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### 2.3.1 Classical basic structure$[C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq {B(L^2( \Omega, \nu ))}]$

In classical systems, the basic structure$[{\mathcal A} \subseteq \overline{\mathcal A} \subseteq B(H)]$ is restricted to the classical basic structure: \begin{align*} [C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq {B(L^2( \Omega, \nu ))}] \end{align*} And we get the following diagram:
(A): Classical basic structure: $[C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq {B(L^2( \Omega, \nu ))}]$ $\require{AMScd}$ $\begin{CD} {\mathcal M}(\Omega) @. {} @. {} \\ @AA{\mbox{ dual}}A @. \\ \quad \fbox{C_0(\Omega)} \quad @>{\subseteq}>\mbox{ subalgebra\cdotweak-closure}> \quad \fbox{ L^\infty(\Omega, \nu )} \quad @>{\subseteq}>\mbox{ subalgebra}> \quad \fbox{ B(L^2(\Omega, \nu ))} \quad \\ @. @VV{\mbox{ pre-dual}}V \\ {} @. L^1(\Omega, \nu ) @. {} \\ \end{CD}$

In what follows, we shall explain this diagram.

### 2.3.1.1 Commutative $C^*$-algebra $C_0(\Omega)$

Let $\Omega$ a locally compact space, for example, it suffices to image $\Omega$ as follows. \begin{align*} & {\mathbb R}(=\mbox{the real line}), \;\;{\mathbb R}^2(=\mbox{plane}), \;\; {\mathbb R}^n(=\mbox{$n$-dimensional Euclidean space}), \;\; \\ \\ & [a, b ](=\mbox{interval}), \;\; \underset{\mbox{ (with discrete metric $d_D$)}}{\mbox{finite set}\Omega(=\{\omega_1,..., \omega_n\})} \end{align*} where the discrete metric $d_D$ is defined by $d_D(\omega, \omega')=1 \;(\omega \not= \omega'), =0\;(\omega = \omega')$
Define the continuous functions space $C_0(\Omega)$ such that \begin{align} C_0(\Omega) = \{f:\Omega \to {\mathbb C} \;|\; \mbox{$f$ is complex-valued continuous on $\Omega$, }\lim_{\omega \to \infty} f(\omega)=0 \} \end{align} where "$\lim_{\omega \to \infty} f(\omega)=0$" means
 (A1): for any positive real $\epsilon >0$, there exists a compact set $K (\subseteq \Omega)$ such that \begin{align*} \{ \omega \;|\; \omega \in \Omega \setminus K, |f(\omega)| > \epsilon \} = \emptyset \end{align*}

Therefore, if $\Omega$ is compact, the condition $\lim_{\omega \to \infty} f(\omega)=0$ is not needed, and thus, $C_0(\Omega)$ is usually denoted by $C(\Omega)$. In this note, even if $\Omega$ is compact, we often denote $C(\Omega)$ by $C_0(\Omega)$. Defining the norm $\| \cdot \|_{C_0(\Omega )}$ in a complex vector space $C_0(\Omega)$ such that

\begin{align} \| f \|_{C_0(\Omega )}= \max_{\omega \in \Omega } |f(\omega)| \end{align} we get the Banach space $\Big( C_0(\Omega) ,\| \cdot \|_{C_0(\Omega )} \Big)$.
For example,

Let $\Omega$ be a locally compact space, and consider the $\sigma$-finite measure space $(\Omega, {\cal B}_{\Omega}, \nu)$, where, ${\cal B}_{\Omega}$ is the Borel field, i.e., the smallest $\sigma$-field that contains all open sets. Further, assume that

 (A1): for any open set $U \subseteq \Omega$, it holds that $0 < \nu (U) {\; \leqq \;} \infty$

 $\fbox{Note 2.1}$ Without loss of generality, we can assume that $\Omega$ is compact by the Stone-Cech compactification. Also, we can assume that $\nu(\Omega)=1$.

