On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics

TL;DR

This paper emphasizes the role of Hilbert-Schmidt operators in noncommutative quantum mechanics and applies this framework to analyze a charged particle in a magnetic field.

Contribution

It introduces a Hilbert-Schmidt operator approach to noncommutative quantum mechanics and demonstrates its application to a charged particle in a magnetic field.

Findings

Hilbert-Schmidt operators are crucial in noncommutative quantum formulations.

The framework effectively describes a charged particle in a magnetic field.

New insights into noncommutative quantum systems are provided.

Abstract

This work gives value to the importance of Hilbert-Schmidt operators in the formulation of a noncommutative quantum theory. A system of charged particle in a constant magnetic field is investigated in this framework.

Equations

$$\mathcal{U}(A;\epsilon )=\{B;B\in \mathfrak{G},||B-A||<\epsilon \},$$

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

$$\pi (A)=\pi (B^{\ast }B)=\pi (B{)}^{\ast }\pi (B)⩾0.$$

$$||\pi (A)||⩽||A||$$

$$||\pi (A)||=\text{mbox}sup\left\{|\lambda |;\lambda \in \sigma (\pi (A))\right\}$$

$$||\pi (A)||⩽\text{mbox}sup\left\{|\lambda |;\lambda \in {\sigma }_{\mathfrak{G}}(A)\right\}⩽||A||$$

$$\rho (A)=\text{mbox}sup\{|\lambda |;\lambda \in {\sigma }_{\mathfrak{G}}(A)\}.$$

$$\rho (A)=\underset{n\to \infty }{\lim }||A^n|{|}^{1/n}=\text{mbox}{inf}_n||A^n|{|}^{1/n}⩽||A||.$$

$${\lambda }^{-1}\sum _{m⩾0}{\left(\frac{A}{\lambda }\right)}^m$$

$$\rho (A)⩽||A^n|{|}^{1/n}$$

$$\rho (A)⩽\text{mbox}{inf}_n||A^n|{|}^{1/n}⩽\underset{n\to \infty }{\lim }\text{mbox}inf||A^n|{|}^{1/n}.$$

$$r_A=\underset{n\to \infty }{\lim }\text{mbox}sup||A^n|{|}^{1/n}.$$

$$R_n(A;\lambda )=n^{-1}\sum _{k=1}^n{\left(11-\frac{{\omega }^{k}A}{\lambda }\right)}^{-1}$$

$$R_n(A;\lambda )={\left(11-\frac{A^n}{{\lambda }^n}\right)}^{-1}.$$

$$\left|\left|{\left(11-\frac{{\omega }^{k}A}{r_A}\right)}^{-1}-{\left(11-\frac{{\omega }^{k}A}{\lambda }\right)}^{-1}\right|\right|$$

$$||(\lambda 11-A{)}^{-1}||⩽|\lambda {|}^{-1}\sum _{n⩾0}||A|{|}^n/|\lambda {|}^n=(|\lambda |-||A||{)}^{-1}.$$

$$\left|\left|{\left(11-\frac{A^n}{r_A^n}\right)}^{-1}-{\left(11-\frac{A^n}{{\lambda }^n}\right)}^{-1}\right|\right|<\epsilon$$

$$||\pi (A)|{|}^2=||\pi (A^{\ast }A)||⩽||A^{\ast }A||⩽||A|{|}^2.$$

$$\text{mbox}ker\pi =\{A\in \mathfrak{G};\pi (A)=0\}$$

$$||\hat{A}||=||{\hat{\pi }}^{-1}(\hat{\pi }(\hat{A}))||⩽||\hat{\pi }(\hat{A})||⩽||\hat{A}||.$$

$$||A||=||{\pi }^{-1}(\pi (A))||⩽||\pi (A)||⩽||A||.$$

$$\left\{\sum _i\pi (A_i){\psi }_i;A_i\in \mathfrak{G},{\psi }_i\in \mathfrak{K}\right\}$$

$$\omega (A^{\ast }A)⩾0$$

$${\omega }_{\Omega }(A)=(\Omega ,\pi (A)\Omega )$$

$${\omega }_{\Omega }(A^{\ast }A)=||\pi (A)\Omega |{|}^2⩾0.$$

$${\omega }_T(A)=(T{\Omega }_{\omega },{\pi }_{\omega }(A){\Omega }_{\omega })$$

$$|\rho (B^{\ast }A){|}^2$$

$$({\pi }_{\omega }(B){\Omega }_{\omega },T{\pi }_{\omega }(A){\Omega }_{\omega })=\rho (B^{\ast }A).$$

$$0$$

$$({\pi }_{\omega }(B){\Omega }_{\omega },T{\pi }_{\omega }(C){\pi }_{\omega }(A){\Omega }_{\omega })$$

Full text

On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics

Isiaka Aremua $\!{}^{\rm a,b}$, Ezinvi Baloïtcha $\!{}^{\rm b}$, Mahouton Norbert Hounkonnou $\!{}^{\rm b}$ and Komi Sodoga $\!{}^{\rm a,b}$

a Université de Lomé, Faculté des Sciences, Département de Physique,

Laboratoire de Physique des Matériaux et de Mécanique Appliquée, 02 BP 1515 Lomé, Togo

   E-mail: iaremua@univ-lome.tg, ksodoga@univ-lome.tg

b University of Abomey-Calavi, International Chair in Mathematical Physics and Applications (ICMPA), 072 B.P. 050 Cotonou, Benin

   E-mail : ezinvi.baloitcha@cipma.uac.bj, norbert.hounkonnou@cipma.uac.bj

This work gives value to the importance of Hilbert-Schmidt operators in the formulation of noncommutative quantum theory. A system of charged particle in a constant magnetic field is investigated in this framework.

Introduction

The theory of Hilbert-Schmidt operators plays a key role in the formulation of the noncommutative quantum mechanics. In the past three decades, the von Neumann algebras [19, 27] underwent a vigorous growth after the discovery of a natural infinite family of pairwise nonisomorphic factors, and the advent of Tomita-Takesaki theory [25] and Connes noncommutative geometry [10] iniated with the classification theorems for von Neumann algebras and structure of extensions of $C^{*}$-algebras [11]. The modular theory of von Neumann algebras was created by M. Tomita [26] in 1967 and perfectioned by M. Takesaki around 1970.

From physical point of view, a charged particle interacting with a constant magnetic field is one of the important problems in quantum mechanics described by the Hamiltonian $\mathcal{H}=\frac{1}{2M}({\bf p}+\frac{e}{c}{\bf A})^{2}$, inspired by condensed matter physics, quantum optics, etc. The Landau problem [18] is related to the motion of a charged particle on the flat plane xy in the presence of a constant magnetic field along the $z$-axis. In metals, the electrons occupy many Landau levels [13] $E_{n}=\hbar\omega_{c}(n+\frac{1}{2})$, each level being infinitely degenerate, with $\omega_{c}=eB/Mc$, the cyclotron frequency, which are those of the one-dimensional harmonic oscillator, and correspond to the kinetic energy levels of electrons.

This physical model represents an interesting application [1] of the Tomita-Takesaki modular theory [26, 25]. Taking into account the sense of the magnetic field, one obtains a pair of commuting Hamiltonians. Both these Hamiltonians can be written in terms of two pairs of mutually commuting oscillator-type creation and annihilation operators, which then generate two mutually commuting von Neumann algebras, commutants of each other. The associated von Neumann algebra of observables displays a modular structure in the sense of the Tomita-Takesaki theory, with the algebra and its commutant referring to the two orientations of the magnetic field.

Hilbert spaces, at the mathematical side, realize the skeleton of quantum theories. Coherent states (CS), defined as a specific overcomplete family of vectors in the Hilbert space describing quantum phenomena [23, 17, 20, 3, 12], constitute an important tool of investigation. In the studies and understanding of noncommutative geometry, CS were proved to be useful objects [14]. Based on the approach developed in [22], Gazeau-Klauder CS were constructed in noncommutative quantum mechanics [6]. Besides, in studying, in the noncommutative plane [15], the behavior of an electron in an external uniform electromagnetic background coupled to a harmonic potential, matrix vector coherent states (MVCS) as well as quaternionic vector coherent states (QVCS) were constructed and discussed.

Our present contribution paper is organized as follows:

• •

First, we formulate the Hilbert-Schmidt operators and the Tomita-Takesaki modular theories in the framework of noncommutative quantum mechanics.

• •

Detailed proofs are given for main frequently used statements in the study of modular theory and Hilbert-Schmidt operators. As application, a construction of CS from the thermal state is achieved as in a previous work [1]. Relevant properties are discussed. Then, a light is put on the Wigner map as an interplay between the noncommutative quantum mechanics formalism [22] and the modular theory based on Hilbert-Schmidt operators.

• •

Finally, the motion of a charged particle on the flat plane xy in the presence of a constant magnetic field along the $z$-axis with a harmonic potential is studied.

1 von Neumann algebras: modular theory, Hilbert-Schmidt operators and coherent states

This section recapitulates fundamental notions and main ingredients of the modular theory used in the sequel. More details on these mathematical structures and their applications may be found in a series of works [19, 27, 26, 25, 8, 21, 11, 1, 2, 7] (and references therein), widely exploited to write this review section.

1.1 Basics on von Neumann algebras

In this paragraph, $\mathfrak{H}$ denotes a Hilbert space over $\mathbb{C}.$ $\mathfrak{H}$ is assumed to be separable, of dimension $N,$ which could be finite or infinite. Denote by $\mathcal{L}(\mathfrak{H})$ the $C^{*}$-algebra of all bounded operators on $\mathfrak{H}.$ The following definitions are in order:

Definition 1.1

**

Let $\mathfrak{G}$ be an algebra. A mapping $A\in\mathfrak{G}\mapsto A^{*}\in\mathfrak{G}$ is called an involution, or adjoint operation, of the algebra $\mathfrak{G},$ if it has the following properties:

$A^{**}=A$ 2. 2.

$(AB)^{*}=B^{*}A^{*},$ with $A,B\in\mathfrak{G},\,A^{*},B^{*}\in\mathfrak{G}$ 3. 3.

$(\alpha A+\beta B)^{*}=\bar{\alpha}A^{*}+\bar{\beta}B^{*},\,\alpha,\beta\in\mathbb{C}$.

($\bar{\alpha}$ is the complex conjugate of $\alpha.$)

Definition 1.2

∗-algebra

An algebra with an involution is called ∗**-algebra and a subset $\mathfrak{B}$ of $\mathfrak{G}$ is called self-adjoint if $A\in\mathfrak{B}$ implies that $A^{*}\in\mathfrak{B}.$

The algebra $\mathfrak{G}$ is a normed algebra if to each $A\in\mathfrak{G}$ there is associated a real number $||A||,$ the norm of $A,$ satisfying the requirements

$||A||\geqslant 0$ and $||A||=0$ if, and only if, $||A||=0,$ 2. 2.

$||\alpha A||=|\alpha|||A||,$ 3. 3.

$||A+B||\leqslant||A||+||B||,$ 4. 4.

$||AB||\leqslant||A||||B||.$

The third of these conditions is called the triangle inequality and the fourth the product inequality. The norm defines a metric topology on $\mathfrak{G}$ which is referred to as the uniform topology. The neighborhoods of an element $A\in\mathfrak{G}$ in this topology are given by

[TABLE]

where $\varepsilon>0.$ If $\mathfrak{G}$ is complete with respect to the uniform topology, then it is called a Banach algebra. A normed algebra with involution which is complete and has the property $||A||=||A^{*}||$ is called a Banach ∗-algebra. Then, follows the definition:

Definition 1.3

**A $C^{*}$-algebra is a Banach ∗-algebra $\mathfrak{G}$ with the property

[TABLE]

for all $A\in\mathfrak{G}.$

Before going further, let us deal, in the following, with some notions about representations and states.

Definition 1.4

∗-Morphism between two ∗-algebras Let $\mathfrak{G}$ and $\mathfrak{B}$ be two ∗-algebras. The ∗-morphism between $\mathfrak{G}$ and $\mathfrak{B}$ is given by the mapping $\pi:A\in\mathfrak{G}\mapsto\pi(A)\in\mathfrak{B}$, satisfying:

$\pi(\alpha A+\beta B)=\alpha\pi(A)+\beta\pi(B)$ 2. 2.

$\pi(AB)=\pi(A)\pi(B)$ 3. 3.

$\pi(A^{*})=\pi(A)^{*}$

for all $A,B\in\mathfrak{G},\alpha\in\mathbb{C}.$

Remark 1.5

**Each ∗-automorphism $\pi$ between two ∗-algebras $\mathfrak{G}$ and $\mathfrak{B}$ is positive because if $A\geqslant 0,$ then $A=B^{*}B$ for some $B\in\mathfrak{G}$. Hence,

[TABLE]

Definition 1.6

**Representation of a $C^{*}$-algebra

A representation of a $C^{*}$-algebra $\mathfrak{G}$ is defined to be a pair $(\mathfrak{H},\pi)$, where $\mathfrak{H}$ is a complex Hilbert space and $\pi$ is a ∗-morphism of $\mathfrak{G}$ into $\mathcal{L}(\mathfrak{H})$. The representation $(\mathfrak{H},\pi)$ is said to be faithful if, and only if, $\pi$ is a ∗-isomorphism between $\mathfrak{G}$ and $\pi(\mathfrak{G})$, i.e., if, and only if, $\mbox{ker}\,\pi=\{0\}.$

Each representation $(\mathfrak{H},\pi)$ of a $C^{*}$-algebra $\mathfrak{G}$ defines a faithful representation of the quotient algebra $\mathfrak{G}_{\pi}=\mathfrak{G}/\mbox{ker}\,\pi.$

Then, follows the proposition on the criteria for faithfulness:

Proposition 1.7

([8], p.44)

Let $(\mathfrak{H},\pi)$ be a representation of the $C^{*}$-algebra $\mathfrak{G}.$ The representation is faithful if, and only if, it satisfies each of the following equivalent conditions:

$\mbox{ker}\,\pi=\{0\}$; 2. 2.

