**Appendix A. Operator Algebra**

We introduce the basic facts on operator algebras. See [26,30,31,34–37] for more details on operator algebras. A set X is called a C∗-algebra if it satisfies the following conditions: (1) X is a Banach space over C.

(2) X is a ∗-algebra, i.e., it is an algebra with involution. The involution ∗ : X→X satisfies (*aX* + *bY*)∗ = *aX*¯ ∗ + ¯*bY*<sup>∗</sup>, (*XY*)∗ = *Y*<sup>∗</sup>*X*<sup>∗</sup>, and *X*∗∗ := ( *X*∗)∗ = *X* for all *a*, *b* ∈ C and *X*,*Y* ∈ X .

(3) The norm of X satisfies *X*∗*X* = *X*<sup>2</sup> for all *X* ∈ X .

We assume that C∗-algebras are unital.

Let X and Y be C∗-algebras. A map *j* : X→Y is called a ∗-homomorphism if it satisfies the following conditions:

(*i*) *j*(*aX*1 + *bX*2) = *aj*(*<sup>X</sup>*1) + *bj*(*<sup>X</sup>*2) for all *a*, *b* ∈ C and *X*1, *X*2 ∈ X .

(*ii*) *j*(*<sup>X</sup>*1*X*2) = *j*(*<sup>X</sup>*1)*j*(*<sup>X</sup>*2) for all *X*1, *X*2 ∈ X . (*iii*) *j*(*X*∗) = *j*(*X*)∗ for all *X* ∈ X . (*iv*)*j*(1)= 1.

A ∗-homomorphims *β* of X is called a ∗-automorphism of X if there exists a ∗- homomorphims *γ* of X such that *β* ◦ *γ* = idX and *γ* ◦ *β* = idX . *Aut*(X ) denotes the set of automorphisms of X . A ∗-homomorphism and a ∗-automorphism are simply called a homomorphism and an automorphism, respectively.

Let *ω* be a linear functional on X .

(*i*) *ω* is positive if *ω*(*X*<sup>∗</sup>*X*) ≥ 0 for all *X* ∈ X .

1.

(*ii*) *ω* is normalized if *ω*(1) =

X ∗ denotes the set of (complex) linear functionals on X . X ∗+ denotes the set of positive linear functionals on X . A linear functional on X is called a state on X if it is positive and normalized. S(X ) denotes the set of states on X . A state *ω* on X is faithful if *ω*(*X*<sup>∗</sup>*X*) = 0 implies *X* = 0. A C∗-algebra W is called a *W*∗-algebra if it is the dual of a Banach space W∗, called the predual of W. The second dual X ∗∗ = (X ∗)∗ of a C∗-algebra X is a *W*∗-algebra and is called the universal enveloping algebra of X . A *W*∗-algebra W is said to be *σ*-finite if it admits at most countably many orthogonal projections. A positive linear functional *ϕ* on W is said to be normal if {*ϕ*(*<sup>A</sup>γ*)}*γ*∈<sup>Γ</sup> converges to *ϕ*(*A*) for all non-decreasing nets {*<sup>A</sup>γ*}*γ*∈<sup>Γ</sup> of positive operators in W convergen<sup>t</sup> to a positive operator *A* ∈ W. A positive linear functional *ϕ* on W is normal if and only if *ϕ* ∈ W∗. *B*(H) denotes the set of bounded linear operators on a Hilbert space H. A *W*∗-algebra M is called a von Neumann algebra on a Hilbert space H if it is a subset of *<sup>B</sup>*(H), and the involution of M coincides with the adjoint operation on *<sup>B</sup>*(H). The predual M∗ of a von Neumann algebra M on a Hilbert space H satisfies

$$\mathcal{M}\_{\*} = \{ \varphi \in \mathcal{M}^{\*} | \exists \rho \in T(\mathcal{H}) \text{ s.t. } \varphi(M) = \text{Tr}[M\rho] \text{ for all } M \in \mathcal{M} \}, \tag{A1}$$

where *T*(H) denotes the set of trace-class operators on H.

