*4.1. Definition*

Since the investigation [17] by Davies and Lewis, instruments have been defined on the predual of a von Neumann algebra. In order to define its C∗-algebraic generalization, the dual space of a C∗-algebra is too big in general. When a von Neumann algebra M on a Hilbert space K is not finite-dimensional, the predual M∗ of M does not coincide with M∗, i.e., M∗ M∗. In addition, in the case where all physically relevant states are contained in M∗, the whole space M∗ is not needed. This does not depend on whether M is treated as a C∗-algebra or a von Neumann algebra. In the C∗-algebraic formulation introduced here, we can naturally use M∗ as a domain of instruments.

Let X be a C∗-algebra and *π* a representation of X on a Hilbert space H. Let M be a von Neumann algebra on a Hilbert space K. Z(M) denotes the center of M. We define the subset *<sup>V</sup>*(*π*) of X ∗ by

$$V(\pi) = \{ \varphi \in \mathcal{X}^\* \mid \exists \rho \in (\pi(\mathcal{X})'')\_\*, \forall X \in \mathcal{X}, \rho(X) = \rho(\pi(X)) \}. \tag{22}$$

A subspace L of X ∗ is said to be central if there exists a central projection *C* of X ∗∗, i.e., *C* ∈ Z(X ∗∗), such that L = *C*X ∗. Central subspaces of X ∗ are characterized as closed invariant subspaces (see [26] (Chapter III, Theorem 2.7)). A central subspace L(= *C*X ∗) is said to be *σ*-finite if its dual L∗(<sup>∼</sup>= *C*X ∗∗) is a *σ*-finite *W*∗-algebra. For every *M*1, *M*2 ∈ V∗ and *ρ* ∈ V, we define *M*1*ρ*, *ρM*2, *M*1*ρM*2 ∈ V by

$$
\langle M\_{\prime}M\_{1}\rho \rangle = \langle MM\_{1}, \rho \rangle,\tag{23}
$$

$$
\langle M, \rho M\_2 \rangle = \langle M\_2 M, \rho \rangle,\tag{24}
$$

$$
\langle M\_{\prime}M\_{1}\rho M\_{2}\rangle = \langle M\_{2}MM\_{1},\rho\rangle,\tag{25}
$$

respectively, for all *M* ∈ V∗. The usefulness of the central subspace can be seen in the following example:

**Example 1** (See [26] (Chapter III) for example)**.** (1) *Let* X *be a C*∗*-algebra and π a representation of* X *on a Hilbert space* H*. There exists a central projection <sup>C</sup>*(*π*) *of* X ∗∗ *such that*

$$\mathcal{V}(\pi) = \mathbb{C}(\pi)\mathcal{X}^\* = \{ \mathbb{C}(\pi)\varrho \mid \varrho \in \mathcal{X}^\* \} = \{ \varrho \in \mathcal{X}^\* \mid \mathbb{C}(\pi)\varrho = \varrho \}.\tag{26}$$

(2) *Let* M *be a von Neumann algebra on a Hilbert space* H*. There exists a central projection C of* M∗∗ *such that* M∗ = *C*M∗*.*

The following theorem is known.

**Theorem 9.** *Let* X *be a C*∗*-algebra and π*1 *and π*2 *representations of* X *on Hilbert spaces* H1 *and* H<sup>2</sup>*, respectively. The following conditions are equivalent:*


The former part of this theorem is shown in [26] (Chapter III, Proposition 2.12). We can show the latter part in a similar way.

We shall define instruments in terms of central subspaces in the fully C∗-algebraic setting. Let M and N be *W*∗-algebras. *<sup>P</sup>*(M∗, N∗) denotes the set of positive linear maps of M∗ into N∗. In addition, for any Banach space L, ·, · denotes the pairing of L∗ and L.

