**3. Sector Theory**

The concept of sector is defined by Ojima [16] as follows:

**Definition 3.** *A sector of* X *is a quasi-equivalence class of a factor state.*

A state on X is called a factor if the center <sup>Z</sup>*ω*(<sup>X</sup> ) = *πω*(<sup>X</sup> ) ∩ *πω*(<sup>X</sup> ) of *πω*(<sup>X</sup> ) is trivial, i.e., <sup>Z</sup>*ω*(<sup>X</sup> ) = C1. Let *π* be a representation of X on a Hilbert space H. We say that a linear functional *ω* on X is *π*-normal if there exists a trace-class operator *σ* on H such that

$$
\omega(X) = \text{Tr}[\pi(X)\sigma] \tag{12}
$$

for all *X* ∈ X .

**Definition 4.** *Let π*1 *and π*2 *be a representation of* X *on Hilbert spaces* H1 *and* H<sup>2</sup>*, respectively.* (1) *π*1 *and π*2 *are quasi-equivalent, written as π*1 ≈ *π*2*, if every <sup>π</sup>*1*-normal state is <sup>π</sup>*2*-normal and vice versa.*

(2) *π*1 *and π*2 *are mutually disjoint, written as π*1 ◦*– π*2*, if no <sup>π</sup>*1*-normal state is <sup>π</sup>*2*-normal and vice versa.*

*Two states ω*1 *and ω*2 *on* X *are quasi-equivalent (mutually disjoint, resp.), written as ω*1 ≈ *ω*2 *(<sup>ω</sup>*1 ◦*– ω*2*, resp.), if πω*1 *and πω*2 *are quasi-equivalent (mutually disjoint, resp.).*

The sector theory based on sector defined above has already been discussed in [16,21]. However, we believe that mathematics related to sector theory should be reexamined in order to develop measurement theory for quantum systems described by C∗-algebras. The following theorem mathematically justifies the definition of sector, which is obvious from [30] (Corollary 5.3.6).

**Theorem 2.** *Two factor states ω*1 *and ω*2 *are either quasi-equivalent or disjoint.*

By the above theorem, two factor states *ω*1 and *ω*2 belong to different sectors if and only if *ω*1 ◦– *ω*2. A sector corresponds to a macroscopic situation where order parameters of the system have definite values. Although the unitary equivalence of states is efficient for pure states, physically important states are not always pure. For example, KMS states in some quantum system with infinite degrees of freedom are of type III. We would like to stress that the unitary equivalence class of a pure state is not appropriate for a unit of the state space. The reason will be discussed later.

Next, we shall define the notion of orthogonality of states. The order relation *ω*1 ≤ *ω*2 for two positive linear functionals *ω*1 and *ω*2 on X is defined by

$$
\omega\_1(X) \le \omega\_2(X) \tag{13}
$$

for all *X* ∈ X+.

**Definition 5.** *Let ω*1, *ω*2 *be positive linear functionals on* X *. We say that ω*1 *and ω*2 *are mutually orthogonal, written as <sup>ω</sup>*1⊥ *ω*2*, if there exists no non-zero positive linear functional ω such that ω* ≤ *ω*1 *and ω* ≤ *ω*2*.*

The following theorem shows the gap between the disjointness and the orthogonality of states.

**Theorem 3** ([31] (Lemma 4.1.19 and Lemma 4.2.8))**.** *Let ω*1, *ω*2 *be positive linear functionals on* X *. Put ω* = *ω*1 + *ω*2*.*

(1) *If ω*1 *and ω*2 *are mutually orthogonal, then there exists an orthogonal projection P* ∈ *πω*(<sup>X</sup> ) *such that*

$$
\omega\_1(X) = \langle \Omega\_{\omega} | P \pi\_{\omega}(X) \Omega\_{\omega} \rangle, \quad \omega\_2(X) = \langle \Omega\_{\omega} | (1 - P) \pi\_{\omega}(X) \Omega\_{\omega} \rangle \tag{14}
$$

*for all X* ∈ X

*.*

*.*

(2) *If ω*1 *and ω*2 *are mutually disjoint, then there exists an orthogonal projection C* ∈ <sup>Z</sup>*ω*(<sup>X</sup> ) *such that*

$$
\omega\_1(X) = \langle \Omega\_{\omega} | C \pi\_{\omega^\vee}(X) \Omega\_{\omega^\vee} \rangle, \quad \omega\_2(X) = \langle \Omega\_{\omega^\vee} | (1 - \mathbb{C}) \pi\_{\omega^\vee}(X) \Omega\_{\omega^\vee} \rangle \tag{15}
$$

