_{n}$ and $B_{n}$ in $\mathcal{A}$ such that
$\pi_{\omega}(A_{n})\Omega_{\omega}\rightarrow\xi$ and
$\pi_{\omega}(B_{n})\Omega_{\omega}\rightarrow\xi^{\prime}$.

$\displaystyle(\mathcal{U}\xi,\mathcal{U}\xi^{\prime})$
$\displaystyle=\lim_{n\rightarrow\infty}(\mathcal{U}\pi_{\omega}(A_{n})\Omega_{\omega},\mathcal{U}\pi_{\omega}(B_{n})\Omega_{\omega})$
$\displaystyle=\lim_{n\rightarrow\infty}(\pi_{\omega}(A_{n})\Omega_{\omega},\pi_{\omega}(B_{n})\Omega_{\omega})$
$\displaystyle=(\xi,\xi^{\prime})$

The fact that $\mathcal{U}$ is bounded and linear and hence continuous is used
in the first and last steps. Hence, since $\mathcal{U}$ preserves the inner
product and has dense range, it is unitary. ∎

One useful application of the uniqueness part of the result is to take an
invariant state $\omega$ for some dynamics $\tau$ (such as a KMS state) and
apply the theorem to the cyclic representation
$(\mathcal{H}_{\omega},\pi_{\omega}\circ\tau^{t},\Omega_{\omega})$. Since

$(\Omega_{\omega},\pi_{\omega}\circ\tau^{t}(A)\Omega_{\omega})=\omega(\tau^{t}(A))=\omega(A)$

we have that this rep is unitarily equivalent to
$(\mathcal{H}_{\omega},\pi_{\omega},\Omega_{\omega})$. Hence there is some
unitary $\mathcal{U}(t)$ such that

$\displaystyle\pi_{\omega}(\tau^{t}(A))$
$\displaystyle=\mathcal{U}(t)\pi_{\omega}(A)\mathcal{U}(t)^{*}$
$\displaystyle\mathcal{U}(t)\Omega_{\omega}$ $\displaystyle=\Omega_{\omega}$

i.e. the automorphism $\tau^{t}$ in unitarily implemented by $\mathcal{U}(t)$
in the GNS rep of the invariant state. The unitary $\mathcal{U}(t)$ is not
unique, but we will see later that with an additional restriction there is a
unique way to choose it, and then $t\mapsto\mathcal{U}(t)$ will be a one-
parameter strongly continuous unitary group. We can apply Stone’s theorem to
get a self-adjoint (over $\mathcal{H}_{\omega}$) generator $L$ such that
$U(t)=e^{itL}$.

Additionally, for a given state the GNS representation may not be faithful, in
which case some of the structure is lost by working in the representation.
However, there is a notion of faithfulness for states.

######  Definition 6.5.

A state $\omega$ over $\mathcal{A}$ is faithful if $\omega(A^{*}A)=0$ implies
that $A=0$.

If a state $\omega$ is faithful, then in the construction of the GNS rep
$\mathcal{I}_{\omega}$ is trivial and hence $\mathcal{H}_{\omega}=\mathcal{A}$
and $\pi_{\omega}(A)X=AX$ for all $X\in\mathcal{A}$. It follows that
$\pi_{\omega}$ is injective. The GNS representation of a faithful state is
faithful. We also have the following useful result [BR2].

######  Proposition 6.6.

A KMS state over a $\mathcal{C}^{*}$-algebra is automatically faithful.

We may safely work in the GNS rep of an equilibrium state without losing any
information.

Finally, we may sometimes wish to work in the GNS rep of a
$\mathcal{C}^{*}$-algebra of the form $\mathcal{B}(\mathcal{H})$ where
$\mathcal{H}$ is a finite dimensional Hilbert space. In this case it is often
more convenient to consider the standard GNS rep, which is unitarily
equivalent but not equal to the canonical GNS rep. It is constructed from
$(\mathcal{B}(\mathcal{H}),\tau^{t}=e^{it[H,\cdot]},\omega=\textrm{tr}(\rho\cdot))$
by taking $\mathcal{H}_{\omega}=\mathcal{B}(\mathcal{H})$ with inner product

$(A,B)_{\mathcal{H}_{\omega}}=\textrm{tr}(A^{*}B),$

$\pi_{\omega}(A)=L(A)$ where $L(A)$ is the standard left representation

$L(A)B=AB$

for $B\in\mathcal{H}_{\omega}$, and $\Omega_{\omega}=\rho^{1/2}$.