$||\pi(A)||=||A||$* for all $A\in\mathfrak{G}$;* 3. 3.

$\pi(A)>0$* for all $A>0$.*

The Proof of this proposition is achieved by the following proposition:

Proposition 1.8

([8], pp.42-43)

Let $\mathfrak{G}$ be a Banach ∗-algebra with identity, $\mathfrak{B}$ a $C^{*}$-algebra, and $\pi$ a ∗-morphism of $\mathfrak{G}$ into $\mathfrak{B}.$ Then $\pi$ is continuous and

[TABLE]

for all $A\in\mathfrak{G}.$ Moreover, if $\mathfrak{G}$ is a $C^{*}$-algebra then the range $\mathfrak{B}_{\pi}=\{\pi(A);A\in\mathfrak{G}\}$ of $\pi$ is a $C^{*}$-subalgebra of $\mathfrak{B}$.

Proof. (see [8] p.43)

First assume $A=A^{*}.$ Then since $\mathfrak{B}$ is a $C^{*}$-algebra and $\pi(A)\in\mathfrak{B}$, one has

[TABLE]

by Theorem 2.2.5(a) (see [8] p.29). Next, define $P=\pi(1\!\!1_{\mathfrak{G}})$ where $1\!\!1_{\mathfrak{G}}$ denotes the identity of $\mathfrak{G}$. It follows from the definition of $\pi$ that $P$ is a projection in $\mathfrak{B}$.

Hence replacing $\mathfrak{B}$ by the $C^{*}$-algebra $P\mathfrak{B}P$, the projection $P$ becomes the identity $1\!\!1_{\mathfrak{B}}$ of the new algebra $\mathfrak{B}$. Moreover, $\pi(\mathfrak{G})\subseteq\mathfrak{B}$. Now it follows from the definitions of a morphism and of the spectrum that $\sigma_{\mathfrak{B}}(\pi(A))\subseteq\sigma_{\mathfrak{G}}(A).$ Therefore,

[TABLE]

by the following Proposition:

Proposition 1.9

([8] p.26)

Let $A$ be an element of a Banach algebra with identity and define the spectral radius $\rho(A)$ of $A$ by

[TABLE]

It follows that

[TABLE]

In particular, the limit exists. Thus the spectrum of $A$ is a nonempty compact set.

Proof. (see [8] p.26)

Let $|\lambda|^{n}>||A^{n}||$ for some $n>0.$ As each $m\in\mathbb{Z}$ can be decomposed as $m=pn+q$ with $p,q\in\mathbb{Z}$ and $0\leqslant q<n$ one again establishes that the series

[TABLE]

is Cauchy in the uniform topology and defines $(\lambda 1\!\!1-A)^{-1}.$ Therefore,

[TABLE]

for all $n>0,$ and consequently

[TABLE]

Thus to complete the proof it sufficies to establish that $\rho(A)\geqslant r_{A},$ where

[TABLE]

There are two cases.

Firstly, assume $0\in r_{\mathfrak{G}}(A),$ i.e., $A$ is invertible. Then $1=||A^{n}A^{-n}||\leqslant||A^{n}||\,||A^{-n}||$ and hence $1\leqslant r_{A}r_{A^{-1}}.$ This implies $r_{A}>0.$ Consequently, if $r_{A}=0$ one must have $0\in r_{\mathfrak{G}}(A)$ and $\rho(A)\geqslant r_{A}.$

Secondly, we may assume $r_{A}>0.$ We will need the following observation. If $A_{n}$ is any sequence of elements such that $R_{n}=(1\!\!1-A_{n})^{-1}$ exists then $1\!\!1-R_{n}=-A_{n}(1\!\!1-A_{n})^{-1}$ and $A_{n}=-(1\!\!1-R_{n})(1\!\!1-(1\!\!1-R_{n}))^{-1}.$ Therefore, $||1\!\!1-R_{n}||\rightarrow 0$ is equivalent to $||A_{n}||\rightarrow 0$ by power series expansion.

Define $S_{A}=\{\lambda;\lambda\in\mathbb{C},|\lambda|\geqslant r_{A}\}.$ We assume that $S_{A}\subseteq r_{\mathfrak{G}}(A)$ and obtain a contradiction. Let $\omega$ be a primitive nth root of unity. By assumption

[TABLE]

is well defined for all $\lambda\in S_{A}.$ But an elementary calculation shows that

[TABLE]

Next one has the continuity estimate

[TABLE]

which is uniform in $k.$ The supremum is finite since $\lambda\mapsto||(\lambda 1\!\!1-A)^{-1}||$ is continuous on $r_{\mathfrak{G}}(A)$ and for $|\lambda|>||A||$ one has

[TABLE]

It follows then that for each $\varepsilon>0$ there is a $\lambda>r_{A}$ such that

[TABLE]

uniformly in $n.$ But $||A^{n}||/\lambda^{n}\rightarrow 0$ and by the above observation $||(1\!\!1-A^{n}/\lambda^{n})^{-1}-1\!\!1||\rightarrow 0.$ This implies that $||(1\!\!1-A^{n}/r^{n}_{A})^{-1}-1\!\!1||\rightarrow 0$ and $||A^{n}||/r^{n}_{A}\rightarrow 0$ by another application of the same observation. This last statement contradicts, however, the definition of $r_{A}$ and hence the proof is complete. $\hfill{\square}$

Finally, if $A$ is not selfadjoint one can combine this inequality with the $C^{*}$-norm property and the product inequality to deduce that

[TABLE]

Thus $||\pi(A)||\leqslant||A||$ for all $A\in\mathfrak{G}$ and $\pi$ is continuous.

The range $\mathfrak{B}_{\pi}$ is a ∗-subalgebra of $\mathfrak{B}$ by definition and to deduce that it is a $C^{*}$-subalgebra we must prove that it is closed, under the assumption that $\mathfrak{G}$ is a $C^{*}$-algebra.

Now introduce the kernel ker $\pi$ of $\pi$ by

[TABLE]

then ker $\pi$ is closed two-sided ∗-ideal. Given $A\in\mathfrak{G}$ and $B\in\mbox{ker}\,\pi$ then $\pi(AB)=\pi(A)\pi(B)=0,\pi(BA)=\pi(B)\pi(A)=0,$ and $\pi(B^{*})=\pi(B)=0.$ The closedness follows from the estimate $||\pi(A)||\leqslant||A||.$ Thus we can form the quotient algebra $\mathfrak{G}_{\pi}=\mathfrak{G}/\mbox{ker}\,\pi$ and $\mathfrak{G}_{\pi}$ is a $C^{*}$-algebra. The elements of $\mathfrak{G}_{\pi}$ are the classes $\hat{A}=\{A+I;I\in\mbox{ker}\,\pi\}$ and the morphism $\pi$ induces a morphism $\hat{\pi}$ from $\mathfrak{G}_{\pi}$ onto $\mathfrak{B}_{\pi}$ by the defintion $\hat{\pi}(\hat{A})=\pi(A).$ The kernel of $\hat{\pi}$ is zero by construction and hence $\hat{\pi}$ is an isomorphism between $\mathfrak{G}_{\pi}$ and $\mathfrak{B}_{\pi}$. Therefore, one can define a morphism $\hat{\pi}^{-1}$ from the ∗-algebra $\mathfrak{B}_{\pi}$ onto the $C^{*}$-algebra $\mathfrak{G}_{\pi}$ by $\hat{\pi}^{-1}(\hat{\pi}(\hat{A}))=\hat{A}$ and then applying the first statement of the proposition to $\hat{\pi}^{-1}$ and $\hat{\pi}$ successively one obtains

[TABLE]

Thus $||\hat{A}||=||\hat{\pi}(\hat{A})||=||{\pi}(A)||.$ Consequently, if $\pi(A_{n})$ converges uniformly in $\mathfrak{B}$ to an element $A_{\pi}$ then $\hat{A_{n}}$ converges in $\mathfrak{G}_{\pi}$ to an element $\hat{A}$ and $A_{\pi}=\hat{\pi}(\hat{A})=\pi(A)$ where $A$ is any element of the equivalence class $\hat{A}.$ Thus $A_{\pi}\in\mathfrak{B}_{\pi}$ and $\mathfrak{B}_{\pi}$ is closed. $\hfill{\square}$

Proof of Proposition 1.7 (see [8] p.44).

The equivalence of condition (1) and faithfulness is by definition. Prove that (1) $\Rightarrow$ (2) $\Rightarrow$ (3) $\Rightarrow$ (1).

(1) $\Rightarrow$ (2) Since ker$\,\pi=\{0\}$, we can define a morphism $\pi^{-1}$ from the range of $\pi$ into $\mathfrak{G}$ by $\pi^{-1}(\pi(A))=A$ and then applying Proposition 1.8 to $\pi^{-1}$ and $\pi$ sucessively one has

[TABLE]

(2) $\Rightarrow$ (3) If $A>0$ then $||A||>0$ and hence $||\pi(A)||>0,$ or $\pi(A)\neq 0.$ But $\pi(A)\geqslant 0$ by Proposition 1.8 and therefore $\pi(A)>0.$ (3) $\Rightarrow$ (1) If condition (1) is false then there is a $B\in$ ker $\pi$ with $B\neq 0$ and $\pi(B^{*}B)=0.$ But $||B^{*}B||\geqslant 0$ and as $||B^{*}B||=||B||^{2}$ one has $B^{*}B>0.$ Thus condition (3) is false. $\hfill{\square}$

Definition 1.10

**Cyclic representation of a $C^{*}$-algebra A cyclic representation of a $C^{*}$-algebra $\mathfrak{G}$ is defined to be a triplet $(\mathfrak{H},\pi,\Omega)$, where $(\mathfrak{H},\pi)$ is a representation of $\mathfrak{G}$ and $\Omega$ is a vector in $\mathfrak{H}$ which is cyclic for $\pi$, in $\mathfrak{H}$. $\Omega$ is called cyclic vector or cyclic vector for $\pi$. If $\mathfrak{K}$ is a closed subspace of $\mathfrak{H}$ then $\mathfrak{K}$ is called a cyclic subspace for $\mathfrak{H}$ whenever the set

[TABLE]

is dense in $\mathfrak{H}.$

Definition 1.11

** State over a $C^{*}$-algebra A linear functional $\omega$ over the $C^{*}$-algebra $\mathfrak{G}$ is defined to be positive if

[TABLE]

for all $A\in\mathfrak{G}.$ A positive linear functional $\omega$ over a $C^{*}$-algebra $\mathfrak{G}$ with $||\omega||=1$ is called a state.

Remark 1.12

**

Every positive element of a $C^{*}$-algebra is of the form $A^{*}A$ and hence positivity of $\omega$ is equivalent to $\omega$ being positive on positive elements. 2. 2.

Considering a representation $(\mathfrak{H},\pi)$ of the $C^{*}$-algebra $\mathfrak{G}$, taking $\Omega\in\mathfrak{H}$ being a nonzero vector and define $\omega_{\Omega}$ by

[TABLE]

for all $A\in\mathfrak{G}$. It follows that $\omega_{\Omega}$ is a linear function over $\mathfrak{G},$ it is also positive since

[TABLE]

$||\omega_{\Omega}||=1$ whenever $||\Omega||=1$ and then, $\pi$ is nondegenerate. In this case $\omega_{\Omega}$ is a state, and is usually called vector state for the representation $(\mathfrak{H},\pi)$.

Definition 1.13

**The cyclic representation $(\mathfrak{H}_{\omega},\pi_{\omega},\Omega_{\omega})$, constructed from the state $\omega$ over the $C^{*}$-algebra $\mathfrak{G}$ , is defined as the canonical cyclic representation of $\mathfrak{G}$ associated with $\omega$.

Next it will be demonstrated that the notions of purity of a state $\omega$ and irreducibility of the representation associated with $\omega$ are intimately related.

Theorem 1.14

([8] p.57) Let $\omega$ be a state over the $C^{*}$-algebra $\mathfrak{G}$ and $(\mathfrak{H}_{\omega},\pi_{\omega},\Omega_{\omega})$ the associated cyclic representation. The following conditions are equivalent:

$(\mathfrak{H}_{\omega},\pi_{\omega})$* is irreducible;* 2. 2.

$\omega$* is pure;* 3. 3.

$\omega$* is an extremal point of the set $E_{\mathfrak{G}}$ of states over $\mathfrak{G}$. Furthermore, there is one-to-one correspondence*

[TABLE]

between positive functionnals $\omega_{T},$ over $\mathfrak{G}$, majorized by $\omega$ and positive operators $T$ in the commutant $\pi^{\prime}_{\omega}$, of $\pi_{\omega},$ with $||T||\leqslant 1.$

Proof.(see [8] pp.57-58) $(1)\Rightarrow(2)$ Assume that $(2)$ is false. Thus there exists a positive functional $\rho$ such that $\rho(A^{*}A)\leqslant\omega(A^{*}A)$ for all $A\in\mathfrak{G}$. But applying the Cauchy-Schwarz inequality one then has

[TABLE]

Thus $\pi_{\omega}(B)\Omega_{\omega}\times\pi_{\omega}(A)\Omega_{\omega}\longmapsto\rho(B^{*}A)$ is a densely defined, bounded, sesquilinear functional, over $\mathfrak{H}_{\omega}\times\mathfrak{H}_{\omega}$, and there exists a unique bounded operator $T$, on $\mathfrak{H}_{\omega}$, such that

[TABLE]

As $\rho$ is not a multiple of $\omega$ operator $T$ is not a multiple of the identity. Moreover,

[TABLE]

and hence $0\leqslant T\leqslant 1\!\!1.$ But

[TABLE]

and therefore $T\in\pi^{\prime}_{\omega}.$ Thus condition $(1)$ is false. $(2)\Rightarrow(1)$ Assume that $(1)$ is false. If $T\in\pi^{\prime}_{\omega}$ then $T^{*}\in\pi^{\prime}_{\omega}$ and $T+T^{*},\,\,(T-T^{*})/i$ are also elements of the commutant. Thus there exists a selfadjoint element $S$ of $\pi^{\prime}_{\omega}$ which is not a multiple of the identity. Therefore there exists a spectral projector $P$ of $S$ such that $0<P<1\!\!1$ and $P\in\pi^{\prime}_{\omega}$. Consider the functional

[TABLE]

This is certainly positive since

[TABLE]

Moreover,

[TABLE]

Thus $\omega$ majorizes $\rho$. It is verified that $\rho$ is not a multiple of $\omega$ and hence $(2)$ is false. This proves the equivalence of the first two conditions stated in the theorem and simultaneously establishes the correspondence described by the last statement.