For every state *ω* on X , there exist a Hilbert space H*<sup>ω</sup>*, a representation *πω* of X on H*ω* and a unit vector Ω*ω* of H*ω* such that

$$
\omega(X) = \langle \Omega\_{\omega} | \pi\_{\omega'}(X) \Omega\_{\omega'} \rangle, \quad X \in \mathcal{X}, \tag{A2}
$$

and H*ω* = *πω*(<sup>X</sup> )<sup>Ω</sup>*<sup>ω</sup>*. Here, a map *π* : X → *B*(H) is called a representation of X on a Hilbert space H if it satisfies *π*(*aX* + *bY*) = *aπ*(*X*) + *bπ*(*Y*), *π*(*XY*) = *<sup>π</sup>*(*X*)*π*(*Y*), and *π*(*X*<sup>∗</sup>) = *π*(*X*)<sup>∗</sup> for all *a*, *b* ∈ C and *X*,*Y* ∈ X . The triple (*πω*, H*<sup>ω</sup>*, <sup>Ω</sup>*ω*) is called the GNS representation of *ω* and is unique up to unitary equivalence.

For any subset *S* of *<sup>B</sup>*(H), we define the commutant *S* of *S* by *S* = {*A* ∈ *B*(H) | ∀*B* ∈ *S*, *AB* = *BA*} and the double commutant *S* of *S* by *S* = (*S*). *πω*(<sup>X</sup> ) and *πω*(<sup>X</sup> ) are then von Neumann algebras on H*<sup>ω</sup>*.

### **Appendix B. The Proof of Theorem 1**

First, we present theorems used to show Theorem 1.

**Theorem A1** ([24–27,31])**.** *Let* X *be a C*∗*-algebra and* H *a Hilbert space. For every CP map T* : X → *<sup>B</sup>*(H)*, there exist a Hilbert space* K*, a representation π of* X *on* K*, and V* ∈ *<sup>B</sup>*(H, K) *such that*

$$T(X) = V^\* \pi(X) V \tag{A3}$$

*for all X* ∈ X *, and that* K = span(*π*(<sup>X</sup> )*V*H)*. If* X *and* H *are separable, then so is* K*.*

The triplet (*<sup>π</sup>*, K, *V*) is called a Stinespring representation of *T*, and is unique up to unitary equivalence.

**Theorem A2** ([26] (Chapter IV, Theorem 5.5))**.** *Let* M1 *and* M2 *be von Neumann algebras on Hilbert spaces* H1 *and* H<sup>2</sup>*, respectively. If π is a normal homomorphism of* M1 *onto* M<sup>2</sup>*, then*

*there exist a Hilbert space* L*, a projection E of* M1 ⊗ *<sup>B</sup>*(L)*, and an isometry U of <sup>E</sup>*(H1 ⊗ L) *onto* H2 *such that*

$$
\pi(M) = \mathsf{U}j\_E(M \otimes 1)\mathsf{U}^\* \tag{A4}
$$

*for all M* ∈ M<sup>1</sup>*, where jE* : *<sup>B</sup>*(H1 ⊗ L) → *<sup>E</sup>B*(H1 ⊗ L)*E is defined by jE*(*X*) = *EXE for all X* ∈ *<sup>B</sup>*(H1 ⊗ L)*.* M1 ⊗ C1 *is then a multiplicative domain of jE.*

As a corollary of Theorem A2, the following holds:

**Corollary A1.** *Let* H1 *and* H2 *be Hilbert spaces. If π is a normal homomorphism of <sup>B</sup>*(H1) *onto <sup>B</sup>*(H2)*, then there exist a Hilbert space* K *and a unitary W of* H1 ⊗ K *onto* H2 *such that*

$$
\pi(X) = \mathcal{W}(X \otimes 1)\mathcal{W}^\* \tag{A5}
$$

*for all X* ∈ *<sup>B</sup>*(H1)*.*

Let X and Y be C∗-algebras. We define a partial order *T*1 ≤ *T*2 on CP(X , Y) by *T*2 − *T*1 ∈ CP(X , Y).

**Theorem A3** ([25] (Theorem 1.4.2))**.** *Let T*1, *T*2 *be elements of* CP(X , *B*(H)) *such that T*1 ≤ *T*2*, and* (*<sup>π</sup>*, K, *V*) *is the Stinespring representation of T*2*. There exists a positive operator R of π*(<sup>X</sup> ) *such that*

$$T\_1(X) = V^\* R \pi(X) V \tag{A6}$$

*for all X* ∈ X *.*

> By using the above theorems, we show Theorem 1.