Define the Banach space $L^r (\Omega, \nu)$ (where, $r = 1, 2, \infty$) by the all complex-valued measurable functions $f: \Omega\to {\mathbb C}$ such that

\begin{align} \|f\|_{L^r (\Omega, \nu)} < \infty \end{align} The norm $\|f\|_{L^r (\Omega, \nu)}$ is defined by \begin{align} \|f\|_{L^r (\Omega, \nu)} = \left\{\begin{array}{ll} {\Big[ \int_{\Omega} |f (\omega)|^r \, \nu(d \omega) \Big]^{1/r}} \quad & \mbox{(when $r = 1, 2$)} \\ \\ \underset{\omega \in \Omega}{\mbox{ ess.sup} }|f (\omega)| & \mbox{(when $r = \infty$)} \end{array}\right. \end{align} where \begin{align*} \mbox{ ess.sup}_{\omega \in \Omega} |f (\omega)| = \sup \{ a \in {\mathbb R} \;|\; \nu(\{ \omega \in \Omega \; :\; |f (\omega)| {\; \geqq \;}a \; \}) >0 \} \end{align*}

$L^r (\Omega, \nu)$ is often denoted by $L^r (\Omega)$ or $L^r (\Omega, {\cal B}_{\Omega}, \nu)$.

Remark 2.11 [$C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq B(L^2( \Omega, \nu ) )$] Consider a Hilbert space $H$ such that \begin{align*} H=L^2( \Omega, \nu ) \end{align*} For each $f \in L^\infty (\Omega )$, define $T_f \in B(L^2( \Omega, \nu ) )$ such that \begin{align*} L^2( \Omega, \nu ) \ni \phi \longrightarrow T_f( \phi) = f \cdot \phi \in L^2( \Omega, \nu ) \end{align*} Then, under the identification: \begin{align} L^\infty (\Omega ) \ni f \underset{\mbox{ identification}}{\longleftrightarrow} T_f \in B(L^2( \Omega, \nu ) ) \tag{2.33} \end{align} we see that \begin{align*} f \in L^\infty (\Omega ) \subseteq B(L^2( \Omega, \nu ) ) \end{align*} and further, we have the classical basic structure: \begin{align} [ C_0(\Omega ) \subseteq L^\infty (\Omega ) \subseteq B(L^2( \Omega, \nu ) ) ] \tag{2.34} \end{align} This will be shown in what follows.
Riese theorem says that \begin{align} C_0(\Omega)^*= \mbox{{${\mathcal M}(\Omega)$}}(=\mbox{{the set of all complex-valued measures on $\Omega$} }) \end{align}

Therefore, for any $F \in C_0(\Omega)$, $\rho \in C_0(\Omega)^*={\mathcal M}(\Omega)$, we have the bi-linear form which is written by the several ways such as

\begin{align} \rho(F) = {}_{\stackrel{{}}{C_0(\Omega)^* }}\Big(\rho, F \Big){}_{\stackrel{{}}{{ C_0(\Omega) } }} = {}_{\stackrel{{}}{{\mathcal M}(\Omega)}}\Big(\rho, F \Big){}_{\stackrel{{}}{{ C_0(\Omega) } }} = \int_\Omega F(\omega) \rho( d \omega ) \tag{2.36} \end{align} Also, the dual norm is calculated as follows. \begin{align} & \|\rho\|_{C_0(\Omega)^* } = \sup \{ |\rho(F) \;|\; \|F\|_{C_0(\Omega)}=1 \} = \sup_{||F||_{C_0(\Omega)}=1}| \int_\Omega F(\omega ) \rho(d\omega)| \nonumber \\ = & \sup_{\Xi, \Gamma \in {\mathcal B}_\Omega} \Big( |Re(\rho(\Xi))-Re(\rho(\Xi^c))|^2 + |Im(\rho(\Gamma))-Im(\rho(\Gamma^c))|^2 \Big)^{1/2} \nonumber \\ = & \|\rho\|_{{\mathcal M}(\Omega)} \tag{2.37} \end{align}

where, $\Xi^c$ is the complement of $\Xi$, and $Re(z)$="the real part of the complex number $z$", $Im(z)$="the imaginary part of the complex number $z$".