**Definition 7** (instrument)**.** *Let* Vin *and* Vout *be σ-finite central subspaces of C*∗*-algebras* X *and* Y*, respectively, and* (*<sup>S</sup>*, F) *a measurable space.* I *is called an instrument for* (X , Vin, Y, Vout, *S*) *if it satisfies the following three conditions:*


$$
\langle \mathcal{M}, \mathcal{Z}(\cup\_j \Delta\_j)\rho \rangle = \sum\_{j=1}^{\infty} \langle \mathcal{M}, \mathcal{Z}(\Delta\_j)\rho \rangle. \tag{27}
$$

When X = Y, an instrument I for (X , Vin, Y, Vout, *S*) is called that, for (X , Vin, Vout, *<sup>S</sup>*). Furthermore, when Vin = Vout = V, an instrument I for (X , Vin, Vout, *S*) is called for (X , V, *<sup>S</sup>*). In particular, an instrument for (M,M∗, *S*) is called for (M, *<sup>S</sup>*). For every instrument I for (Vin, Vout, *S*) and normal state *ϕ* on V∗in, we define the probability measure I *ϕ* on (*<sup>S</sup>*, F) by I *ϕ*(Δ) = I(Δ)*ϕ* for all Δ ∈ F. For every instrument I for (X , Vin, Y, Vout, *<sup>S</sup>*), the dual map I∗ : V∗out×F →V∗inof I is defined by

$$
\langle M, \mathcal{T}(\Delta)\rho \rangle = \langle \mathcal{T}^\*(M, \Delta), \rho \rangle \tag{28}
$$

for all *ρ* ∈ Vin, *M* ∈ V∗out and Δ ∈ F.

**Definition 8.** *An instrument* I *for* (X , Vin, Y, Vout, *S*) *is said to be completely positive (CP) if the map* V∗out *M* → I∗(*<sup>M</sup>*, Δ) ∈ V∗in *is CP for all* Δ ∈ F*.*

For every map J : V∗out ×F →V∗in satisfying the following three conditions, there uniquely exists an instrument I for (X , Vin, Y, Vout, *S*) such that J = I∗:

(1) For every Δ ∈ F, the map V∗out *M* → J (*<sup>M</sup>*, Δ) ∈ V∗in is normal, positive, and linear. (2) J (1, *S*) = 1.

(3) For every *ρ* ∈ Vin, *M* ∈ V∗out and mutually disjoint sequence {<sup>Δ</sup>*j*}*j*∈<sup>N</sup> of F,

$$
\langle \mathcal{J}(M, \cup\_{\dot{\jmath}} \Delta\_{\dot{\jmath}}), \rho \rangle = \sum\_{j=1}^{\infty} \langle \mathcal{J}(M, \Delta\_{\dot{\jmath}}), \rho \rangle. \tag{29}
$$

From now on, I denotes the dual map I∗ of an instrument I for (X , Vin, Y, Vout, *<sup>S</sup>*). The dual map of an instrument for (X , Vin, Y, Vout, *S*) is also called an instrument for (X , Vin, Y, Vout, *<sup>S</sup>*).

### *4.2. Central Decomposition of State via CP Instrument*

Let V be a *σ*-finite central subspace of the dual space of a C∗-algebra X and (*<sup>S</sup>*, F) a measurable space. Let *C* : F→Z(V∗) be a projection valued measure (PVM). A CP instrument I*C* for (X , V, *S*) is defined by

$$\mathcal{I}\_{\mathbb{C}}(\Delta)\rho = \mathbb{C}(\Delta)\rho\tag{30}$$

for all *ρ* ∈ V and Δ ∈ F. **Theorem 10.** I*C satisfies the following conditions:*


$$\mathcal{L}\_{\mathbb{C}}(\Delta)\mathcal{Z}\_{\mathbb{C}}(\Gamma) = \mathcal{Z}\_{\mathbb{C}}(\Delta \cap \Gamma) \tag{31}$$

*for all* Δ, Γ ∈ F*.*

(3) *For every ρ* ∈ V+ := V∩X ∗+ *and* Δ ∈ F*,* <sup>I</sup>*C*(Δ)*ρ and* <sup>I</sup>*C*(Δ*<sup>c</sup>*)*ρ are mutually disjoint.*

(4) *For every* Δ ∈ F*,* <sup>I</sup>*C*(Δ) *is* V∗*-bimodule map, i.e., for every* Δ ∈ F*, ρ* ∈ V *and M*1, *M*2 ∈ V∗*,*

$$
\mathcal{L}\_{\mathbb{C}}(\Delta)(M\_1 \rho M\_2) = M\_1(\mathcal{L}\_{\mathbb{C}}(\Delta)\rho)M\_2. \tag{32}
$$