The equivalence of conditions (2) and (3) is performed as follows. Suppose that $\omega$ is an extremal point of $E_{\mathfrak{G}}$ and $\omega\neq 0.$ Then, we must have $||\omega||=1.$ Thus $\omega$ is a state and we must deduce that it is pure. Suppose the contrary; then there is a state $\omega_{1}\neq\omega$ and a $\lambda$ with $0<\lambda<1$ such that $\omega\geqslant\lambda\omega_{1}.$ Define $\omega_{2}$ by $\omega_{2}=(\omega-\lambda\omega_{1})/(1-\lambda);$ then $||\omega_{2}||=(||\omega||-\lambda||\omega_{1}||)/(1-\lambda)=1$ and $\omega_{2}$ is also a state. But $\omega=\lambda\omega_{1}+(1-\lambda)\omega_{2}$ and $\omega$ is not extremal, which is a contradiction. $\hfill{\square}$

In the following, some notions on von Neumann algebra are provided. To specify the Hilbert space upon which a von Neumann algebra $\mathfrak{A}$ acts, one often uses the notation $\{\mathfrak{A},\mathfrak{H}\}$ to denote the von Neumann algebra $\mathfrak{A}.$

Definition 1.15

**von Neumann algebra

Let $\mathfrak{H}$ be a Hilbert space. For each subset $\mathfrak{A}$ of $\mathcal{L}(\mathfrak{H})$, let $\mathfrak{A^{\prime}}$ denote the set of all bounded operators on $\mathfrak{H}$ commuting with every operator in $\mathfrak{A}$. Clearly, $\mathfrak{A^{\prime}}$ is a Banach algebra of operators containing the identity operator $I_{\mathfrak{H}}$ on $\mathfrak{H}.$

• (i)

A von Neumann algebra is a ∗-subalgebra $\mathfrak{A}$ of $\mathcal{L}(\mathfrak{H})$ such that $\mathfrak{A}=\mathfrak{A}^{\prime\prime}$.

• (ii)

$\mathfrak{A}^{\prime}$ denotes the commutant of $\mathfrak{A}$, the set of all elements in $\mathcal{L}(\mathfrak{H})$ which commute with every element of $\mathfrak{A}$.

• (iii)

A von Neumann algebra always contains the identity operator $I_{\mathfrak{H}}$ on $\mathfrak{H}$. It is called a factor if $\mathfrak{A}\cap\mathfrak{A}^{\prime}=\mathbb{C}I_{\mathfrak{H}}$.

• (iv)

If a subset $\mathcal{S}$ of $\mathcal{L}(\mathfrak{H})$ is invariant under the ∗-operation, then $\mathcal{S^{\prime\prime}}$, the double commutant of $\mathcal{S}$, is the smallest von Neumann algebra containing $\mathcal{S}$, and it is called the von Neumann algebra generated by $\mathcal{S}$.

We also have the following definition:

Definition 1.16

**A von Neumann algebra $\mathfrak{A}\subset\mathcal{L}(\mathfrak{H})$ is a $C^{*}$-algebra acting on the Hilbert space $\mathfrak{H}$ that is closed under the weak-operator topology: $A_{n}\stackrel{{\scriptstyle n\rightarrow+\infty}}{{\longrightarrow}}A$ iff $\langle\xi|A_{n}\eta\rangle\stackrel{{\scriptstyle n\rightarrow+\infty}}{{\longrightarrow}}\langle\xi|A\eta\rangle,\forall\xi,\eta\in\mathfrak{H},$ or equivalently under the $\sigma$-weak topology: $A_{n}\stackrel{{\scriptstyle n\rightarrow+\infty}}{{\longrightarrow}}A$ iff for all sequences $(\xi_{k}),(\zeta_{k})$ in $\mathfrak{H}$ such that $\displaystyle\sum_{k=1}^{+\infty}||\xi_{k}||^{2}<+\infty$ and $\displaystyle\sum_{k=1}^{+\infty}||\zeta_{k}||^{2}<+\infty$ we have $\displaystyle\sum_{k=1}^{+\infty}\langle\xi_{k}|A_{n}\zeta_{k}\rangle\stackrel{{\scriptstyle n\rightarrow+\infty}}{{\longrightarrow}}\displaystyle\sum_{k=1}^{+\infty}\langle\xi_{k}|A\zeta_{k}\rangle.$

Case of $\mathcal{L}(\mathfrak{H})$

• (i)

$\mathcal{L}(\mathfrak{H})$ is a von Neumann algebra and even a factor since $\mathcal{L}(\mathfrak{H})^{\prime}=\mathbb{C}1\!\!1\quad(1\!\!1=I_{\mathfrak{H}})$.

• (ii)

The Hilbert space adjoint operation defines an involution on $\mathcal{L}(\mathfrak{H})$ and with respect to these operations and this norm, $\mathcal{L}(\mathfrak{H})$ is a $C^{*}$-algebra. In particular, the $C^{*}$-norm property follows from $||A||^{2}\leqslant||A^{*}||||A||=||A||^{2}$.

• (iii)

Any uniformly closed subalgebra $\mathfrak{M}$ of $\mathcal{L}(\mathfrak{H})$ which is self-adjoint is also a $C^{*}-$algebra.

Next, it comes the following definition:

Definition 1.17

**Closure-Orthogonal projection

If $\mathfrak{M}$ is a subset of $\mathcal{L}(\mathfrak{H})$ and $\mathfrak{K}$ is a subset of $\mathfrak{H},$ $[\mathfrak{M}\mathfrak{K}]$ denotes the closure of the linear span of elements of the form $A\xi,$ where $A\in\mathfrak{M}$ and $\xi\in\mathfrak{K}.$ $[\mathfrak{M}\mathfrak{K}]$ also denotes the orthogonal projection onto $[\mathfrak{M}\mathfrak{K}].$

• (iv)

A ∗-subalgebra $\mathfrak{M}\subseteq\mathcal{L}(\mathfrak{H})$ is said to be nondegenerate if $[\mathfrak{M}\mathfrak{H}]=\mathfrak{H}$.

• (v)

If $\mathfrak{M}\subseteq\mathcal{L}(\mathfrak{H})$ contains the identity operator, then, it is automatically nondegenerate.

A nondegenerate ∗-algebra contains the identity operator; If a subalgebra of $\mathcal{L}(\mathfrak{H})$ is invariant under the *-operation, then it is called a ∗-subalgebra of $\mathcal{L}(\mathfrak{H})$ or a ∗-algebra of operators on $\mathfrak{H}$.

We have the following proposition (see [25] pp.72-73):

Proposition 1.18

([25], p.72)

The subset $\mathfrak{M}$ of $\mathcal{L}(\mathfrak{H})$ is a von Neumann algebra on $\mathfrak{H}$.

Proof. (see [25], p.72)

Let $\{\mathfrak{M}_{i},\mathfrak{H}_{i}\}_{i\in I}$ be a family of von Neumann algebras. Let $\mathfrak{H}$ denote the direct sum $\displaystyle\sum^{\oplus}_{i\in I}\mathfrak{H}_{i}$ of Hilbert spaces $\{\mathfrak{H}_{i}\}_{i\in I}.$ Each vector $\xi=\{\xi_{i}\}_{i\in I}$ in $\mathfrak{H}$ is denoted by $\displaystyle\sum^{\oplus}_{i\in I}\xi_{i}.$ For each bounded sequence $\{x_{i}\}_{i\in I}$ in $\displaystyle\prod_{i\in I}\mathfrak{M}_{i},$ one defines an operator $x$ on $\mathfrak{H}$ by

[TABLE]

Then, $x$ is a bounded operator on $\mathfrak{H}$ denoted by $\displaystyle\sum^{\oplus}_{i\in I}x_{i}.$ Let $\mathfrak{M}$ be the set of all such $x.$

Particularly, taking $\mathfrak{M}$ as a subset of $\mathcal{L}(\mathfrak{H})$, the proof is completed. $\hfill{\square}$

Definition 1.19

** Cyclic and separating vector

The modular theory of von Neumann algebras is such that to every von Neumann algebra $\mathfrak{M}\subset\mathcal{L}(\mathfrak{H}),$ and to every vector $\xi\in\mathfrak{H}$ that is cyclic

[TABLE]

i.e. the set $\{A\Phi;A\in\mathfrak{M}\}$ ($\mathfrak{M}$ denoting a set of bounded operators on $\mathfrak{H}$) is dense in $\mathfrak{H};$ and separating i.e. for $\mathcal{A}\in\mathfrak{M},$

[TABLE]

Moreover, a vector $\psi\in\mathfrak{H}$ is said separating for a von Neumann algebra $\mathfrak{A}$ if $A\psi=B\psi,\,A,B\in\mathfrak{A},$ if and only if $A=B.$

We have the following definitions:

Definition 1.20

**Separating subset

Let $\mathfrak{A}$ be a von Neumann algebra on a Hilbert space $\mathfrak{H}.$ A subset $\mathfrak{K}\subseteq\mathfrak{H}$ is separating for $\mathfrak{A}$ if for any $A\in\mathfrak{A},\,A\xi=0$ for all $\xi\in\mathfrak{K}$ implies $A=0.$

Definition 1.21

**Cyclic and separating subset of a von Neumann algebra

Let $\{\mathfrak{A},\mathfrak{H}\}$ be a von Neumann algebra. A subset $\mathfrak{M}$ of $\mathfrak{H}$ is called separating (resp.cyclic) for $\mathfrak{A}$ if $a\xi=0,\,a\in\mathfrak{A},$ for every $\xi\in\mathfrak{M}$ implies $a=0$ (resp. the smallest invariant subspace $[\mathfrak{A}\mathfrak{M}]$ under $\mathfrak{A}$ containing $\mathfrak{M}$ in the whole space $\mathfrak{H}$).

Recall that a subset $\mathfrak{K}\subseteq\mathfrak{H}$ is cyclic for $\mathfrak{M}$ if $[\mathfrak{M}\mathfrak{K}]=\mathfrak{H}.$ There is a dual relation between the properties of cyclic for the algebra and separating for the commutant.

We have the following propositions:

Proposition 1.22

([8] p.85)

Let $\mathfrak{A}$ be a von Neumann algebra on $\mathfrak{H}$ and $\mathfrak{K}\subseteq\mathfrak{H}$ a subset. The following conditions are equivalent:

• (1)

$\mathfrak{K}$* is cyclic for $\mathfrak{A}$;*

• (2)

$\mathfrak{K}$* is separating for $\mathfrak{A^{\prime}}.$*

Proof. (see [8] p.85)

(1) $\Rightarrow$ (2) Assume that $\mathfrak{K}$ is cyclic for $\mathfrak{A}$ and choose $A^{\prime}\in\mathfrak{A^{\prime}}$ such that $A^{\prime}\mathfrak{K}=\{0\}.$ Then, for any $B\in\mathfrak{A}$ and $\xi\in\mathfrak{K},\,A^{\prime}B\xi=BA^{\prime}\xi=0,$ hence $A^{\prime}[\mathfrak{A}\mathfrak{K}]=0$ and $A^{\prime}=0.$

(2) $\Rightarrow$ (1) Suppose that $\mathfrak{K}$ is separating for $\mathfrak{A^{\prime}}$ and set $P^{\prime}=[\mathfrak{A}\mathfrak{K}].$ $P^{\prime}$ is then a projection in $\mathfrak{A^{\prime}}$ and $(1\!\!1-P^{\prime})\mathfrak{K}=\{0\}.$ Hence $1\!\!1-P^{\prime}=0$ and $[\mathfrak{A}\mathfrak{K}]=\mathfrak{H}.$ $\hfill{\square}$

Definition 1.23

**The weak and $\sigma$-weak topologies If $\xi,\eta\in\mathfrak{H},$ then $A\mapsto|(\xi,A\eta)|$ is a seminorm on $\mathcal{L}(\mathfrak{H})$. The locally convex topology on $\mathcal{L}(\mathfrak{H})$ defined by these seminorms is called the weak topology. The seminorms defined by the vector states $A\mapsto|(\xi,A\xi)|$ suffice to define this topology because $\mathfrak{H}$ is complex and one has the polarization identity

[TABLE]

Let $\{\xi_{n}\},\{\eta_{n}\}$ be two sequences from $\mathfrak{H}$ such that

[TABLE]

Then for $A\in\mathcal{L}(\mathfrak{H})$

[TABLE]

Hence $A\mapsto|\displaystyle\sum_{n}(\xi_{n},A\eta_{n})|$ is a seminorm on $\mathcal{L}(\mathfrak{H}).$ The locally convex topology on $\mathcal{L}(\mathfrak{H})$ induced by these seminorms is called the $\sigma$-weak topology.