### 2.3.1.2 Commutative $W^*$-algebra $L^\infty (\Omega, \nu)$

Further, we see that \begin{align*} L^1(\Omega, \nu )^* = L^\infty (\Omega , \nu ) \qquad \mbox{ in the same sense, } \qquad L^1(\Omega, \nu ) = L^\infty (\Omega , \nu )_* \end{align*} Also, }it is clear that \begin{align*} C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \end{align*}

For any $f \in L^\infty (\Omega, \nu )$, there exist $f_n \in C_0(\Omega), n=1,2,..$ such that

\begin{align*} \left\{\begin{array}{ll} \nu(\{ \omega \in \Omega \;|\; \lim_{n \to \infty } f_n (\omega ) \not= f(\omega) \}=0 \\ \\ |f_n (\omega )|\le \|f \|_{L^\infty (\Omega, \nu )} \quad (\forall \omega \in \Omega, \forall n=1,2,3,... ) \end{array}\right. \end{align*}

Therefore, we see \begin{align*} \lim_{n \to \infty } | \Big\langle \phi, (f-f_n) \phi \Big\rangle_{L^2(\Omega, \nu )} | \le & \lim_{n \to \infty } \int_\Omega |f_n(\omega ) - f(\omega ) | \cdot | \phi ( \omega )|^2 \nu (d \omega ) \\ = & 0 \qquad \qquad (\forall \phi \in L^2 ( \Omega , \nu )) \end{align*} Hence, \begin{align*} \mbox{the weak closure of $C_0(\Omega )$ is equal to $L^\infty ( \Omega, \nu )$ } \end{align*} Then, we have the classical basic structure: \begin{align} [ C_0(\Omega ) \subseteq L^\infty (\Omega ) \subseteq B(L^2( \Omega, \nu ) ) ] \tag{2.38} \end{align}
Theorem 2.12 [Gelfand theore] Consider a general basic structure: \begin{align*} [ {\mathcal A} \subseteq \overline{\mathcal A} \subseteq B(H) ] \end{align*} where it is assumed that ${\mathcal A}$ is commutative. Then, there exists a measure space $(\Omega, {\mathcal B}_\Omega, \nu )$ (where $\Omega$ is a locally compact space) such that \begin{align*} {\mathcal A}=C_0(\Omega ), \;\; \overline{\mathcal A}=L^\infty (\Omega, \nu ), \;\; B(H)=B(L^2 (\Omega, \nu )) \end{align*} where $\Omega$ is called a spectrum .

### 2.3.2 Classical basic structure$[C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq {B(L^2( \Omega, \nu ))}]$ and State space

Consider the classical basic structure $[C_0(\Omega ) \subseteq L^\infty (\Omega, \nu ) \subseteq {B(L^2( \Omega, \nu ))}]$. Then, we see the following diagram:
(D):Classical basic structure and State space \begin{align} & \begin{array}{rlrlll} \underset{{\mbox{ $C^*$-pure state}}}{ {\underset{(\approx \Omega )}{{\mathcal M}_{+1}^p (\Omega )}} } \subset \underset{{\mbox{ $C^*$-mixed state}}}{\underset{\mbox{ (probability measure)}}{{{\mathcal M}_{+1} (\Omega )} }}\subset & {\mathcal M}(\Omega ) &&&& \\ & \Big\uparrow \mbox{ dual} &&&& \\ & \fbox{$C_0(\Omega )$} & \xrightarrow[\underset{\mbox{ weak-closure}}{\mbox{ subalgebra}}]{\subseteq} & \fbox{$L^\infty ( \Omega)$} & \xrightarrow[\mbox{ subalgebra}]{\subseteq} \fbox{${B(L^2( \Omega))}$} & \\ & && \;\;\; \Big\downarrow \;\mbox{ pre-dual} && \end{array} \tag{2.39} \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \underset{{\mbox{ $W^*$-mixed state}}}{\underset{\mbox{ (probability density function)}}{{L^1_{+1}( \Omega, \nu )}}} \subset L^1( \Omega, \nu ) \nonumber \end{align}
In the above, the mixed state space ${\frak S}^m(C_0(\Omega)^*)$ is characterized as \begin{align} {\frak S}^m(C_0(\Omega)^*) = & \{ \rho \in {\mathcal M}(\Omega) \;:\; \rho \ge 0, ||\rho||_{{\mathcal M}(\Omega)}=1\} \nonumber \\ = & \{ \rho \in {\mathcal M}(\Omega) \;:\; \rho \mbox{ is a probability measure on $\Omega$ }\} \nonumber \\ =: & {\mathcal M}_{+1} (\Omega ) \tag{2.40} \end{align} Also, the pure state space ${\frak S}^p(C_0(\Omega)^*)$ is \begin{align} & {\frak S}^p(C_0(\Omega)^*) \nonumber \\ = & \{ \rho=\delta_{\omega_0} \in {\frak S}^p(C_0(\Omega)^*) \;:\; \delta_{\omega_0} \mbox{ is the point measure at $\omega_0(\in \Omega)$}, \omega_0 \in \Omega\} \nonumber \\ \equiv & {\mathcal M}_{+1}^p(\Omega) \tag{2.41} \end{align} Here, the point measure $\delta_{\omega_0}\in {\mathcal M}(\Omega)$ is defined by \begin{align*} \int_\Omega f(\omega ) \delta_{\omega_0} (d \omega ) = f(\omega_0) \quad( \forall f \in C_0(\Omega )) \end{align*}
Therefore, \begin{align} {\mathcal M}_{+1}^p(\Omega)= {\frak S}^p(C_0(\Omega)^*) \ni \delta_\omega \underset{\mbox{ identification}}{\longleftrightarrow} \omega \in \Omega \tag{2.42} \end{align} Under this identification, we consider that \begin{align*} {\frak S}^p(C_0(\Omega)^*) = \Omega \end{align*}
Also, it is well known that \begin{align*} L^1(\Omega, \nu )^*= L^\infty (\Omega, \nu ) \end{align*} Therefore, the $W^*$-mixed state space is characterized by \begin{align} L^1_{+1}(\Omega, \nu ) & = \{ f \in L^1(\Omega, \nu ) \; : \; f \ge 0, \;\; \int_\Omega f (\omega ) \nu (d \omega ) = 1 \} \nonumber \\ & = \mbox{the set of all probability density functions on $\Omega$} \tag{2.43} \end{align}