*Conversely, if an instrument* I *for* (V, *S*) *satisfies the conditions* (2) *and* (4)*, then there exists a spectral measure C* : F→Z(V∗) *such that* I = I*C.*

**Proof.** We can easily check (1), (2), and (4). (3) is shown by using Theorem 9.

The converse is also obvious as follows. We define a map *C* : F→V∗ by *C*(Δ) = I(1, Δ) for all Δ ∈ F. For every Δ ∈ F, *ρ* ∈ V and *M* ∈ V∗, we have

$$
\langle M, \mathcal{Z}(\Delta)\rho \rangle = \langle 1, \mathcal{Z}(\Delta)(\rho M) \rangle = \langle \mathbb{C}(\Delta), \rho M \rangle = \langle MC(\Delta), \rho \rangle. \tag{33}
$$

*<sup>M</sup>*, <sup>I</sup>(Δ)*ρ* = *C*(Δ)*<sup>M</sup>*, *ρ*is also shown in the same way. Therefore, we have [*C*(Δ), *<sup>M</sup>*], *ρ* = 0 for all Δ ∈ F, *ρ* ∈ V and *M* ∈ V∗. When *ϕ* is normal faithful state on V∗ and *ρ* = *ϕ*([*C*(Δ), *<sup>M</sup>*])<sup>∗</sup>, ([*C*(Δ), *<sup>M</sup>*])∗[*C*(Δ), *<sup>M</sup>*], *ϕ* = 0, so that [*C*(Δ), *M*] = 0 for all Δ ∈ F and *M* ∈ V∗. We obtain *C*(Δ) ∈ Z(V∗) for all Δ ∈ F.

By the conditions (2) and (4),

$$
\begin{split}
\langle\mathcal{C}(\Delta\cap\Gamma),\rho\rangle &= \langle1,\mathcal{Z}(\Delta\cap\Gamma)\rho\rangle = \langle1,\mathcal{Z}(\Delta)\mathcal{Z}(\Gamma)\rho\rangle = \langle\mathcal{C}(\Delta),\mathcal{Z}(\Gamma)\rho\rangle \\ &= \langle1,\mathcal{Z}(\Gamma)(\rho\mathcal{C}(\Delta))\rangle = \langle\mathcal{C}(\Gamma),\rho\mathcal{C}(\Delta)\rangle = \langle\mathcal{C}(\Delta)\mathcal{C}(\Gamma),\rho\rangle.
\end{split}
\tag{34}
$$

Thus, *C* : F→Z(V∗) is a PVM, and we have I = I*C*.

An instrument I for (X , Vin, Y, Vout, *S*) is said to be subcentral if, for every *ρ* ∈ Vin,+ and Δ ∈ F, <sup>I</sup>*C*(Δ)*ρ* and <sup>I</sup>*C*(Δ*<sup>c</sup>*)*ρ* are mutually disjoint. The condition (3) in Theorem 10 is a special case of the subcentrality of instruments. P(X , V) denotes the subset {I*C*|*<sup>C</sup>* : F→Z(V∗) is a PVM.} of the set of instruments defined on V. An instrument I for (X , V, *S*) is said to be central if it is an element of P(X , V) and is the maximum in P(X , V), where the maximum is due to the (pre)order ≺ on instruments defined as follows: For instruments I1, I2 for (X , Vin, Y, Vout, *<sup>S</sup>*1) and (X , Vin, Y, Vout, *<sup>S</sup>*2), respectively, I1 ≺ I2 if <sup>I</sup>1(F)*ρ* ⊂ <sup>I</sup>2(F)*ρ* for all *ρ* ∈ S(X ) ∩ Vin, where <sup>I</sup>*i*(F*i*)*ρ*, *i* = 1, 2, is the subset of (Vin)+ defined by <sup>I</sup>*i*(F*i*)*ρ* = {I*i*(<sup>Δ</sup>*i*)*ρ* | Δ*i* ∈ F*i*}. By Theorem 10, we have the following theorem.