**Notations: ** In the sequel,

• •

$\mathfrak{A}_{+}$ denotes the positive part of the von Neumann algebra $\mathfrak{A}$ or the set of positive elements of the von Neumann algebra $\mathfrak{A}$;

• •

$\mathfrak{A}_{*}$ denotes the predual of a von Neumann algebra. It is the space of all $\sigma$-weakly continuous linear functionals on $\mathfrak{A}$;

• •

$\mathfrak{L}(\mathfrak{H})_{1}$ denotes the unit ball of $\mathfrak{L}(\mathfrak{H})$. $\mathfrak{L}(\mathfrak{H})_{1}$ is norm dense in the unit ball of the norm closure of $\mathfrak{L}(\mathfrak{H})$, and it is taken as a $C^{*}$-algebra (see [8] p.74).

Definition 1.24

**

Let $\varphi:\mathfrak{A}\rightarrow\mathbb{C}$ be a bounded linear functional on $\mathfrak{A},$ which is denoted by $\langle\varphi;A\rangle,\,A\in\mathfrak{A}.$

$\varphi$ is called a state on this algebra if it also satisfies the two conditions:

• (a)

$\langle\varphi;A^{*}A\rangle\geqslant 0,\quad\forall A\in\mathfrak{A}$

• (b)

$\langle\varphi;I_{\mathfrak{H}}\rangle=1.$

The state $\varphi$ is called a vector state if there exists a vector $\phi\in\mathfrak{H}$ such that

[TABLE]

Such a state is also normal.

Definition 1.25

**

A state $\omega$ on a von Neumann algebra $\mathfrak{A}$ is faithful if $\omega(A)>0$ for all nonzero $A\in\mathfrak{A}_{+}.$

Remark 1.26

** (see [8], Example 2.5.5 p.85) Let $\mathfrak{A}=\mathcal{L}(\mathfrak{H})$ with $\mathfrak{H}$ separable. Every normal state $\omega$ over $\mathfrak{A}$ is of the form

[TABLE]

where $\rho$ is a density matrix. If $\omega$ is faithful then $\omega(E)>0$ for each rank one projector, i.e., $||\rho^{1/2}\psi||>0$ for each $\psi\in\mathfrak{H}\setminus\left\{0\right\}.$ Thus $\rho$ is invertible (in the densely defined self-adjoint operators on $\mathfrak{H}$). Conversely, if $\omega$ is not faithful then $\omega(A^{*}A)=0$ for some nonzero $A$ and hence $||\rho^{1/2}A^{*}\psi||=0$ for all $\psi\in\mathfrak{H},$ i.e., $\rho$ is not invertible. This establishes that $\omega$ is faithful if, and only if, $\rho$ is invertible.

Lemma 1.27

([8] p.76) Let $\{A_{\alpha}\}$ be an increasing set in $\mathcal{L}(\mathfrak{H})_{+}$ with an upper bound in $\mathcal{L}(\mathfrak{H})_{+}$. Then $\{A_{\alpha}\}$ has a least upper bound (l.u.b.) A, and the net converges $\sigma$-strongly to $A.$

Proof. (see [8] p.76) Let $\mathfrak{K}_{\alpha}$ be the weak closure of the set of $A_{\beta}$ with $\beta>\alpha.$ Since $\mathcal{L}(\mathfrak{H})_{1}$ is weakly compact, there exists an element A in $\bigcap_{\alpha}\mathfrak{K}_{\alpha}.$ For all $A_{\alpha}$ the set of $B\in\mathcal{L}(\mathfrak{H})_{+}$ such that $B\geqslant A_{\alpha}$ is $\sigma$-weakly closed and contains $\mathfrak{K}_{\alpha},$ hence $A\geqslant A_{\alpha}.$ Thus, $A$ majorizes $\{A_{\alpha}\}$ and lies in the weak closure of $\{A_{\alpha}\}$. If $B$ is another operator majorizing $\{A_{\alpha}\}$, then it majorizes its weak closure; thus $B\geqslant A$ and $A$ is the least upper bound of $\{A_{\alpha}\}.$ Finally, if $\xi\in\mathfrak{H}$ then

[TABLE]

Since the strong and $\sigma$-strong topology coincide on $\mathcal{L}(\mathfrak{H})_{1},$ this ends the proof. $\hfill{\square}$

Proposition 1.28

([8] p.68) Let Tr be the usual trace on $\mathcal{L}(\mathfrak{H})$, and let $\mathcal{T}(\mathfrak{H})$ be the Banach space of trace-class operators on $\mathfrak{H}$ equipped with the trace norm $T\mapsto\mbox{Tr}(|T|)=||T||_{\tiny{\mbox{Tr}}}$. Then it follows that $\mathcal{L}(\mathfrak{H})$ is the dual $\mathcal{T}(\mathfrak{H})^{*}$ of $\mathcal{T}(\mathfrak{H})$ by the duality

[TABLE]

The weak∗ topology on $\mathcal{L}(\mathfrak{H})$ arising from this duality is just the $\sigma$-weak topology.

Proof.(see [8] pp.68-69) Due to the inequality $|\mbox{Tr}(AT)|\leqslant||A||\,||T||_{\tiny{\mbox{Tr}}},\,\mathcal{L}(\mathfrak{H})$ is the subspace of $\mathcal{T}(\mathfrak{H})^{*}$ by the duality described in the proposition. Conversely, assume $\omega\in\mathcal{T}(\mathfrak{H})^{*}$ and consider a rank one operator $E_{\varphi,\psi}$ defined for $\varphi,\psi\in\mathfrak{H}$ by

[TABLE]

One has $E^{*}_{\varphi,\psi}=E_{\psi,\varphi}$ and $E_{\varphi,\psi}E_{\psi,\varphi}=||\psi||^{2}E_{\varphi,\varphi}.$ Hence

[TABLE]

It follows that

[TABLE]

Hence there exists, by the Riesz representation theorem, an $A\in\mathcal{L}(\mathfrak{H})$ with $||A||\leqslant||\omega||$ such that

[TABLE]

Consider $\omega_{0}\in\mathcal{T}(\mathfrak{H})^{*}$ defined by

[TABLE]

then

[TABLE]

Now for any $T\in\mathcal{T}(\mathfrak{H})$ there exist bounded sequences $\{\psi_{n}\}$ and $\{\varphi_{n}\}$ and a sequence $\{\alpha_{n}\}$ of complex numbers such that

[TABLE]

and

[TABLE]

The latter series converges with respect to the trace norm and hence

[TABLE]

Thus $\mathcal{L}(\mathfrak{H})$ is just the dual of $\mathcal{T}(\mathfrak{H}).$ The weak*∗* topology on $\mathcal{L}(\mathfrak{H})$ arising from this duality is given by the seminorms

[TABLE]

Now, for $T$ as in (1.51), one has

[TABLE]

Thus the seminorms are equivalent to the seminorms defining the $\sigma$-weak topology. $\hfill{\square}$

It follows the theorem below:

Theorem 1.29

([8], p.76) Let $\omega$ be a state on a von Neumann algebra $\mathfrak{A}$ acting on a Hilbert space $\mathfrak{H}.$ The following conditions are equivalent:

• (1)

$\omega$* is normal;*

• (2)

$\omega$* is $\sigma$-weakly continuous;*

• (3)

there exists a density matrix $\rho,$ i.e., a positive trace-class operator $\rho$ on $\mathfrak{H}$ with $Tr(\rho)=1,$ such that

[TABLE]

Proof. (See [8] pp.76 to 78). (3) $\Rightarrow$ (2) follows from Proposition 1.28 and (2) $\Rightarrow$ (1) from Lemma 1.27. Next show (2) $\Rightarrow$ (3). If $\omega$ is $\sigma$-weakly continuous there exist sequences $\{\xi_{n}\},\,\{\eta_{n}\}$ of vectors such that $\displaystyle\sum_{n}||\xi_{n}||^{2}<\infty,\,\displaystyle\sum_{n}||\eta_{n}||^{2}<\infty,$ and $\omega(A)=\displaystyle\sum_{n}(\xi_{n},A\eta_{n}).$ Define $\tilde{\mathfrak{H}}=\displaystyle\bigoplus_{n=1}^{\infty}\mathfrak{H}_{n}$ and introduce a representation $\pi$ of $\mathfrak{A}$ on $\tilde{\mathfrak{H}}$ by $\pi(A)(\displaystyle\bigoplus_{n}\psi_{n})=\displaystyle\bigoplus_{n}(A\psi_{n}).$ Let $\xi=\displaystyle\bigoplus_{n}\xi_{n},\,\eta=\displaystyle\bigoplus_{n}\eta_{n}$ and then $\omega(A)=(\xi,\pi(A)\eta).$ Since $\omega(A)$ is real for $A\in\mathfrak{A}_{+}$ (with $\mathfrak{A}_{+}$ denoting the positive part of the von Neumann algebra $\mathfrak{A}$ or the set of positive elements of the von Neumann algebra $\mathfrak{A}$), we have

[TABLE]

Hence, by Theorem 1.14 there exists a positive $T\in\pi(\mathfrak{A})^{\prime}$ with $0\leqslant T\leqslant 1\!\!1/2$ such that

[TABLE]

Now $\psi\in\tilde{\mathfrak{H}}$ has the form $\psi=\displaystyle\bigoplus_{n}\psi_{n},$ and therefore

[TABLE]

The right side of this relation can be used to extend $\omega$ to a $\sigma$-weakly continuous positive linear functional $\tilde{\omega}$ on $\mathcal{L}(\mathfrak{H}).$ Since $\tilde{\omega}(1\!\!1)=1,$ it is a state. Thus, by Proposition 1.28 there exists a trace-class operator $\rho$ with $Tr(\rho)=1$ such that

[TABLE]

Let $P$ be the rank one projector with range $\xi$; then

[TABLE]

Thus $\rho$ is positive. Turn now to the proof of (1) $\Rightarrow$ 2. Assume that $\omega$ is a normal state on $\mathfrak{A}.$ Let $\{B_{\alpha}\}$ be an increasing net of elements in $\mathfrak{A}_{+}$ such that $||B_{\alpha}||\leqslant 1$ for all $\alpha$ and such that $A\mapsto\omega(AB_{\alpha})$ is $\sigma$-strongly continuous for all $\alpha.$ One can use Lemma 1.27 to define $B$ by

[TABLE]

Then $0\leqslant B\leqslant 1\!\!1$ and $B\in\mathfrak{A}.$ But for all $A\in\mathfrak{A}$ we have

[TABLE]

Hence

[TABLE]

But $\omega$ is normal. Therefore $\omega(B-B_{\alpha})\rightarrow 0$ and $\omega(\cdot B_{\alpha})$ tends to $\omega(\cdot B)$ in norm. As $\mathfrak{A}_{*}$ is a Banach space, $\omega(\cdot B)\in\mathfrak{A}_{*}.$ Now, applying Zorn’s lemma, we can find a maximal element $P\in\mathfrak{A}_{+}\cap\mathfrak{A}_{1}$ such that $A\mapsto\omega(AP)$ is $\sigma$-strongly continuous. If $P=1\!\!1$ the theorem is proved. Assume ad absurdum that $P\neq 1\!\!1.$ Put $P^{\prime}=1\!\!1-P$ and choose $\xi\in\mathfrak{H}$ such that $\omega(P^{\prime})<(\xi,P^{\prime}\xi).$ If $\{B_{\alpha}\}$ is an increasing net in $\mathfrak{A}_{+}$ such that $B_{\alpha}\leqslant P^{\prime},$ $\omega(B_{\alpha})\geqslant(\xi,B_{\alpha}\xi),$ and $B={\mbox{l.u.b.}}_{\alpha}\,B_{\alpha}=\sigma$-strong $\lim_{\alpha}B_{\alpha},$ then $B\in\mathfrak{A}_{+},B\leqslant P^{\prime},$ and $\omega(B)=\mbox{sup}\,\omega(B_{\alpha})\geqslant\mbox{sup}\,(\xi,B_{\alpha}\xi)=(\xi,B\xi).$ Hence, by Zorn’s lemma, there exists a maximal $B\in\mathfrak{A}_{+}$ such that $B\leqslant P^{\prime}$ and $\omega(B)\geqslant(\xi,B\xi).$ Take $Q=P^{\prime}-B.$ Then, $Q\in\mathfrak{A}_{+},Q\neq 0$, since $\omega(P^{\prime})<(\xi,P^{\prime}\xi),$ and if $A\in\mathfrak{A}_{+},\,A\leqslant Q,\,A\neq 0,$ then $\omega(A)<(\xi,A\xi)$ by the maximality of $B.$

For any $A\in\mathfrak{A}$ one has

[TABLE]

Hence $(QA^{*}AQ)/||A||^{2}||Q||\leqslant Q$ and $\omega(QA^{*}AQ)<(\xi,QA^{*}AQ\xi)$. Combining this with the Cauchy-Schwartz inequality one finds

[TABLE]

Thus both $A\mapsto\omega(AQ)$ and $A\mapsto\omega(A(P+Q))$ are $\sigma$-strongly continuous. Since $P+Q\leqslant 1\!\!1,$ this contradicts the maximality of $P.$ $\hfill{\square}$

Proposition 1.30

([8] p.86) Let $\mathfrak{A}$ be a von Neumann algebra on a Hilbert space $\mathfrak{H}.$ Then the following four conditions are equivalent:

• (1)

$\mathfrak{A}$* is $\sigma$-finite;*

• (2)

there exists a countable subset of $\mathfrak{H}$ which is separating for $\mathfrak{A}$;

• (3)

there exists a faithful normal state on $\mathfrak{A}$;

• (4)

$\mathfrak{A}$* is isomorphic with a von Neumann algebra $\pi(\mathfrak{A})$ which admits a separating and cyclic vector.*