### 6.2 Normal States

Having moved to a GNS representation for a given state, one can ask whether
other states can also be represented as vectors in this representation. In
general, two arbitrary states cannot be represented in the same GNS Hilbert
space, and those that can exhibit certain behaviour.

######  Definition 6.7.

Let $\mathcal{A}$ be a $\mathcal{C}^{*}$ algebra and $\omega,\eta$ two states.
The state $\eta$ is said to be $\omega$-normal if there is a positive trace-
one operator $\rho_{\eta}$ in $\mathcal{B}(\mathcal{H}_{\omega})$ such that

$\eta(A)=\textrm{tr}(\rho_{\eta}\pi_{\omega}(A)).$

In other words, $\eta$ can be written as a trace state in the $\omega$-GNS
rep.

######  Proposition 6.8.

If $\eta$ is $\omega$-normal, then there exists a vector
$\Omega_{\eta}\in\mathcal{H}_{\omega}$ such that

$\eta(A)=(\Omega_{\eta},\pi_{\omega}(A)\Omega_{\eta}).$

This should be thought of as the non-commutative analog of a measure $\eta$
being absolutely continuous with respect to another measure $\omega$. In that
situation integrals against $\eta$ can be computed as integrals against
$\omega$ (by weighting by the Radon-Nikodym derivative). Similarly here, the
value of $\eta$ on an operator can be computed in the $\omega$-GNS rep, by
taking an inner product against an appropriate vector. We will see later that
there is an object built out of $\Omega_{\eta}$ which plays the role of the
Radon-Nikodym derivative.

We denote by $\mathcal{N}_{\omega}$ the set of all states which are
$\omega$-normal. This set can be thought of physically as the states that are
“not too different" to the state $\omega$. We can now compare states by
comparing these sets of relatively normal states.

######  Definition 6.9.

Let $\omega$ and $\eta$ be two states. If
$\mathcal{N}_{\omega}=\mathcal{N}_{\eta}$ then we say they are quasi-
equivalent. If $\mathcal{N}_{\omega}\cap\mathcal{N}_{\eta}=\phi$ then we say
they are disjoint.

This allows us to refine our description of the set of KMS states.

######  Proposition 6.10.

If $\omega_{1}$ and $\omega_{2}$ are two extremal states in the set of
$\tau,\beta$-KMS states for some fixed $\tau$ and $\beta$ then they are either
equal or disjoint.

Physically, this corresponds to pure phases being “far apart" in the sense
that one cannot represent two different pure phases both as vectors states in
any GNS representation. The set of normal states is also an important concept
used in Araki’s perturbation theory of KMS states, which is the subject of the
next section.

### 6.3 Araki Perturbation Theory

In Araki’s perturbation theory of KMS states we start with a
$\mathcal{C}^{*}$-dynamical system $(\mathcal{A},\tau^{t})$ and a fixed
temperature $\beta^{-1}$. We assume that $\omega$ is some fixed
$\tau,\beta$-KMS state. We then perturb the dynamics by a bounded self-adjoint
operator and study the new equilibrium states. The following proposition
describes the type of perturbation we are using.

######  Proposition 6.11.

Let $(\mathcal{A},\tau^{t})$ be a $\mathcal{C}^{*}$-dynamical system with
generator $\delta$. Let $V\in\mathcal{A}$ be self adjoint. Define $\delta_{V}$
by

$\displaystyle\mathcal{D}(\delta_{V})$ $\displaystyle=\mathcal{A}$
$\displaystyle\delta_{V}(A)$ $\displaystyle=i[V,A].$

Then $\delta+\delta_{V}$ generates a one-parameter group of ∗-automorphisms,
$\tau_{V}^{t}$. This dynamics has the perturbative expansion

$\tau_{V}^{t}(A)=\tau^{t}(A)+\sum_{n\geq
1}i^{n}\int_{0}^{t}\int_{0}^{t_{1}}...\int_{0}^{t_{n-1}}[\tau^{t_{n}}(V),[...[\tau^{t_{1}}(V),\tau^{t}(A)]]]dt_{n}...dt_{2}dt_{1}.$

If we move to the GNS rep, where our unperturbed dynamics is unitarily
implemented, we also have a unitary implementation for the perturbed dynamics.