**Theorem 11.** *Let* (*<sup>S</sup>*, F) *be a measurable space,* V *a σ-finite central subspace of the dual of a C*∗*-algebra* X *, and C* : F→Z(V∗) *a PVM.* I*C is central if and only if the abelian W*∗*-algebra generated by* {*C*(Δ)|Δ ∈ F} *is isomorphic to* <sup>Z</sup>(V∗)*.*

### **5. Operational Requirement and Macroscopic Distinguishability**

In this section, we discuss the characterization of CP instruments. We deepen our conceptual understanding of measurement theory by referring to the mathematics of sector theory. In sector theory, we explained that a sector is a macroscopic unit. As an application of sector theory to measurement theory, we follow the macroscopic distinction made by the disjointness of states. That is, in contrast to the usual understanding of measurement, our understanding is that a measurement is a physical process that realizes macroscopically distinguishable situations when different values are output. In past investigations, the concept of CP instrument has been justified by clarifying the statistical properties that a measuring apparatus should satisfy from an operational point of view in the (extended) Schrödinger picture. We first review this here. Next, we proceed to characterize CP instruments from the perspective of the macroscopic distinguishability of states, which is related to sector theory.

Here, we assume that the system **S** is described by a C∗-algebra X and that Vin a *σ*-finite central subspace of X ∗. We consider a measuring apparatus **<sup>A</sup>**(*x*) with output variable *x* to measure the system **S**, where *x* takes values in a measurable space (*<sup>S</sup>*, F). In the following, we consider three assumptions from an operational point of view. They are modified from [19,32] in the C∗-algebraic setting.

### **Assumption 1. <sup>A</sup>**(*x*) *statistically specifies the following two components:*

(1)*the probability measure* Pr{*x* ∈ <sup>Δ</sup>*<sup>ω</sup>*}*,* Δ ∈ F*, on* (*<sup>S</sup>*, F)*for every initial state ω* ∈ S(X ) ∩ Vin*.* (2) *the state <sup>ω</sup>*{*x*∈<sup>Δ</sup>} *(on a C*∗*-algebra* Y*) after the measurement under the condition that ω is an initial state and output values not contained in* Δ *are ignored. For every ω* ∈ S(X ) ∩ Vin *and* Δ ∈ F*, <sup>ω</sup>*{*x*∈<sup>Δ</sup>} *is unique whenever* Pr{*x* ∈ <sup>Δ</sup>*ω*} = 0*, or is indefinite otherwise.*

From now on, we consider only the case of X = Y for simplicity. The joint probability distribution of the successive measurement of **<sup>A</sup>**(*x*) and **<sup>A</sup>**(*y*) in this order in a state *ω* ∈ Vin ∩ S(X ) is given by

$$\Pr\{\mathbf{x}\in\Delta,\mathbf{y}\in\Gamma\|\omega\}=\Pr\{\mathbf{x}\in\Delta\|\omega\}\Pr\{\mathbf{y}\in\Gamma\|\omega\_{\{\mathbf{x}\in\Delta\}}\}\tag{35}$$

for all Δ ∈ F and Γ ∈ F.

**Assumption 2.** *For every* Δ ∈ F*, measuring apparatus* **<sup>A</sup>**(*y*) *whose output variable y takes values in a measurable space* (*S*, F)*, and* Γ ∈ F*, the map* S(X ) ∩ Vin *ω* → Pr{*x* ∈ Δ, *y* ∈ <sup>Γ</sup>*ω*} *is affine, that is,*

$$\Pr\{\mathbf{x}\in\Delta,\mathbf{y}\in\Gamma\|\|a\omega\_1+(1-a)\omega\_2\}=a\Pr\{\mathbf{x}\in\Delta,\mathbf{y}\in\Gamma\|\omega\_1\}+(1-a)\Pr\{\mathbf{x}\in\Delta,\mathbf{y}\in\Gamma\|\omega\_2\}\tag{36}$$

*for all α* ∈ [0, 1] *and ω*1, *ω*2 ∈ S(X ) ∩ Vin*.*

The affine property of joint distributions of successive measurements characterizes the instrument as shown in the following theorem.