**Proof of Theorem 1.** Put *P* = *<sup>T</sup>*(1). Suppose *P* = 0 without loss of generality. We define a unital normal CP map *T* on *B*(H) by

$$T'(X) = \frac{1}{||P||}T(X) + \left(1 - \frac{P}{||P||}\right)^{\frac{1}{2}}X\left(1 - \frac{P}{||P||}\right)^{\frac{1}{2}}\tag{A7}$$

for all *X* ∈ *<sup>B</sup>*(H). By Theorem A1, there exist a separable Hilbert space K, a normal representation *π* of X on K, and an isometry *V* ∈ *<sup>B</sup>*(H, K) such that K = span(*π*(<sup>X</sup> )*V*H) and that

$$T'(X) = (V')^\* \pi'(X) V' \tag{A8}$$

for all *X* ∈ *<sup>B</sup>*(H). Since

$$\frac{1}{||P||}T(X^\*X) \le T'(X^\*X) \tag{A9}$$

for all *X* ∈ *<sup>B</sup>*(H), by Theorem A3, there exists a positive operator *R* of *π*(<sup>X</sup> ) such that

$$\frac{1}{\|P\|}T(X) = (V')^\*\pi'(X)R'V'\tag{A10}$$

for all *X* ∈ *<sup>B</sup>*(H). By Corollary A1, there exist a separable Hilbert space L1 and a unitary operator *W* ∈ *<sup>B</sup>*(H⊗L1, K) such that

$$
\pi'(X) = \mathcal{W}'(X \otimes 1)\mathcal{W}'^\* \tag{A11}
$$

for all *X* ∈ *<sup>B</sup>*(H). There then exists a positive operator *R* on L1 such that *RW* = *W*(1 ⊗ *<sup>R</sup>*).

Let L2 be an infinite-dimensional separable Hilbert space, *v* a unit vector in L2, and *y* a unit vector in L1. We define an isometry *U* : H ⊗ C*y* ⊗ C*v* →H⊗L1 ⊗ L2 by

$$\mathcal{U}l\_0(\mathbf{x}\otimes\mathbf{y}\otimes\boldsymbol{\upsilon})=(\mathcal{W}')^\*V'\mathbf{x}\otimes\boldsymbol{\upsilon}\tag{A12}$$

for all *x* ∈ H. Since H ⊗ C*y* ⊗ C*v* and *<sup>U</sup>*0(H ⊗ C*y* ⊗ C*v*) satisfy dim((H ⊗ C*y* ⊗ <sup>C</sup>*v*)<sup>⊥</sup>) = dim((*<sup>U</sup>*0(H ⊗ C*y* ⊗ <sup>C</sup>*v*))<sup>⊥</sup>) as subspaces of H⊗L1 ⊗ L2, there exists a unitary operator *U* on H⊗L1 ⊗ L2 such that *<sup>U</sup>*|H⊗<sup>C</sup>*y*⊗C*v* = *U*0. We put K = L1 ⊗ L2 and *ξ* = *y* ⊗ *v*, and define a positive operator *R* on K by *R* = *PR* ⊗ 1. For every *X* ∈ *B*(H) and *x*1, *x*2 ∈ H, we obtain

$$\begin{split} \langle \mathbf{x}\_{1} | T(X) \mathbf{x}\_{2} \rangle &= \| P \| \langle \mathbf{x}\_{1} | (V')^{\*} W' (X \otimes R^{\prime\prime}) (W')^{\*} V' \mathbf{x}\_{2} \rangle \\ &= \| P \| \langle (W')^{\*} V' \mathbf{x}\_{1} \otimes v | (X \otimes R^{\prime\prime} \otimes 1) [(W')^{\*} V' \mathbf{x}\_{2} \otimes v] \rangle \\ &= \langle \mathcal{U} (\mathbf{x}\_{1} \otimes y \otimes v) | (X \otimes R) [\mathcal{U} (\mathbf{x}\_{2} \otimes y \otimes v)] \rangle \\ &= \operatorname{Tr} [\mathcal{U}^{\*} (X \otimes R) \mathcal{U} (| \mathbf{x}\_{2} \rangle \langle \mathbf{x}\_{1} | \otimes | \mathbf{g}' \rangle \langle \mathbf{g}' |)] \\ &= \operatorname{Tr} [\operatorname{Tr}\_{\mathcal{K}} [\mathcal{U}^{\*} (X \otimes R) \mathcal{U} (1 \otimes | \mathbf{g}' \rangle \langle \mathbf{g}' |)] | \mathbf{x}\_{2} \rangle \langle \mathbf{x}\_{1} |] \\ &= \langle \mathbf{x}\_{1} | \operatorname{Tr}\_{\mathcal{K}} [\mathcal{U}^{\*} (X \otimes R) \mathcal{U} (1 \otimes | \mathbf{g}' \rangle \langle \mathbf{g}' |)] | \mathbf{x}\_{2} \rangle, \end{split} \tag{A13}$$

which completes the proof of (1).