Proof. (see [8] p.86) (1) $\Rightarrow$ (2) Let $\{\xi_{\alpha}\}$ be a maximal family of vectors in $\mathfrak{H}$ such that $[\mathfrak{A^{\prime}}\xi_{\alpha}]$ and $[\mathfrak{A^{\prime}}\xi_{\alpha^{\prime}}]$ are orthogonal whenever $\alpha\neq\alpha^{\prime}.$ Since $[\mathfrak{A^{\prime}}\xi_{\alpha}]$ is a projection in $\mathfrak{A}$ (in fact the smallest projection in $\mathfrak{A}$ containing $\xi_{\alpha}),$ $\{\xi_{\alpha}\}$ is countable. But by the maximality,

[TABLE]

Thus $\{\xi_{\alpha}\}$ is cyclic for $\mathfrak{A^{\prime}}.$ Hence $\{\xi_{\alpha}\}$ is separating for $\mathfrak{A}$ by Proposition 1.22. (2) $\Rightarrow$ (3) Choose a sequence $\xi_{n}$ such that the set $\{\xi_{n}\}$ is separating for $\mathfrak{A}$ and such that $\displaystyle\sum_{n}||\xi_{n}||^{2}=1.$ Define $\omega$ by

[TABLE]

$\omega$ is $\sigma$-weakly continuous, hence normal, by using Theorem 1.29. If $\omega(A^{*}A)=0$ then $0=(\xi_{n},A^{*}A\xi_{n})=||A\xi_{n}||^{2}$ for all $n,$ hence $A=0.$ (3) $\Rightarrow$ (4) Let $\omega$ be a faithful normal state on $\mathfrak{A}$ and $(\mathfrak{H},\pi,\Omega)$ the corresponding cyclic representation. Since $\pi(\mathfrak{A})$ is a von Neumann algebra, if $\pi(A)\Omega=0$ for an $A\in\mathfrak{A}$ then $\omega(A^{*}A)=||\pi(A)\Omega||^{2}=0,$ hence $A^{*}A=0$ and $A=0.$ This proves that $\pi$ is faithful and $\Omega$ separating for $\pi(\mathfrak{A}).$ (4) $\Rightarrow$ (1) Let $\Omega$ be the separating (and cyclic) vector for $\pi(\mathfrak{A}),$ and let $\{E_{\alpha}\}$ be a family of mutually orthogonal projections in $\mathfrak{A}.$ Set $E=\displaystyle\sum_{\alpha}E_{\alpha}.$ Then

[TABLE]

by Lemma 1.27. Since $\displaystyle\sum_{\alpha}||\pi(E_{\alpha})\Omega||^{2}<+\infty,$ only a countable number of the $\pi(E_{\alpha})\Omega$ is nonzero, and thus the same is true for the $E_{\alpha}.$ $\hfill{\square}$

1.2 Hilbert space of Hilbert-Schmidt operators

Here we recall some definitions provided in [1, 2, 21] (and references therein).

Definition 1.31

**The trace of a linear operator A linear operator $A$ defined on the separable Hilbert space $\mathfrak{H}$ is said to be of trace class if the series $\displaystyle\sum_{k}\langle e_{k}|Ae_{k}\rangle$ converges and has the same value in any orthonormal basis $\{e_{k}\}$ of $\mathfrak{H}.$ The sum

[TABLE]

is called the trace of $A.$

Definition 1.32

**Trace norm Consider the class of Hilbert-Schmidt operators. For every such operator $A$, the trace norm is given by

[TABLE]

Remark 1.33

**If $A$ is any operator of trace class, then $A^{*}$ is also of trace class:

[TABLE]

Definition 1.34

**Hilbert-Schmidt operator Given a bounded operator, having the decomposition $A=\displaystyle\sum_{k}|\phi_{k}\rangle\lambda_{k}\langle\phi_{k}|$, where $\{\phi_{k}\}$ is an othonormal basis of $\mathfrak{H}$, and $\lambda_{1},\lambda_{2},\dots$ positive numbers, A is called a Hilbert-Schmidt operator if

[TABLE]

Remark 1.35

**If the series (1.81) is infinte, its convergence implies that $\lambda_{k}\rightarrow 0$ when $k\rightarrow+\infty.$ Consequently, $\lambda^{2}_{k}\geqslant\lambda_{k}$ for sufficiently large value of $k.$ Hence, $\displaystyle\sum_{k}\lambda^{2}_{k}$ converges when $\displaystyle\sum_{k}\lambda_{k}$ converges. This shows that any completely continuous operator $A$ satisfying (1.81) is a Hilbert-Schmidt operator.

Definition 1.36

**Hilbert-Schmidt norm For any Hilbert-Schmidt operator $A,$ the quantity

[TABLE]

exists, and is called Hilbert-Schmidt norm of $A.$

Definition 1.37

**Let $\mathcal{B}_{2}(\mathfrak{H})$, $\mathcal{B}_{2}(\mathfrak{H})\subset\mathcal{L}(\mathfrak{H})$ the set of all bounded operators on $\mathfrak{H}$, be the Hilbert space of Hilbert-Schmidt operators on $\mathfrak{H}=L^{2}(\mathbb{R})$, with the scalar product

[TABLE]

where $\{\Phi_{k}\}^{\infty}_{k=0}$ is an orthonormal basis of $\mathfrak{H}$.

$\mathcal{B}_{2}(\mathfrak{H})\simeq\mathfrak{H}\otimes\bar{\mathfrak{H}}$ (where $\bar{\mathfrak{H}}$ denotes the dual of $\mathfrak{H}$) and basis vectors of $\mathcal{B}_{2}(\mathfrak{H})$ are given by

[TABLE]

Remark 1.38

**In the notation $B_{2}(\mathfrak{H})\simeq\mathfrak{H}\otimes\overline{\mathfrak{H}}$, $\,\mathfrak{H}\otimes\overline{\mathfrak{H}}$ is taken here as the completude of the algebraic tensor product of $\mathfrak{H}$ by $\overline{\mathfrak{H}}$ which is a prehilbert space containing finite sums of the type $\displaystyle\sum_{j,k=0}^{n}\lambda_{jk}|\phi_{j}\rangle\otimes|\overline{\phi_{k}\rangle}$, where a basis of $\mathfrak{H}\otimes\overline{\mathfrak{H}}$ is $\{|\phi_{j}\rangle\otimes|\overline{\phi_{k}\rangle}\}_{j,k=0}^{\infty}.$ Then, $B_{2}(\mathfrak{H})$, being the Hilbert space of Hilbert-Schmidt operators on $\mathfrak{H}$ is isomorphic to $\mathfrak{H}\otimes\overline{\mathfrak{H}}$, since the separable Hilbert spaces are taken two by two isomorphic each other. Setting $|\phi_{j}\rangle\otimes|\overline{\phi_{k}\rangle}=|\phi_{j}\rangle\langle\phi_{k}|,$ $B_{2}(\mathfrak{H})$ admits for orthonormal basis $\{\phi_{jk}\}_{j,k=0}^{\infty}$ such that $\phi_{jk}:=|\phi_{j}\rangle\langle\phi_{k}|$.

Definition 1.39

**Let $A$ and $B$ two operators on $\mathfrak{H}$. The operator $A\vee B$ is such that

[TABLE]

For bounded linear operators $A$ and $B$, $A\vee B$ defines a linear operator on $B_{2}(\mathfrak{H})$. Indeed, $\forall A,B\in\mathcal{L}(\mathfrak{H}),$ (the space of bounded linear operators on $\mathfrak{H}$), since $B_{2}(\mathfrak{H})\subset\mathcal{L}(\mathfrak{H}),$ we get $\forall X\in B_{2}(\mathfrak{H}),$ $X\in\mathcal{L}(\mathfrak{H})$. Then, $AXB^{*}\in\mathcal{L}(\mathfrak{H})$, i.e. $(A\vee B)\in\mathcal{L}(\mathfrak{H}).$ Thus, $A\vee B$ defines a bounded linear operator on $B_{2}(\mathfrak{H})$.

From the scalar product in $\mathcal{B}_{2}(\mathfrak{H}),$

[TABLE]

it comes

[TABLE]

Since for any $X\in\mathcal{B}_{2}(\mathfrak{H})$,

[TABLE]

we have

[TABLE]

1.3 Modular theory and Hilbert-Schmidt operators

This paragraph is devoted to Tomita-Takesaki modular theory [26, 25, 24] of von Neumann algebras [19, 27]. Recall that the origins of Tomita-Takesaki modular theory lie in two unpublished papers of M. Tomita in 1967 [26] and a slim volume by M. Takesaki. As one of the most important contributions in the operator algebras, this theory finds many applications in mathematical physics.

We provide some key ingredients from [8, 1, 2, 7, 25] as needed for this section. First, let us deal with some notions from [8]:

Let $\mathfrak{A}$ be a von Neumann algebra on a Hilbert space $\mathfrak{H}$ and $\mathfrak{A^{\prime}}$ its commutant. Let $\Phi\in\mathfrak{H}$ be a unit vector which is cyclic and separating for $\mathfrak{A}.$ This is the case, if $\mathfrak{A}$ is a $\sigma$-finite von Neumann algebra, and by applying Proposition 1.22. 2. 2.

The mapping $A\in\mathfrak{A}\mapsto A\Omega\in\mathfrak{H}$ then establishes a one-to-one linear correspondence between $\mathfrak{A}$ and a dense subspace $\mathfrak{A}\Omega$ of $\mathfrak{H}$. Let $S_{0}$ and $F_{0}$ two antilinear operators on $\mathfrak{A}\Omega$ and $\mathfrak{A^{\prime}}\Omega,$ respectively. By Proposition 1.22, $\Omega$ is cyclic and separating for $\mathfrak{A}$ and $\mathfrak{A^{\prime}}$. Therefore the two antilinear operators $S_{0}$ and $F_{0}$, given by

[TABLE]

are both well defined on the dense domains on $D(S_{0})=\mathfrak{A}\Omega$ and $D(F_{0})=\mathfrak{A^{\prime}}\Omega.$ Then follows the definition:

Definition 1.40

** Define $S$ and $F$ as the closures of $S_{0}$ and $F_{0},$ respectively, i.e.,

[TABLE]

where the bar denotes the closure. Let $\Delta$ be the unique, positive, selfadjoint operator and $J$ the unique antiunitary operator occurring in the polar decomposition

[TABLE]

of $S$. $\Delta$ is called the modular operator associated with the pair $\{\mathfrak{A},\Omega\}$ and $J$ the modular conjugation.

The following proposition provides connections between $S,F,\Delta$ and $J$:

Proposition 1.41

([8] p.89) The following relations are valid:

[TABLE]

Proof. (see [8] pp.89-90) $\Delta=S^{*}S=FS,$ and $S=J\Delta^{1/2}$ by Definition 1.40. Using the fact that for any $\psi\in D(\bar{S}_{0})$ there exists a closed operator $Q$ on $\mathfrak{H},$with $S^{*}_{0}=\bar{F}_{0},F^{*}_{0}=\bar{S}_{0},$ such that

[TABLE]

where $\mathfrak{A^{\prime}}D(Q)\subseteq D(Q),\,QQ^{\prime}\supseteq Q^{\prime}Q$ for all $Q^{\prime}\in\mathfrak{A^{\prime}}$, with $S_{0}=S^{-1}_{0}$, it follows by closure that $S=S^{-1},$ and hence

[TABLE]

so that $J^{2}\Delta^{1/2}=J\Delta^{-1/2}J^{*}$. Since $J\Delta^{-1/2}J^{*}$ is a positive operator, and by the uniqueness of the polar decomposition one deduces that

[TABLE]

and then

[TABLE]

But this implies that

[TABLE]

and

[TABLE]

$\hfill{\square}$ 3. 3.

The principal result of the Tomita-Takesaki theory [25, 8] is that the following relations

[TABLE]

hold for all $t\in\mathbb{R}$. 4. 4.

Definition 1.42

**Modular automorphism group Let $\mathfrak{A}$ be a von Neumann algebra, $\omega$ a faithful, normal state on $\mathfrak{A}$, $(\mathfrak{H}_{\omega},\pi_{\omega},\Omega_{\omega})$ the corresponding cyclic representation, and $\Delta$ the modular operator associated with the pair $(\omega(\mathfrak{A}),\Omega_{\omega})$. The Tomita-Takesaki theorem establishes the existence of a $\sigma$-weakly continuous one-parameter group $t\mapsto\sigma^{\omega}_{t}$ of ∗-automorphisms of $\mathfrak{A}$ through the definition

[TABLE]

The group $t\mapsto\sigma^{\omega}_{t}$ is called the modular automorphism group associted with the pair $(\mathfrak{A},\omega)$. 5. 5.

Definition 1.43

$C^{*}$-dynamical system A $C^{*}$-dynamical system $(\mathfrak{G},\alpha)$ is a $C^{*}$-algebra $\mathfrak{G}$ equipped with a group homomorphism $\alpha:G\rightarrow\mbox{Aut}(\mathfrak{G})$ that is strongly continuous i.e., $g\mapsto||\alpha_{g}(x)||$ is a continuous map for all $x\in\mathfrak{G}$. A von Neumann dynamical system $(\mathfrak{A},\alpha)$ is a von Neumann algebra acting on the Hilbert space $\mathfrak{H}$ equipped with a group homomorphism $\alpha:G\rightarrow\mbox{Aut}(\mathfrak{A})$ that is weakly continuous i.e., $g\mapsto\langle\xi|\alpha_{g}(x)\eta\rangle$ is continuous for all $x\in\mathfrak{A}$ and all $\xi,\eta\in\mathfrak{H}.$

Definition 1.44

**A state $\omega$ on a one-parameter $C^{*}$-dynamical system $(\mathfrak{G},\alpha)$ is a $(\alpha,\beta)$-KMS state, for $\beta\in\mathbb{R},$ if for all pairs of elements $x,y$ in a norm dense $\alpha$-invariant ${*}$-subalgebra of $\alpha$-analytic elements of $\mathfrak{G}$, then $\omega(x\alpha_{i\beta}(y))=\omega(yx).$

Remark 1.45

**In the case of a von Neumann dynamical system $(\mathfrak{A},\alpha),$ a $(\alpha,\beta)$-KMS state must be normal (i.e., for every increasing bounded net of positive elements $x_{\lambda}\rightarrow x$, we have $\omega(x_{\lambda})\rightarrow\omega(x)$). Besides, given $\alpha:\mathbb{R}\rightarrow\mbox{Aut}(A),$ an element $x\in\mathfrak{G}$ is $\alpha$-analytic if there exists a holomorphic extension of the map $t\mapsto\alpha_{t}(x)$ to an open horizontal strip $\{z\in\mathbb{C}||\mbox{Im}\,z|<r\}$, with $r>0,$ in the complex plane. The set of $\alpha$-analytic elements is always $\alpha$-invariant (i.e., for all $x$ is analytic, $\alpha(x)$ is analytic)∗-subalgebra of $\mathfrak{G}$ that is norm dense in the $C^{*}$ case and weakly dense in the von Neumann case. 6. 6.