Remark 2.13 [The case that $\Omega$ is finite: $C_0(\Omega)=L^\infty(\Omega,\nu)$, ${\mathcal M}(\Omega ) = L^1(\Omega, \nu )$] Let $\Omega$ be a finite set $\{\omega_1,\omega_2,...,\omega_n \}$ with the discrete metric $d_D$ and the counting measure $\nu$. Here, the counting measure $\nu$ is defined by

$\nu( D )= \sharp [D] (= \mbox{"the number of the elements of D"} )$ Then, we see that $C_0(\Omega ) = \{ F : \Omega \to {\mathbb C} \;|\; \mbox{ F is a complex valued function on \Omega} \} = L^\infty (\Omega, \nu )$ And thus, we see that $\rho \in {\mathcal M}_{+1}(\Omega ) \;\; \Longleftrightarrow \;\; \rho = \sum_{k=1}^n p_k \delta_{\omega_k } \;\;( \sum_{k=1}^n p_k=1, p_k \ge 0)$ and $f \in L^1_{+1}(\Omega, \nu ) \;\; \Longleftrightarrow \;\; \sum_{k=1}^n f(\omega_k ) =1. \;\; f(\omega_k )\ge 0$ In this sense, we have the following identifications: ${\mathcal M}_{+1}(\Omega ) = L^1_{+1}(\Omega, \nu ) \quad (\mbox{ or, } {\mathcal M}(\Omega ) = L^1 (\Omega, \nu ) )$ After all, we have the following identification: $C_0(\Omega )= L^\infty(\Omega) = {\mathbb C}^n \qquad {\mathcal M}(\Omega )= L^1(\Omega) = {\mathbb C}^n \tag{2.44}$ where the norm $\| \cdot \|_{C_0(\Omega )}$ in the former is defined by \begin{align} \| z \|_{C_0(\Omega )}= \max_{k=1,2,...,n} |z_k | \qquad \forall z= \left[\begin{array}{l} z_1 \\ z_2 \\ \vdots \\ x_n \end{array}\right] \in {\mathbb C}^n \tag{2.45} \end{align} and the norm $\| \cdot \|_{{\mathcal M}(\Omega )}$ in the latter is defined by \begin{align} \| z \|_{{\mathcal M}(\Omega )}= \sum_{k=1}^n |z_k | \qquad \forall z= \left[\begin{array}{l} z_1 \\ z_2 \\ \vdots \\ x_n \end{array}\right] \in {\mathbb C}^n \tag{2.46} \end{align}