Next, we show (2). By Theorem A1, there exist a separable Hilbert space K1, a normal representation *π* of X on K and *V* ∈ *<sup>B</sup>*(H, K) such that K1 = span(*π*(<sup>X</sup> )*V*H) and that

$$T(X) = V^\* \pi(X) V \tag{A14}$$

for all *X* ∈ *<sup>B</sup>*(H). By Corollary A1, there exist a separable Hilbert space K2 and a unitary operator *W* ∈ *<sup>B</sup>*(K1, H⊗K2) such that

$$
\pi(X) = \mathcal{W}(X \otimes 1)\mathcal{W}^\* \tag{A15}
$$

for all *X* ∈ *<sup>B</sup>*(H). Let {*yi*}dim(K2) *i*=1 be a complete orthonormal system of K2. For every 1 ≤ *i* ≤ dim(K2), we define *Ki* ∈ *B*(H) by

$$<\langle \mathbf{x}\_1 | \mathbf{K}\_i \mathbf{x}\_2 \rangle = \langle \mathbf{x}\_1 \otimes y\_i | W^\* V \mathbf{x}\_2 \rangle \tag{A16}$$

for all *x*1, *x*2 ∈ H. For every 1 ≤ *i* ≤ dim(K2), *X* ∈ *B*(H) and *x*1, *x*2 ∈ H, we have

$$
\begin{split}
\langle x\_{1}|K\_{i}^{\*}XK\_{i}x\_{2}\rangle &= \langle K\_{i}x\_{1}|XK\_{i}x\_{2}\rangle = \sum\_{j=1}^{\dim(\mathcal{H})} \langle K\_{i}x\_{1}|z\_{j}\rangle \langle z\_{j}|XK\_{i}x\_{2}\rangle \\ &= \sum\_{j=1}^{\dim(\mathcal{H})} \langle K\_{i}x\_{1}|z\_{j}\rangle \langle X^{\*}z\_{j}|K\_{i}x\_{2}\rangle \\ &= \sum\_{j=1}^{\dim(\mathcal{H})} \langle W^{\*}V\mathbf{x}\_{1}|z\_{j}\otimes y\_{i}\rangle \langle X^{\*}z\_{j}\otimes y\_{i}|W^{\*}V\mathbf{x}\_{2}\rangle \\ &= \sum\_{j=1}^{\dim(\mathcal{H})} \langle W^{\*}V\mathbf{x}\_{1}|(|z\_{j}\rangle\langle z\_{j}|\otimes|y\_{i}\rangle\langle y\_{i}|)(X\otimes 1)W^{\*}V\mathbf{x}\_{2}\rangle \\ &= \langle W^{\*}V\mathbf{x}\_{1}|(X\otimes|y\_{i}\rangle\langle y\_{i}|)W^{\*}V\mathbf{x}\_{2}\rangle = \langle \mathbf{x}\_{1}|V^{\*}W(X\otimes|y\_{i}\rangle\langle y\_{i}|)W^{\*}V\mathbf{x}\_{2}\rangle.
\end{split}
\tag{A17}
$$

Therefore, for every *X* ∈ *B*(H) and *x*1, *x*2 ∈ H, we obtain

$$\begin{split} \langle \mathbf{x}\_{1} | T(\mathbf{X}) \mathbf{x}\_{2} \rangle &= \langle \mathbf{x}\_{1} | V^{\*} W(\mathbf{X} \otimes 1) W^{\*} V \mathbf{x}\_{2} \rangle \\ &= \sum\_{i=1}^{\dim(\mathcal{K}\_{2})} \langle \mathbf{x}\_{1} | V^{\*} W(\mathbf{X} \otimes |y\_{i}\rangle \langle y\_{i}|) W^{\*} V \mathbf{x}\_{2} \rangle \\ &= \sum\_{i=1}^{\dim(\mathcal{K}\_{2})} \langle \mathbf{x}\_{1} | K\_{i}^{\*} X K\_{i} \mathbf{x}\_{2} \rangle = \langle \mathbf{x}\_{1} | \left( \sum\_{i=1}^{\dim(\mathcal{K}\_{2})} K\_{i}^{\*} X K\_{i} \right) \mathbf{x}\_{2} \rangle, \end{split} \tag{A18}$$

which completes the proof of (2).

The proof of (1) in the above theorem refers to that of [18] (Theorem 5.1). The results of this appendix are related to the theory of Hilbert modules [38–43].