The modular automorphism group associated with $\omega$ is only the one parameter automorphism group that satisfies the Kubo-Martin-Schwinger (KMS)-condition with respect to the state $\omega,$ at inverse temperature $\beta,$ i.e.,

[TABLE]

and for all $x,y\in\mathfrak{A},$ there exists a function $F_{x,y}:\mathbb{R}\times[0,\beta]\rightarrow\mathbb{C}$ such that:

[TABLE]

1.3.1 Kubo-Martin-Schwinger (KMS) state

Let $\alpha_{i},\,i=1,2,\dots,N$ be a sequence of non-zero, positive numbers, satisfying : $\displaystyle\sum^{N}_{i=1}\alpha_{i}=1.$ Let

[TABLE]

$\{\zeta_{i}\}^{\infty}_{i=0}$ being an orthonormal basis of $\mathfrak{H},$ and the vectors $\{X_{ij}=|\zeta_{i}\rangle\langle\zeta_{j}|,i,j=1,2,\dots,N\}$ forming an orthonormal basis of $\mathcal{B}_{2}(\mathfrak{H}),$

[TABLE]

In particular, the vectors

[TABLE]

are one-dimensional projection operators on $\mathfrak{H}.$ Then, we have the following properties:

• (i)

Proposition 1.46

$\Phi$* defines a vector state $\varphi$ on the von Neumann algebra $\mathfrak{A}_{l}$ corresponding to the operators given with $A$ in the left of the identity operator $I_{\mathfrak{H}}$ on $\mathfrak{H},$ i.e., $\mathfrak{A}_{l}=\left\{A_{l}=A\vee I|A\in\mathcal{L}(\mathfrak{H})\right\}.$*

Proof. Indeed, for any $A\vee I\in\mathfrak{A}_{l},$ since $\mathcal{B}_{2}(\mathfrak{H})\subset\mathcal{L}(\mathfrak{H})$ and $\mathfrak{A}\subset\mathcal{L}(\mathfrak{H})$, from the Remark 1.26 and the equality (1.37) together, the state $\varphi$ on $\mathfrak{A}_{l}$ may be defined by

[TABLE]

$\hfill{\square}$

• (ii)

Proposition 1.47

The state $\varphi$ is faithful and normal.

Proof. The state $\varphi$ is normal by Theorem 1.29, using the fact that $\rho_{\varphi}$ is a density matrix and since we have

[TABLE]

Its faithfulness comes from Proposition 1.30 by use of the equivalence (2) $\Leftrightarrow$ (3), since $\mathfrak{A}_{l}\subset\mathcal{L}(\mathfrak{H})$ using the Eq.(1.37) in the Remark 1.26, we have, with ${\mathbb{P}}=|\zeta_{i}\rangle\langle\zeta_{i}|,$

[TABLE]

where the $\{\zeta_{i}\}^{N}_{i=1}$ form an orthonormal basis set of $\mathfrak{H}$. $\Phi$ is separating for $\mathfrak{A}_{l}$, by use of Theorem 1.29, and the relation

[TABLE]

Thereby, $\langle\varphi;(A\vee I)^{*}(A\vee I)\rangle=0$ if and only if $A\vee I=0.$ $\hfill{\square}$

• (iii)

Proposition 1.48

The vector $\Phi$ is cyclic and separating for $\mathfrak{A}_{l}.$

Proof. If $X\in\mathcal{B}_{2}(\mathfrak{H})$ is orthogonal to all $(A\vee I)\Phi,A\in\mathcal{L}(\mathfrak{H}),$ then

[TABLE]

Taking $A=X_{kl}$, it follows from the above equality, $\langle\zeta_{l}|X^{*}\zeta_{k}\rangle=0$ and since this holds for all $k,l,$ we get $X=0.$ Indeed, let $\Phi=\displaystyle\sum^{N}_{i=1}\alpha^{\frac{1}{2}}_{i}{\mathbb{P}}_{i}=\displaystyle\sum^{N}_{i=1}\alpha^{\frac{1}{2}}_{i}X_{ii}=\displaystyle\sum^{N}_{i=1}\alpha^{\frac{1}{2}}_{i}|\zeta_{i}\rangle\langle\zeta_{i}|,$ then by definition, see Eq.(1.85),

[TABLE]

such that the orthogonality implies

[TABLE]

Now taking $A=X_{kl}=|\zeta_{k}\rangle\langle\zeta_{l}|,$ it follows that

[TABLE]

From (1.134) and (1.135) together, it follows

[TABLE]

Thereby,

[TABLE]

Therefore, we have

[TABLE]

implying that the set $\left\{(A\vee I)\Phi,A\in\mathfrak{A}_{l}\right\}$ is dense in $\mathcal{B}_{2}(\mathfrak{H})$, proving from the Definition 1.19 that $\Phi$ is cyclic for $\mathfrak{A}_{l}.$ $\hfill{\square}$

The fact that $\Phi$ is separating for $\mathfrak{A}_{l}$ is obtained through the relation

[TABLE]

Proof. Let $A,B\in\mathfrak{A}_{l},$ such that $(A\vee I)\Phi=(B\vee I)\Phi,$ and take $X\neq 0,\,X\in\mathcal{B}_{2}(\mathfrak{H})$. We have

[TABLE]

Taking $(A\vee I)\Phi=(B\vee I)\Phi$, the equality $\langle X|\left\{(A\vee I)-(B\vee I)\right\}\Phi\rangle_{2}=0$ leads to

[TABLE]

which completes the proof. $\hfill{\square}$

In the same way, $\Phi$ is also cyclic for $\mathfrak{A}_{r}=\left\{A_{r}=I\vee A|A\in\mathcal{L}(\mathfrak{H})\right\},$ which corresponds to the operators given with $A$ in the right of the identity operator $I_{\mathfrak{H}}$ on $\mathfrak{H},$ hence separating for $\mathfrak{A}_{r},$ i.e. $(I\vee A)\Phi=(I\vee B)\Phi\Longleftrightarrow I\vee A=I\vee B.$

Then, starting to the above setup, to the pair $\{\mathfrak{A},\varphi\}$ is associated:

• •

a one parameter unitary group $t\mapsto\Delta^{-\frac{i}{t}\beta}_{\varphi}\in\mathcal{L}(\mathfrak{H})$

• •

and a conjugate-linear isometry $J_{\varphi}:\mathfrak{H}\rightarrow\mathfrak{H}$ that:

[TABLE]

[TABLE]

[TABLE]

Denote the automorphisms by $\alpha_{\varphi}(t)$, and deal with operators $A\in\mathfrak{A}$ with $\mathfrak{A}\subset\mathcal{L}(\mathfrak{H})$. Then, taking into account the Definition 1.25 and the Remark 1.26, from the expresssion (1.107), the automorphisms, in this case, satisfy the following relation:

[TABLE]

The KMS condition with respect to the automorphism group $\alpha_{\varphi}(t),t\in\mathbb{R},$ is obtained for any two $A,B\in\mathfrak{A},$ such that the function

[TABLE]

has an extension to the strip $\{z=x+iy|t\in\mathbb{R},y\in[0,\beta]\}\subset\mathbb{C}$ such that $F_{A,B}(z)$ is analytic in the strip $(0,\beta)$ and continuous on its boudaries. In addition, it also satisfies the boundary condition, at an inverse temperature $\beta$

[TABLE]

Setting the generator of the one-parameter group by ${\bf H}_{\varphi}$, the operators $\Delta^{-\frac{i}{t}\beta}_{\varphi}$ verify the relation

[TABLE]

Before introducing the von Neumann algebra generated by the unitary operators, let us consider the following:

Definition 1.49

** Consider the unitary operator $U(x,y)$ on $\mathfrak{H}=L^{2}(\mathbb{R})$ given by

[TABLE]

$x,y,\xi\in\mathbb{R}$, with $U(x,y)=e^{-i(xQ+yP)}$, where $Q,P$ are the usual position and momentum operators given on $\mathfrak{H}=L^{2}(\mathbb{R})$, with $[Q,P]=i\mathbb{I}_{\mathfrak{H}},$ and the Wigner transform, given by

[TABLE]

where $X\in\mathcal{B}_{2}(\mathfrak{H}),x,y\in\mathbb{R}$. $\mathcal{W}$ is unitary.

Indeed, given $X_{1},X_{2}\in\mathcal{B}_{2}(\mathfrak{H})$,

[TABLE]

On $\tilde{\mathfrak{H}}=L^{2}(\mathbb{R}^{2},dxdy)$, $\forall(x,y)\in\mathbb{R}^{2}$, consider the operators

[TABLE]

and let $\mathfrak{A}_{i},i=1,2$, be the von Neumann algebra generated by the unitary operators [25] $\{U_{i}(x,y)/(x,y)\in\mathbb{R}^{2}\}$. Then, it follows that:

Proposition 1.50

([1])

• (i)

The algebra $\mathfrak{A}_{1}$ is the commutant of the algebra $\mathfrak{A}_{2}$ (i.e. each element of $\mathfrak{A}_{1}$ commutes with every element of $\mathfrak{A}_{2}$) and vice versa with a factor, i.e,

[TABLE]

Considering the antiunitary operator $J_{\beta}$ (i.e. $\langle J\phi|J\psi\rangle=\langle\psi|\phi\rangle,\forall\phi,\psi\in\mathfrak{H}=L^{2}(\mathbb{R})$) such that:

[TABLE]

it comes

[TABLE]

The relation (1.165) and the property (i) provide the modular structure of the triplet $\{\mathfrak{A}_{1},\mathfrak{A}_{2},J_{\beta}\}$.

• (ii)

The map **[1]**

[TABLE]

is closable and has the polar decomposition

[TABLE]

where $J_{\beta}$ is the antiunitary operator, with

[TABLE]

[TABLE]

Indeed, $J_{\beta}$ is by definition an antiunitary operator. Then, it is self-adjoint, symmetric and consequently closable. $\Delta^{\frac{1}{2}}_{\beta}$ also being self-adjoint by definition, is closable too. From (1.166), the map $S_{\beta}$ given as the product of two closable operators, is then closable.

Proof of (1.166). The proof is achieved as follows: The vectors $\Psi_{jk},\,j,k=0,1,2,\cdots,\infty$, form an orthonormal basis of $\tilde{\mathfrak{H}}=L^{2}(\mathbb{R}^{2},dxdy)$. We have   $\Phi_{\beta}=\displaystyle\sum_{i=0}^{\infty}\lambda^{\frac{1}{2}}_{i}\Psi_{ii}.$ Applying $U_{1}(x,y)$ to both sides leads to

[TABLE]

Since $\displaystyle\sum_{j,k=0}^{\infty}|\Psi_{jk}\rangle\langle\Psi_{jk}|=I_{\tilde{\mathfrak{H}}}$, we get

[TABLE]

From the relations

[TABLE]

we get

[TABLE]

Using the fact that $\phi_{i},\;i=0,1,2,\cdots,\infty,$ form a basis of $\mathfrak{H}=L^{2}(\mathbb{R})$, we have

[TABLE]

Thus $\langle\Psi_{jk}|U_{1}(x,y)\Psi_{ii}\rangle_{\tilde{\mathfrak{H}}}=(2\pi)^{\frac{1}{2}}\delta_{ik}\overline{\Psi_{ji}(x,y)}$. From (1.3.1), it comes

[TABLE]

i.e.

[TABLE]

Let us calculate $U_{1}(x,y)^{*}\Phi_{\beta}$. We have

[TABLE]

where

[TABLE]

Putting (1.176) in (1.175) leads to

[TABLE]

From (1.173), we have

[TABLE]

Applying $S_{\beta}$ to both sides of the equality gives

[TABLE]

Since $S_{\beta}\left[U_{1}(x,y)\Phi_{\beta}\right]=U_{1}(x,y)^{*}\Phi_{\beta}$, then $S_{\beta}\overline{\Psi_{ji}(x,y)}=\Psi_{ji}(x,y).$ Thus,

[TABLE]

which rewrites

[TABLE]

From the relations (1.183) and (1.184) together, it follows that

[TABLE]

for all$\,\Psi_{ij}\in\tilde{\mathfrak{H}},\;i,j=0,1,2,\cdots,\infty$. $\hfill{\square}$

Proof of (1.167). Consider the operator $J_{\beta}$ with $J_{\beta}\Psi_{ji}=\Psi_{ij}$. We have

[TABLE]

Besides,

[TABLE]

Thus $J_{\beta}\Phi_{\beta}=\Phi_{\beta}$. Therefore, $J_{\beta}$ is such that

[TABLE]

From (1.166) and

[TABLE]

we get

[TABLE]

Using this above relations yields

[TABLE]

From (1.185),

[TABLE]

i.e.