######  Proposition 6.12.

$\pi_{\omega}(\tau_{V}^{t}(A))=\Gamma_{V}^{t}\pi_{\omega}(\tau^{t}(A)){\Gamma_{V}^{t}}^{*}$

where

$\Gamma_{V}^{t}=\mathbbm{1}+\sum_{n\geq
1}i^{n}\int_{0}^{t}\int_{0}^{t_{1}}...\int_{0}^{t_{n-1}}\pi_{\omega}\left(\tau^{t_{1}}(V)\right)...\pi_{\omega}\left(\tau^{t_{n}}(V)\right)dt_{1}...dt_{n}.$

Now, under this perturbation we want to see what we can say about the new KMS
states [BR2].

######  Theorem 6.13 (Araki Perturbation).

Let $(\mathcal{A},\tau)$ be a $\mathcal{C}^{*}$-dynamical system, $\omega$ a
$(\tau,\beta)$-KMS state and $V$ a self adjoint element of $\mathcal{A}$.
Furthermore, let $(\mathcal{H}_{\omega},\pi_{\omega},\Omega)$ be the GNS
representation of $\omega$ and $U(t)=e^{itL}$ be a unitary implementation of
$\tau^{t}$ in this representation.

  1. (1)

There is a unique $\tau_{V},\beta$-KMS state which is $\omega$-normal. Call it
$\omega_{V}$.

  2. (2)

The states $\omega$ and $\omega_{V}$ are quasi-equivalent.

  3. (3)

If $\\{V_{\alpha}\\}$ is a net of perturbations converging to $0$ in norm then

$\lim_{\alpha}\omega_{V_{\alpha}}=\omega.$

Furthermore, we have the following explicit formulas.

  * •

In the $\omega$-GNS representation, the perturbed dynamics $\tau_{V}$ is given
by

$\pi_{\omega}(\tau_{V}^{t}(A))=e^{it(L+\pi_{\omega}(V))}\pi_{\omega}(A)e^{-it(L+\pi_{\omega}(V))}.$

  * •

$\Omega$ is in the domain of $e^{-\beta(L+\pi_{\omega}(V))/2}$ and the vector
representative of $\omega_{V}$ in $\mathcal{H}_{\omega}$ is

$\Omega_{V}=\frac{e^{-\beta(L+\pi_{\omega}(V))/2}\Omega}{\|e^{-\beta(L+\pi_{\omega}(V))/2}\Omega\|}.$

The function

$z\mapsto e^{-z(L+\pi(V))}\Omega_{0}$

is analytic in the strip $0<\textit{Re}z<\beta/2$ and bounded and continuous
on its closure.

###### Proof. (of formulas).

Let $\mathcal{U}_{V}^{t}=e^{it(L+\pi_{\omega}(V))}$ and
$X^{t}=\mathcal{U}_{V}^{t}\mathcal{U}^{-t}$. We compute

$\displaystyle\frac{dX^{t}}{dt}$
$\displaystyle=e^{it(L+\pi_{\omega}(V))}(i(L+\pi_{\omega}(V)))e^{-itL}+e^{it(L+\pi_{\omega}(V))}(-iL)e^{-itL}$
$\displaystyle=i\mathcal{U}_{V}^{t}\pi_{\omega}(V)\mathcal{U}^{-t}$
$\displaystyle=iX^{t}\pi_{\omega}(\tau^{t}(V))$ and $\displaystyle X^{0}$
$\displaystyle=\mathbbm{1}.$

Hence $X^{t}$ is the unique solution of

$X^{t}=\mathbbm{1}+i\int_{0}^{t}X^{s}\pi_{\omega}(\tau^{s}(V))ds.$

Iterating this we obtain that $X^{t}=\Gamma_{V}^{t}$ and hence
$\Gamma_{V}^{t}\mathcal{U}^{t}=\mathcal{U}_{V}^{t}$. To conclude we have that

$\pi_{\omega}(\tau_{V}^{t}(A))=\Gamma_{V}^{t}\pi_{\omega}(\tau^{t}(A)){\Gamma_{V}^{t}}^{*}=\Gamma_{V}^{t}\mathcal{U}^{t}\pi_{\omega}((A)){\mathcal{U}^{t}}^{*}{\Gamma_{V}^{t}}^{*}=\mathcal{U}_{V}^{t}\pi_{\omega}(A){\mathcal{U}_{V}^{t}}^{*}.$

Next we will assume that $V$ is an analytic element for $\tau$ in order to
define

$\Gamma_{V}^{is}=\mathbbm{1}+\sum_{n\geq
1}(-1)^{n}\int_{0}^{s}\int_{0}^{s_{1}}...\int_{0}^{s_{n-1}}\pi_{\omega}\left(\tau^{is_{n}}(V)\right)...\pi_{\omega}\left(\tau^{is_{1}}(V)\right)ds_{1}ds_{2}...ds_{n}.$

This converges uniformly and we have that
$\Gamma_{V}^{i\beta/2}\Omega=e^{-\beta(L+\pi_{\omega})/2}\Omega$, noting that
$L\Omega=0$. ∎

Using this, we can also look at the structure of the set of KMS states under
the perturbation, and see that it is preserved.