**Theorem 12.** *Let* **<sup>A</sup>**(*x*) *be a measuring apparatus satisfying Assumption 1. Suppose that there exists a σ-finite central subspace* Vout *of* X *such that* {*<sup>ω</sup>*{*x*∈<sup>Δ</sup>}|*<sup>ω</sup>* ∈ S(X ) ∩ Vin, Δ ∈ F} ⊂ Vout*. The following conditions are equivalent:*

(1) **<sup>A</sup>**(*x*) *satisfies Assumption 2.*

(2) *There exists an instrument* I *for* (Vin, Vout, *S*) *such that*

$$\Pr\{\mathfrak{x}\in\Delta\|\omega\}=\|\mathcal{T}(\Delta)\omega\|\tag{37}$$

*for all ω* ∈ S(X ) ∩ Vin *and* Δ ∈ F*, and that*

$$
\omega\_{\{\mathbf{x}\in\Lambda\}} = \frac{\mathcal{Z}(\Lambda)\omega}{||\mathcal{Z}(\Lambda)\omega||}\tag{38}
$$

*whenever* Pr{*x* ∈ <sup>Δ</sup>*ω*} = 0*.*

The complete positivity of instrument is based on the general description of the dynamics of open systems. In Section 2, we discussed the dynamics of open systems state/representation-independently. We consider the following assumption that is called the trivial extendability.

**Assumption 3.** *For any quantum system* **S** *that is described by a C*∗*-algebra* Y *and does not interact with an apparatus* **<sup>A</sup>**(*x*) *nor* **S***,* **<sup>A</sup>**(*x*) *can be extended into an apparatus* **<sup>A</sup>**(*x* ) *measuring the composite system* **S** + **S** *with the following statistical properties:*

$$\Pr\{\mathbf{x}' \in \Delta \|\boldsymbol{\omega} \otimes \boldsymbol{\varphi}\} = \Pr\{\mathbf{x} \in \Delta \|\boldsymbol{\omega}\},\tag{39}$$

$$(\omega \odot \mathfrak{p})\_{\{\mathbf{x}' \in \Lambda\}} = \omega\_{\{\mathbf{x} \in \Lambda\}} \odot \mathfrak{p} \tag{40}$$

*for all ω* ∈ Vin ∩ S(X )*, ϕ* ∈W∩S(Y) *and* Δ ∈ F*, where* W *is a central subspace of* Y∗*.*

Let M and N be von Neumann algebras. For every *σ* ∈ N∗, we define a map id ⊗ *σ* : M ⊗N→M by *ρ* ⊗ *σ*, *X* = *ρ*,(id ⊗ *σ*)( *X*) for all *ρ* ∈ M∗ and *X* ∈ M ⊗ N . Athatsatisfies3isdescribedCPinstrument.

 measuring apparatus Assumption by a In the von Neumann algebraic setting, a measuring process is defined as follows.

**Definition 9** (Measuring process [19] (Definition 3.2))**.** *Let* M *be a von Neumann algebra on a Hilbert space* H*, and* (*<sup>S</sup>*, F) *a measurable space. A 4-tuple* M = ( K, *σ*, *E*, *U*) *is called a measuring process for* (M, *S*) *if it satisfies the following conditions:*

(1) K *is a Hilbert space,*

(2) *σ is a normal state on <sup>B</sup>*(K)*,*

(3) *E* : F → *B*(K) *is a spectral measure,*


$$\mathcal{L}\_{\mathbb{M}}(X,\Delta) = (\text{id}\otimes\sigma)[\![\![L^\*(X\otimes E(\Delta))\![L]\!]\!]\!]\tag{41}$$

*for all X* ∈ *B*(H) *and* Δ ∈ F*.*

As shown in [18], every CP instrument for (*B*(H), *S*) is defined by a measuring process. By contrast, in the case where M is a non-atomic injective von Neumann algebra, it is shown in [19] that there exist CP instruments for (M, *S*) which cannot be defined by any measuring processes. Furthermore, a necessary and sufficient condition for a CP instrument to be defined by a measuring process is given in [19].

In the context of measurement, we do not always care about sectors as a macroscopic unit, but we actively utilize the macroscopic distinction based on the disjointness. We introduce two kinds of subcentral lifting property for instruments as follows.