[TABLE]

$\hfill{\square}$

Proposition 1.51

([1]) If $\displaystyle\{\lambda_{n}\}_{n=0}^{\infty}$ is a sequence of non-zero positive numbers such that $\displaystyle\sum_{n=0}^{\infty}\lambda_{n}=1$, then the vector

[TABLE]

is cyclic (that is the set of vectors $\{A\Phi/A\in\mathfrak{A}_{1}\}$ is dense in $\tilde{\mathfrak{H}}$) and separating (i.e. if $A\Phi=0$, for all $A\in\mathfrak{A}_{1}$ then $A=0$) for $\mathfrak{A}_{1}.$

Proof. Let $X\in\mathcal{B}_{2}(\tilde{\mathfrak{H}})$ and consider the operator $\mathcal{W}[U(x,y)\vee I_{\tilde{\mathfrak{H}}}]\mathcal{W}^{-1}\in\mathfrak{A}_{1}$. Taking $U(x,y)\vee I_{\tilde{\mathfrak{H}}},$ we have, since $\mathcal{W}$ is unitary,

[TABLE]

and the complex conjugate of (1.189) given by

[TABLE]

Then, integrating over $\mathbb{C}$ the modulus squared

[TABLE]

with respect to $x,y,$ we get

[TABLE]

Since $\mathcal{W}$ is unitary, we may write

[TABLE]

This implies that the set $\left\{\mathcal{W}[U(x,y)\vee I_{\tilde{\mathfrak{H}}}]\mathcal{W}^{-1}\Phi,\mathcal{W}[U(x,y)\vee I_{\tilde{\mathfrak{H}}}]\mathcal{W}^{-1}\in\mathfrak{A}_{1}\right\}$ is dense in $\tilde{\mathfrak{H}}$, proving from the Definition 1.19 that $\Phi$ is cyclic for $\mathfrak{A}_{1}.$ $\hfill{\square}$

The fact that $\Phi$ is separating for $\mathfrak{A}_{1}$ is obtained through the relation

[TABLE]

Proof. Let $U(x,y),U^{\prime}(x,y)$ such that $\mathcal{W}(U(x,y)\vee I_{\tilde{\mathfrak{H}}})\mathcal{W}^{-1}\Phi=\mathcal{W}(U^{\prime}(x,y)\vee I_{\tilde{\mathfrak{H}}})\mathcal{W}^{-1}\Phi.$ Take $X\neq 0,\,X\in\mathcal{B}_{2}(\tilde{\mathfrak{H}})$ and set $\Phi=\displaystyle\sum^{N}_{i=1}\lambda^{\frac{1}{2}}_{i}|\zeta_{i}\rangle\langle\zeta_{i}|$. We have

[TABLE]

Then, we have

[TABLE]

which completes the proof. $\hfill{\square}$

1.4 Modular theory-Thermal state

Here, we give two examples of thermal states as known from the literature. For more details, see [26, 25, 10, 8, 24, 7, 1, 2].

Let $\alpha_{i},i=1,2,\dots,N$ be a sequence of non-zero, positive numbers, satisfying $\displaystyle\sum_{i=1}^{N}\alpha_{i}=1.$ Then, the thermal state is defined as:

[TABLE]

where $\mathbb{P}_{i}=X_{ii}=|\zeta_{i}\rangle\langle\zeta_{i}|$ is defined as in (1.113)-(1.115), with $\{X_{ij}=|\zeta_{i}\rangle\langle\zeta_{j}|,i,j=1,2,\dots,N\}$ forming an orthonormal basis of $\mathcal{B}_{2}(\mathfrak{H})$. 2. 2.

The thermal equilibrium state $\Phi$ at inverse temperature $\beta,$ corresponding to the harmonic oscillator Hamiltonian

[TABLE]

where the density matrix is

[TABLE]

is

[TABLE] 3. 3.

Let the two von Neumann algebras be given by

[TABLE]

where $\mathfrak{A}_{l}$ corresponds to the operators given with $A$ in the left, and $\mathfrak{A}_{r}$ corresponds to the operators given with $A$ in the right of the identity operator $I_{\mathfrak{H}}$ on $\mathfrak{H},$ respectively. $\Phi$ defines a vector state $\varphi$, called KMS state, on the von Neumann algebra $\mathfrak{A}_{l}$. For any $A\vee I\in\mathfrak{A}_{l},$ one has the state $\varphi$ on $\mathfrak{A}_{l}$ given by

[TABLE]

with ${\mathbb{P}}_{n}=|\phi_{n}\rangle\langle\phi_{n}|,$ where

[TABLE]

and

[TABLE]

1.5 Coherent states built from the harmonic oscillator thermal state

Before dealing with the CS construction, we shall first extract few facts and notations about the modular structures emerging for von Neumann algebras in the study of an electron in a magnetic field as needed for the developement of this paragraph. For details see [1] and references therein.

1.5.1 Electron in a magnetic field

Considering the quantum Hilbert space $\mathfrak{H}=L^{2}(\mathbb{R})$ of the Hamiltonian $H_{\mbox{osc}}$ in (1.222), take $\mathcal{B}_{2}(\mathfrak{H})\simeq\mathfrak{H}\otimes\bar{\mathfrak{H}}$ the space of Hilbert-Schmidt operators on $\mathfrak{H}$ with an orthonormal basis given by

[TABLE]

Taking the classical Hamiltonian describing an electron placed in the $xy$ plane and subjected to a constant magnetic field [1],

[TABLE]

let the following quantum Hamiltonians

[TABLE]

with the magnetic field aligned along the negative $z$ axis, $\overrightarrow{A}=\frac{1}{2}(y,-x,0),$ where the quantized observables given on $\tilde{\mathfrak{H}}=L^{2}(\mathbb{R}^{2},dxdy)$ by

[TABLE]

and

[TABLE]

with the magnetic field aligned along the positive $z$ axis, $\overrightarrow{A}=\frac{1}{2}(-y,x,0),$ with the quantized observables given on $\tilde{\mathfrak{H}}=L^{2}(\mathbb{R}^{2},dxdy)$ by

[TABLE]

Since $[H_{1},H_{2}]=0$, the eigenvectors $\Psi_{nl}$ of $H_{1}$ can be so chosen that they are also the eigenvectors of $H_{2}$ as follows:

[TABLE]

so that $H_{2}$ lifts the degeneracy of $H_{1}$ and vice versa. Next, from the Definition 1.49, it is established that [1]

[TABLE]

where the $\phi_{nl}$ are the basis vecrtors given in (1.229) and the $\Psi_{nl}$ the normalized eigenvectors in (1.235). Then, from (1.236), the vectors $\Psi_{nl},n,l=0,1,2,\dots,\infty$ form a basis of $\tilde{\mathfrak{H}}=L^{2}(\mathbb{R}^{2},dxdy).$

Note that, in the sequel, the CS will be constructed from the thermal state $\Phi_{\beta},$ identified with the vector $\Phi$ given in (1.224), denoted as a ket state $|\Phi_{\beta}\rangle,$ the normalized eigenvectors (1.236) $\Psi_{nl}$ also denoted $|\Psi_{nl}\rangle$ as proceeded in [1].

1.5.2 Coherent states built from the thermal state

Take the cyclic vector $\Phi$ of the von Neumann algebra $\mathfrak{A}_{1}$ generated by the unitary operator (1.162)

[TABLE]

where $\mathcal{W}$ and $U(x,y)=e^{-i(xQ+yP)}$ are defined by (1.157) and (LABEL:map1), and consider Proposition 1.51 with the thermal state $\Phi_{\beta},$ instead of $\Phi,$ such that

[TABLE]

The CS, denoted $|z,\bar{z},\beta\rangle^{\mbox{\tiny{KMS}}},$ built from the thermal state in ket notation $|\Phi_{\beta}\rangle$ (see [1]), are given by

[TABLE]

where the annihilation and creation operators, $A_{1}$ and $A^{\dagger}_{1}$ with [1]

[TABLE]

act on $|\Psi_{n,l}\rangle$, eigenstates of the oscillator Hamiltonian in one dimension (1.231) with eigenvalues $E_{n}=\omega(n+\frac{1}{2})$, as follows:

[TABLE]

where the states $|\Psi_{nl}\rangle$ are here denoted $|\Psi_{n,l}\rangle$ by commodity.

Proposition 1.52

([1]) The CS $|z,\bar{z},\beta\rangle^{\mbox{\tiny{KMS}}}$ satisfy the resolution of the identity condition

[TABLE]

Proof. Consider the unitary operator $U(x,y)=e^{-i(xQ+yP)},\forall(x,y)\in\mathbb{R}^{2}$. Let

[TABLE]

Show that, for any normalized state $\phi\in\mathfrak{H}=L^{2}(\mathbb{R})$:

[TABLE]

[TABLE]

such that $\forall\psi,\xi\in\mathfrak{H}$, we have

[TABLE]

Then,

[TABLE]

Since  $\frac{1}{2\pi}\int_{\mathbb{R}}e^{iy(q-q^{\prime})}dy=\delta(q-q^{\prime})$, we get

[TABLE]

Thus

[TABLE]

Using the isometry property of $\mathcal{W}$, the states $|z,\bar{z},\beta\rangle^{\mbox{\tiny{KMS}}}$ satisfy the following resolution of the identity

[TABLE]

Proof. By definition of the states $|z,\bar{z},\beta\rangle^{\mbox{\tiny{KMS}}}$ and using (1.174), we have:

[TABLE]

Thereby

[TABLE]

Integrating the two members of the Eq.(1.252) over $\mathbb{C}$, and using the Wigner map $\mathcal{W},$ we get

[TABLE]

$\hfill{\square}$

From $(A^{\dagger}_{1})|\Psi_{0n}\rangle=\sqrt{n!}|\Psi_{nn}\rangle$ and $U_{1}(z)|\Psi_{0n}\rangle=e^{-\frac{|z|^{2}}{2}}\sum_{k=0}^{\infty}\frac{(zA^{\dagger}_{1})^{k}}{k!}|\Psi_{0n}\rangle$, it comes that:

[TABLE]

Proof of the first equality of (1.5.2). The operators $P$ and $Q$ verify the following relations:

[TABLE]

We establish that

[TABLE]

Multiplying the first and second equalities of (1.260), by $\frac{1}{\sqrt{2}}$ and $\frac{-\imath}{\sqrt{2}}$, respectively, provides:

[TABLE]

Setting $u=-x$ and $u=-y$, in the first and second relations of (1.261), respectively, gives with replacing $P$ by $P_{1}$, and $Q$ by $Q_{1}$, respectively:

[TABLE]

From $A^{\dagger}_{1}=\frac{Q_{1}-\imath P_{1}}{\sqrt{2}}$, set $z=\frac{-x+\imath y}{\sqrt{2}}$. Summing both equalities of (1.262) gives:

[TABLE]

Since $U_{1}(z)=e^{zA^{\dagger}_{1}-\bar{z}A_{1}}$, with $zA^{\dagger}_{1}-\bar{z}A_{1}=\imath(P_{1}x+Q_{1}y)$, it follows: $U_{1}(z)=e^{\imath(P_{1}x+Q_{1}y)}$. This latter equality with (1.263) together lead to:

[TABLE]

From (1.264), we have

[TABLE]

such that, by recursion, we get

[TABLE]

$\hfill{\square}$

**Proof of the second equality of (1.5.2).

**Considering $A^{\dagger}_{1}=\frac{1}{\sqrt{2}}(Q_{1}-\imath P_{1})$, where $z=\frac{1}{\sqrt{2}}(y-\imath x)$, with

[TABLE]

provides

[TABLE]

which completes the proof. $\hfill{\square}$

Since

[TABLE]

setting $|z;n\rangle=U_{1}(z)|\Psi_{0n}\rangle$ and from (1.5.2), it comes

[TABLE]

Set $|z;n\rangle=U_{2}(z)|\Psi_{n0}\rangle$ with $(A^{\dagger}_{2})^{n}|\Psi_{n0}\rangle=\sqrt{n!}|\Psi_{nn}\rangle\,$ and

[TABLE]

Taking $|z;0\rangle=U_{2}(z)|\Psi_{00}\rangle$ leads to

[TABLE]

Then,   $|z;n\rangle=\frac{1}{\sqrt{n!}}(A^{\dagger}_{2})^{n}|z;0\rangle$. In (1.272), we get

[TABLE]

Using the relation $U_{1}(x,y)^{*}=U_{1}(-x,-y)$, the CS $|-z,-\bar{z},\beta\rangle^{\mbox{\tiny{KMS}}}$ are obtained as follows:

[TABLE]

Indeed, by definition   $|z,\bar{z},\beta\rangle^{\mbox{\tiny{KMS}}}=U_{1}(z)|\Phi_{\beta}\rangle:=U_{1}(x,y)|\Phi_{\beta}\rangle$ such that

[TABLE]

where $e^{-zA^{\dagger}_{1}+\bar{z}A_{1}}|\Phi_{\beta}\rangle:=|-z,-\bar{z^{\prime}},\beta\rangle^{\mbox{\tiny{KMS}}}$, leading to (1.276). The CS (1.276) satisfy a resolution of the identity analogue to (1.249), i.e.,

[TABLE]

Proof. Similar to the proof of (1.249). $\hfill{\square}$

2 Noncommutative quantum harmonic oscillator Hilbert space

Without loss of generality, we restrict our developments to the noncommutative quantum mechanics formalism [10, 22, 6, 15] for the physical system of harmonic oscillator. We focus on the application of Hilbert-Schmidt operators, bounded operators on the noncommutative classical configuration space denoted by

[TABLE]

This space is isomorphic to the boson Fock space $\mathcal{F}=\displaystyle\{|n\rangle\}_{n=0}^{\infty},$ where the annihilation and creation operators $a,a^{\dagger}$ obey the Fock algebra $[a,a^{\dagger}]=1$. The physical states of the system represented on $\mathcal{H}_{q}$ known as the set of Hilbert-Schmidt operators, is equivalent to the Hilbert space of square integrable function, with

[TABLE]

where $\mathcal{B}(\mathcal{H}_{c})$ is the set of bounded operators on $\mathcal{H}_{c}$. $\mathcal{H}_{q}$ is defined as the set of bounded operators, with the form $|\cdot\rangle\langle\cdot|$, acting on the classical configuration space $\mathcal{H}_{c}$, with a general element of the quantum Hilbert space, in ”bra-ket” notation given by

[TABLE]

with $\left\{|n,m):=|n\rangle\langle m|\right\}_{n,m=0}^{\infty}$ a basis of $\mathcal{H}_{q}$ endowed with the inner product

[TABLE]

Considering the unitary Wigner map $\mathcal{W}:\mathcal{B}_{2}(\mathfrak{H})\rightarrow L^{2}(\mathbb{R}^{2},dxdy)$ let us discuss a correspondence between $L^{2}(\mathbb{R}^{2},dxdy)$ and $\mathcal{B}_{2}(\mathfrak{H})$.