######  Corollary 6.14.

The map from the set of $(\tau,\beta)$-KMS states to $(\tau_{V},\beta)$-KMS
states given by

$\omega\mapsto\omega_{V}$

is an isomorphism and maps extremal points to extremal points.

This means that the physical phase structure remains the same under this type
of bounded perturbation.

##  7 Modular Theory

### 7.1 von Neumann Algebras

Modular theory provides a very useful set of tools for doing computations in
the GNS representation. To define the necessary structure to develop the
modular theory, we need a special class of $\mathcal{C}^{*}$-algebras called
von Neumann algebras. These are concrete algebras of operators over Hilbert
spaces.

######  Definition 7.1.

Let $\mathcal{H}$ be a Hilbert space and let $\mathscr{M}$ be a subset of
$\mathcal{B}(\mathcal{H})$. The commutant of $\mathscr{M}$, denoted
$\mathscr{M}^{\prime}$, is the set of all bounded linear operators on
$\mathcal{H}$ which commute with every element of $\mathscr{M}$. $\mathscr{M}$
is a Von Neumann Algebra if

$\mathscr{M}^{\prime\prime}=\mathscr{M}.$

This algebraic definition is equivalent to an analytic property.

######  Proposition 7.2 (Bicommutant Theorem ).

Let $\mathscr{M}$ be a non-degenerate ∗-algebra of operators on a Hilbert
space $\mathcal{H}$. $\mathscr{M}$ is a von Neumann algebra if and only if
$\mathscr{M}$ is closed in the weak operator topology, i.e. the smallest
topology which makes all the linear functionals of the form

$A\mapsto\langle x,Ay\rangle$

continuous for all $x,y\in\mathcal{H}$.

Note that the weak operator topology is weaker than the norm topology, which
makes every von Neumann algebra automatically a $\mathcal{C}^{*}$ algebra.

In our setting, we have algebras of the form
$\pi_{\omega}(\mathcal{A})\subset\mathcal{B}(\mathcal{H}_{\omega})$, where
$\mathcal{A}$ is a $\mathcal{C}^{*}$ algebra, arising from GNS
representations. These may not be von Neumann algebras, but there is a
canonical way to enlarge them to get a von Neumann algebra.

######  Proposition 7.3.

Let $\mathscr{M}$ be a sub-algebra. $\mathscr{M}^{(n)}=\mathscr{M}^{(n+2)}$
for $n\geq 1$

###### Proof.

$\mathscr{M}^{\prime}$ is the set of $B\in\mathcal{B}(\mathcal{H})$ such that
$[A,B]=0\forall A\in\mathscr{M}$. So if $A\in\mathscr{M}$ then $[A,B]=0\forall
B\in\mathscr{M}^{\prime}$ which implies that $A\in\mathscr{M}^{\prime\prime}$.
Therefore we always have the inclusion
$\mathscr{M}\subset\mathscr{M}^{\prime\prime}$. It follows from this that
$(\mathscr{M}^{\prime\prime})^{\prime}=\mathscr{M}^{\prime\prime\prime}\subset\mathscr{M}^{\prime}$.
As well,
$\mathscr{M}^{\prime}\subset(\mathscr{M}^{\prime})^{\prime\prime}=\mathscr{M}^{\prime\prime\prime}$.
By induction, $\mathscr{M}^{(n)}=\mathscr{M}^{(n+2)}$ for $n\geq 1$. ∎

Given any sub algebra $\mathscr{M}\subset\mathcal{B}(\mathcal{H})$ we can turn
it into a von Neumann algebra by taking the bicommutant. This has the effect
of “adding in some elements” and guarantees the von Neumann algebra property
since $\mathscr{M}^{(4)}=\mathscr{M}^{\prime\prime}$. For this reason
$\mathscr{M}^{\prime\prime}$ is called the enveloping von Neumann algebra. The
enveloping von Neumann algebra is also the weak closure, making it the minimal
algebra with this property.

Following [BR1] Chapter 2, our next step is to define certain operators on
$\mathcal{H}$. We will do this by defining them on a domain of the form
$\mathscr{M}\Omega$ or $\mathscr{M}^{\prime}\Omega$ for some vector $\Omega$