*for all X* ∈ X

The topology of S(X ) used here is the restriction of the weak<sup>∗</sup>-topology of X ∗ to S(X ). That is to say, it is generated by the basis B = {*<sup>O</sup>ω*({*Xi*,*εi*}*<sup>n</sup> <sup>i</sup>*=<sup>1</sup>) | *ω* ∈ S(X ), *n* ∈ N, *X*1, ··· , *Xn* ∈ X ,*ε*1, ··· ,*ε<sup>n</sup>* > <sup>0</sup>}, where *<sup>O</sup>ω*({*Xi*,*εi*}*<sup>n</sup> <sup>i</sup>*=<sup>1</sup>) = {*ω* ∈ *S*(*X*) | ∀*i* = 1, ··· , *n*, |*ω*(*Xi*) − *ω* (*Xi*)| < *<sup>ε</sup>i*}. Then, S(X ) is a compact convex set, and we use the Borel field B(S(X )) of S(X ) generated by this topology. A positive linear functional *ω* on X is called a barycenter of a regular Borel measure *μ* on S(X ) if

$$
\omega = \int\_{S(\mathcal{X})} \rho \, d\mu(\rho). \tag{16}
$$

*μ* is then called a barycentric measure of *ω*.

**Definition 6.** *A regular Borel measure μ on* S(X ) *is orthogonal if* 

$$\int\_{\Delta} \rho \, d\mu(\rho) \quad \perp \quad \int\_{\Delta^c} \rho \, d\mu(\rho) \tag{17}$$

*for all* Δ ∈ B(S(X ))*.* O*ω*(S(<sup>X</sup> )) *denotes the set of orthogonal measures on* S(X ) *with barycenter ω.*

The following theorem characterizes orthogonal measures of a state.

**Theorem 4** ([31] (Theorem 4.1.25))**.** *Let* X *be a unital C*∗*-algebra and ω a state on* X *. There is a one-to-one correspondence between the following three sets:*

(*i*) *the orthogonal measures μ* ∈ O*ω*(S(<sup>X</sup> ))*;*

(*ii*) *the abelian von Neumann subalgebras* B *of πω*(<sup>X</sup> ) *;*

(*iii*) *the orthogonal projections P on* H*ω such that P* Ω*ω* = Ω*ω and P πω*(<sup>X</sup> )*P* ⊆ { *P πω*(<sup>X</sup> )*P*} *.* *If μ,* B *and P are in correspondence, one has the following conditions:*

(1) B = (*πω*(<sup>X</sup> ) ∪ {*P*})*;*

(2) *P is the orthogonal projection onto* BΩ*ω;*

(3) *μ*(*<sup>X</sup>* 1 ··· *X n*) = <sup>Ω</sup>*ω*|*πω*(*<sup>X</sup>*1)*<sup>P</sup>πω*(*<sup>X</sup>*2)*<sup>P</sup>* ··· *<sup>P</sup>πω*(*Xn*)<sup>Ω</sup>*ω;*

(4) B *is* ∗*-isomorphic to the range of the map κμ* : *L*∞(S(X ), *μ*) *f* → *κμ*(*f*) ∈ *πω*(<sup>X</sup> ) *defined by*

$$
\langle \Omega\_{\omega} | \kappa\_{\mu}(f) \pi\_{\omega}(X) \Omega\_{\omega} \rangle = \int\_{S(\mathcal{X})} f(\rho) \, \underset{\omega}{\mathcal{X}}(\rho) \, d\mu\_{\omega}(\rho) \tag{18}
$$

*for all X* ∈ X *and f* ∈ *L*∞(S(X ), *μ*)*, where X*ˆ ∈ *C*(S(X )) *is defined by <sup>X</sup>*<sup>ˆ</sup>(*ρ*) = *ρ*(*X*) *for all ρ* ∈ S(X )*. κμ satisfies*

$$
\kappa\_{\mu}(\hat{X})\pi\_{\omega}(Y)\Omega\_{\omega} = \pi\_{\omega}(Y)P\pi\_{\omega}(X)\Omega\_{\omega},\tag{19}
$$