Proposition 2.1

Given the Hilbert space $\mathfrak{H}=L^{2}(\mathbb{R})$, the inverse of the map $\mathcal{W}$ is defined on the dense set of vectors $f\in L^{2}(\mathbb{R}^{2},dxdy)$ as follows:

[TABLE]

where the integral is defined weakly, $|\phi\rangle\langle\psi|$ is an element of $\mathcal{B}_{2}(\mathfrak{H})\simeq\mathfrak{H}\otimes\overline{\mathfrak{H}}$ and $f=\mathcal{W}(|\phi\rangle\langle\psi|)$.

Proof. Let us derive the inverse of the map $\mathcal{W}$ on $L^{2}(\mathbb{R}^{2},dxdy)$ where the group $G$ and the Duflo-Moore operator $C$ with domain $\mathcal{D}(C^{-1})$ given in [3] are identified here to $\mathbb{R}^{2}$, $I_{\mathfrak{H}}$, the identity operator on $\mathfrak{H}=L^{2}(\mathbb{R})$, respectively, with $\mathcal{D}(C^{-1})=\mathfrak{H}$ and $\mathcal{D}(C^{-1})^{\dagger}=\overline{\mathfrak{H}}$. Consider an element in $\mathcal{B}_{2}(\mathfrak{H})\simeq\mathfrak{H}\otimes\overline{\mathfrak{H}}$ of the type $|\phi\rangle\langle\psi|$, with $\phi,\psi\in\mathfrak{H}$ and let $f=\mathcal{W}(|\phi\rangle\langle\psi|)$. For $\phi^{\prime},\psi^{\prime}\in\mathfrak{H}$, we have from the definition of $\mathcal{W}$ in (LABEL:map1)

[TABLE]

By the orthogonality relation, we get

[TABLE]

The relation $|\langle\phi^{\prime}|\phi\rangle\langle\psi|\psi^{\prime}\rangle|\leqslant||\phi^{\prime}||||\psi^{\prime}||||\phi||||\psi||$ implies

[TABLE]

Then, (2.9) holds for all $\phi^{\prime},\psi^{\prime}\in\mathfrak{H}$. Then, we obtain

[TABLE]

which completes the proof. $\hfill{\square}$

3 Application

Consider the motion of an electron in the $xy$-plane, subjected to a constant magnetic field pointing along the positive $z-$direction, i.e., in the symmetric gauge ${\bf A}^{\uparrow}=\left(-\frac{B}{2}y,\frac{B}{2}x\right),$ in the presence of a harmonic potential, described by the following Hamiltonian [4]

[TABLE]

where $\omega_{c}=\frac{eB}{2c}$ is the cyclotron frequency, $\Omega^{2}=\omega^{2}_{0}+\frac{\omega^{2}_{c}}{4}$, with the commutation relations:

[TABLE]

Next, introduce the dimensionless complex variables, related to the chiral decomposition of the physical model, given by

[TABLE]

such that they satisfy, with

[TABLE]

the relations

[TABLE]

Set

[TABLE]

satisfying the commutation relations

[TABLE]

Then, taking $\tilde{\Omega}_{\pm}=\tilde{\Omega}\pm\frac{\tilde{\omega}_{c}}{2},$ where [15]

[TABLE]

the Hamiltonian $H_{\theta}$ is obtained as follows:

[TABLE]

$N_{\pm}$ being the number operators such that $H_{\theta}$ writes

[TABLE]

The Hilbert spaces $\mathcal{H}_{q,\pm}$ are given by $\mathcal{H}_{q,\pm}=span\{|n_{\pm}\rangle\langle m_{\pm}|\}^{\infty}_{{n_{\pm},m_{\pm}}=0}$, with $\mathbb{I}_{\mathcal{H}_{q},\pm}$ their corresponding identity operators. The system $\{A_{\pm},A^{\dagger}_{\pm}\}$ forms an irreducible set of operators on the chiral boson Fock space $\mathcal{F}=\{|n_{\pm}\rangle\}^{\infty}_{{n_{\pm}}=0}$, and has the following realization on the states

[TABLE]

of the Hilbert space $\mathcal{H}_{q,+}\otimes\mathcal{H}_{q,-}$:

[TABLE]

[TABLE]

The operators $A_{\pm}$ act on the right by $A^{\dagger}_{\pm}$ by conjugation of (3.13) and (3.15). We have

[TABLE]

with $|0,0\rangle\langle 0,0|$ standing for the vacuum state of $\mathcal{H}_{q,+}\otimes\mathcal{H}_{q,-}.$

Then, the eigenvalues of the Hamiltonian $H_{\theta}$ are derived from the relation

[TABLE]

as follows:

[TABLE]

Given a state $|m\rangle\langle n|$ on $\mathcal{H}_{q}\simeq\mathfrak{H}\otimes\overline{\mathfrak{H}},$ the left $a_{L}$ and right $b_{R}$ annihilation operators act as follows:

[TABLE]

where

[TABLE]

[TABLE]

$Q,P$ are the usual position and momentum operators given on $\mathfrak{H}=L^{2}(\mathbb{R})$, with $[Q,P]=i\mathbb{I}_{\mathfrak{H}}.$

3.1 Coherent states construction

Using the operators $\{A_{\pm},A^{\dagger}_{\pm}\}$, the eigenstates $|z_{\pm}\rangle$ satisfy

[TABLE]

with the complex eigenvalues $z_{\pm}$,

[TABLE]

given in terms of the chiral Fock basis. Provided the Baker-Campbell-Hausdorff identity

[TABLE]

the CS of the Hamiltonian (3.11) in the noncommutative plane, denoted by $|z_{+},z_{-}),$ are defined by

[TABLE]

where we have used [6]:

[TABLE]

with $D_{R}=e_{R}^{-z_{+}A^{\dagger}_{+}+\bar{z}_{+}A_{+}}$ and $D_{L}=e_{L}^{-\bar{z}_{+}A_{+}+z_{+}A^{\dagger}_{+}}.$ The lower indices $R,L$ of the exponential operators refer to the right and left actions, respectively.

These CS satisfy the resolution of the identity [15]

[TABLE]

where $\mathbb{I}_{q}$ stands for the identity on $\mathcal{H}_{q}$ given by [22]:

[TABLE]

and the identity operator on $\mathcal{H}_{q,+}\otimes\mathcal{H}_{q,-}$ is:

[TABLE]

Proof.

In order to provide an equivalence between (3.31) and (3.32), let us consider the following relations

[TABLE]

where $w=z+u$ with $d^{2}w=d^{2}u$, and $e^{u\partial_{z}}f(z)=f(z+u)$. Then, set

[TABLE]

and

[TABLE]

We have

[TABLE]

Let

[TABLE]

We obtain

[TABLE]

which supplies, by performing a radial parametrization,

[TABLE]

Besides,

[TABLE]

implying

[TABLE]

Then,

[TABLE]

allowing to obtain (3.34) under the form:

[TABLE]

which completes the proof. $\hfill{\square}$

3.2 Density matrix and diagonal elements

Considering that the quantum system obeys the canonical distribution [9, 12, 4], let us take the partition function $\mathcal{Z}$ as that of a composite system made of two independent systems such that it is the product of the partition functions of the components, i.e. $\mathcal{Z}=\mathcal{Z}_{+}\mathcal{Z}_{-}$. The diagonal elements of the density operator $\hat{\rho}=\frac{1}{\mathcal{Z}}e^{-\beta H_{\theta}}$, where $H_{\theta}$ is given in (3.10) with eigenvalues $E_{{n}_{+},{n}_{-}}$ in (3.19), are then derived, in the CS $|z_{+},z_{-})$ (3.29) representation, as

[TABLE]

where $\mathcal{Z}=\mathcal{Z}_{+}\mathcal{Z}_{-}$.

Since

[TABLE]

then,

[TABLE]

Thereby,

[TABLE]

$\bar{n}=\left[e^{\beta\hbar\tilde{\Omega}_{+}}-1\right]^{-1}$ and ${\bar{n}}^{*}=\left[e^{\beta\hbar\tilde{\Omega}_{-}}-1\right]^{-1}$ are the corresponding thermal expectation values of the number operator (i. e. the Bose-Einstein distribution functions for oscillators with angular frequencies $\tilde{\Omega}_{+}$ and $\tilde{\Omega}_{-},$ respectively), also called the thermal mean occupancy for harmonic oscillators with the angular frequencies $\tilde{\Omega}_{+}$ and $\tilde{\Omega}_{-},$ respectively.

Performing the variable changes $r_{+}=\left[1-e^{-\beta\hbar\tilde{\Omega}_{+}}\right]^{1/2}|z_{+}|$ and $r_{-}=\left[1-e^{-\beta\hbar\tilde{\Omega}_{-}}\right]^{1/2}|z_{-}|$ with $\frac{d^{2}z}{\pi}=rdr\frac{d\varphi}{\pi},\,r\in[0,\infty),\varphi\in(0,2\pi]$, we obtain

[TABLE]

with here $n_{\pm}=0$, where we have used the following integral

[TABLE]

ensuring that the normalization condition of the density matrix is accomplished. The right-hand side of (3.63) corresponds to the product of two harmonic oscillators Husimi distributions [16].

3.3 Lowest Landau levels and reproducing kernel

Let us make a relationship between the quantum numbers $n_{\pm}\in\mathbb{N}$ which label the energy levels per sector and the quantum numbers $n,m$ where $n$ labels the levels and $m$ describes the degeneracy [5]. Fixing $n_{-}=0$ (resp. $n=0$), one obtains a state corresponding to the quantum number $m$ in the lowest Landau level (LLL) given by

[TABLE]

where $l_{0}=\sqrt{\frac{1}{eB}}\equiv 1$ (with $\hbar=1,e=1$) is the scale of lengths associated with the Landau problem.

Equivalently, fixing $n_{+}=0$ (resp. $m=0$), one gets a state centered at the origin ($m=0$) in the Landau level $n$ given by

[TABLE]

Consider the projector onto the LLL given by

[TABLE]

In the LLL $|0,m)$, the state $|0,\bar{\tilde{z}}_{+})$, where $z_{+}=x^{1}_{+}-ix^{2}_{+}$, is such that

[TABLE]

and also

[TABLE]

The matrix elements of the projector $\mathbb{P}_{0}$ are obtained as

[TABLE]

Let $|\psi)\in\mathcal{H}_{q}$, a state given on the LLL by

[TABLE]

We obtain that $|\psi)$ is analytic up to the Landau gaussian factor $e^{-\frac{|\tilde{z}_{+}|^{2}}{2}}$ as follows:

[TABLE]

with $d\nu(z,\bar{z})=\frac{e^{-|z|^{2}}}{2\pi}\frac{d{z}\wedge d\bar{z}}{i}$. Next, let us define the projection operator

[TABLE]

which is an integral operator with the reproducing kernel

[TABLE]

for $L^{2}_{hol}(\mathbb{C},d\nu(z,\bar{z}))$[3]. $L^{2}_{hol}(\mathbb{C},d\nu(z,\bar{z}))$ is the subspace of the Hilbert space $L^{2}(\mathbb{C},d\nu(z,\bar{z}))$ of $d\nu$-square integrable holomorphic functions on $\mathbb{C}$ in the variable $z$. Then, given an operator $\mathcal{O}$ on $L^{2}(\mathbb{C},d\nu(z,\bar{z}))$ and $f\in L^{2}_{hol}(\mathbb{C},d\nu(z,\bar{z})),$ we have

[TABLE]

3.4 Statistical properties

Let us consider the operators given on $\mathcal{H}_{q}\otimes\mathcal{H}_{q}$ by

[TABLE]

[TABLE]

From (3.20), we obtain in a state $|\tilde{n}\rangle\langle\tilde{m}|\otimes|m\rangle\langle n|\in\mathcal{H}_{q}\otimes\mathcal{H}_{q}$

[TABLE]

We get the following expressions:

[TABLE]

[TABLE]

leading to the following uncertainties:

[TABLE]

Concluding remarks

We have first dealt with some preliminaries about definitions, and remarkable properties on Hilbert-Schmidt operators and the Tomita-Takesaki modular theory. Then, the construction of CS built from the thermal state has been achieved and discussed, with the resolution of the identity. Besides, some detailed proofs have been provided in the study of the modular theory and Hilbert-Schmidt operators. The relation between the noncommutative quantum mechanics formalism and the modular theory, both using Hilbert-Schmidt operators, has been evidenced by the use of the Wigner map as an interplay between them. The formalism has been illustrated with the physical model of a charged particle on the flat plane xy in the presence of a constant magnetic field along the z-axis with a harmonic potential. CS have been constructed. Then, the density matrix, the projection onto the lowest Landau level (LLL) and main statistical properties have been discussed on the CS basis.

Acknowledgments

This work is supported by TWAS Research Grant RGA No. 17-542 RG/MATHS/AF/AC_G -FR3240300147. The ICMPA-UNESCO Chair is in partnership with Daniel Iagolnitzer Foundation (DIF), France, supporting the development of mathematical physics in Africa.