 (20)

*for all X*,*Y* ∈ X *.*

By Theorems 3 and 4, we have the following theorem:

**Theorem 5** ([31] (Proposition 4.2.9))**.** *Let ω be a state on* X *and μ a barycentric measure of ω. The following conditions are equivalent.* (1)*Forevery*Δ∈B(S(X))*,*

$$
\int\_{\Delta} \rho \, d\mu(\rho) \, \vartriangleleft \int\_{\Delta^c} \rho \, d\mu(\rho).
$$

(2) *μ is orthogonal, and κμ*(*L*<sup>∞</sup>(S(<sup>X</sup> ), *μ*)) *is a von Neumann subalgebra of the center* <sup>Z</sup>*ω*(<sup>X</sup> ) *of πω*(<sup>X</sup> )*.*

For every *ω* ∈ S(X ), *μω* denotes the orthogonal measure with barycenter *ω* corresponding to the center <sup>Z</sup>*ω*(<sup>X</sup> ) of *πω*(<sup>X</sup> ). *μω* is called the central measure of *ω*. The following theorem shows that the central measure gives the unique integral decomposition into mutually different sectors.

**Theorem 6** ([31] (Theorem 4.2.11))**.** *The central measure μω of a state ω on* X *is pseudosupported by the set* <sup>S</sup>*f*(<sup>X</sup> ) *of factor states on* X *, i.e., μω*(Δ) = 0 *for all* Δ ∈ B(S(X )) *such that* Δ ∩ <sup>S</sup>*f*(<sup>X</sup> ) = ∅*. If* X *is separable, then μω is supported by* <sup>S</sup>*f*(<sup>X</sup> )*.*

That is to say, the concept of sector is applicable to any states via their central measures. *L*∞(S(X ), *μω*) then describes the observable algebra that distinguishes sectors in *ω* and is ∗-isomorphic to <sup>Z</sup>*ω*(<sup>X</sup> ). The ∗-isomorphism *κω* := *κμω* : *L*∞(S(X ), *μω*) → <sup>Z</sup>*ω*(<sup>X</sup> ), defined by

$$<\langle \Omega\_{\omega} | \kappa\_{\omega}(f) \pi\_{\omega}(X) \Omega\_{\omega} \rangle = \int\_{S(X)} f(\rho) \, \underset{\omega}{\mathcal{X}}(\rho) \, d\mu\_{\omega}(\rho) \tag{21}$$

for all *X* ∈ X and *f* ∈ *L*∞(S(X ), *μω*), justifies this statement. By the definition, all elements of the center <sup>Z</sup>*ω*(<sup>X</sup> ) of *πω*(<sup>X</sup> ) are compatible with those of *πω*(<sup>X</sup> ). The following theorem is also shown.

**Theorem 7** ([31] (Theorem 4.2.5))**.** *Let ω be a state on* X *and μ an orthogonal measure with barycenter ω corresponding to a maximal abelian von Neumann subalgebra (MASA) of πω*(<sup>X</sup> )*. Then, μ is pseudosupported by the set* S*e*(<sup>X</sup> ) *of pure states on* X *. If* X *is separable, then μ is supported by* S*e*(<sup>X</sup> )*.*

An orthogonal measure corresponding to a MASA of *πω*(<sup>X</sup> ) gives an irreducible decomposition of the state. In general, MASA of *πω*(<sup>X</sup> ) is not unique. The situation where MASA of *πω*(<sup>X</sup> ) is unique is special. This is the reason why the unitary equivalence class of a pure state is not appropriate for a unit of the state space. It is known that *πω*(<sup>X</sup> ) is a type I von Neumann algebra if *πω*(<sup>X</sup> ) is abelian. The following theorem characterizes such a situation in the context of orthogonal decompositions of states.

**Theorem 8** ([31] (Theorem 4.2.3))**.** *Let ω be a state on* X *, and P the projection operator on* H*ω whose range is πω*(<sup>X</sup> )Ω*<sup>ω</sup>. The following conditions are equivalent:* (1) *πω*(<sup>X</sup> ) *is abelian;* (2)*<sup>P</sup>πω*(<sup>X</sup> )*Pgeneratesanabelianalgebra.*