**Definition 10.** *An instrument* I *for* (X , V, *S*) *is said to have the first subcentral lifting property if there exists a central subspace* W *of the dual space of a C*∗*-algebra* Y(⊃ X ) *and an instrument* I *for* (X , V, Y, W, *S*) *satisfying the following two conditions:*

(1) *For every ω* ∈ S(X ) ∩ V *and* Δ ∈ F*,* I (Δ)*ω* ◦*–* I (Δ*<sup>c</sup>*)*<sup>ω</sup>.*

(2) *For every ω* ∈ S(X ) ∩ V*, X* ∈ X *and* Δ ∈ F*,* [I (Δ)*ω*]( *X*)=[ <sup>I</sup>(Δ)*ω*]( *<sup>X</sup>*)*.*

**Definition 11.** *An instrument* I *for* (X , V, *S*) *is said to have the second subcentral lifting property if there exists a central subspace* W *of the dual space of a C*∗*-algebra* Y(⊃ X ) *and an instrument* I *for* (Y, W, *S*) *satisfying the following two conditions:*

(1) *For every ϕ* ∈ S(Y) ∩ W *and* Δ ∈ F*,* I (Δ)*ϕ* ◦*–* I (Δ*<sup>c</sup>*)*ϕ.*

(2) *For every ω* ∈ S(X ) ∩ V*, there exists ω* ∈ S(Y) ∩ W *such that ω*(*X*) = *ω*(*X*) *and* [I (Δ)*ω*]( *Y*)=[ <sup>I</sup>(Δ)*ω*]( *Y*) *for all X*,*Y* ∈ X *and* Δ ∈ F*.*

Both subcentral lifting properties characterize the measurement obtained by restricting a measurement, which realizes the disjointness of states (after the measurement) of a larger system corresponding to different output values, to the target system. On the other hand, the difference between these two properties may be obvious from the definitions.

An instrument I for (X , Vin, Y, Vout, *S*) is said to be finite if there exists a finite subset *S*0 of *S* and a map *T* : *S*0 → *<sup>P</sup>*(Vin, Vout) such that

$$\mathcal{Z}(\Delta) = \sum\_{s \in \mathbb{S}\_0 \cap \Lambda} T(s) \tag{42}$$

for all Δ ∈ F.

**Theorem 13.** *Every finite instrument for* (X , V, *S*) *has the first subcentral lifting property and the second subcentral lifting property.*

**Proof.** Let I be a finite instrument for (X , V, *<sup>S</sup>*), a finite subset *S*0 of *S*, and a map *T* : *S*0 → *P*(V) satisfying Equation (42) for all Δ ∈ F. For every Δ ∈ F, a linear map I (Δ) : V→V⊗ *<sup>l</sup>*<sup>1</sup>(*<sup>S</sup>*0) is defined by

$$\widetilde{\mathcal{L}}(\Delta)\omega = \sum\_{s \in \mathcal{S}\_0 \cap \Lambda} T(s)\omega \otimes \delta\_s \tag{43}$$

for all *ω* ∈ V. Then, I is a finite instrument for (X , V, X ⊗min *l*∞(*<sup>S</sup>*0), V ⊗ *<sup>l</sup>*<sup>1</sup>(*<sup>S</sup>*0), *<sup>S</sup>*). Then, I satisfies I (Δ)*ω* ◦– I (Δ*<sup>c</sup>*)*ω* for all *ω* ∈ S(X ) ∩ V and Δ ∈ F. Furthermore, every *ω* ∈ S(X ) ∩ V, *X* ∈ X and Δ ∈ F, [I (Δ)*ω*](*<sup>X</sup>* ⊗ <sup>1</sup>)=[I(Δ)*ω*](*X*). Therefore, I has the first subcentral lifting property. 

Next, we define a finite instrument I for (X ⊗min *l*∞(*<sup>S</sup>*0), V ⊗ *<sup>l</sup>*<sup>1</sup>(*<sup>S</sup>*0), *S*) by

I (Δ)*ϕ* = I (Δ)(*j*(*ϕ*)) (44)

for all Δ ∈ F and *ϕ* ∈V⊗ *<sup>l</sup>*<sup>1</sup>(*<sup>S</sup>*0), where *j* : V ⊗ *<sup>l</sup>*<sup>1</sup>(*<sup>S</sup>*0) → V is a linear map defined by

$$[j(\varphi)](X) = \varphi(X \otimes 1) \tag{45}$$

for all *X* ∈ X . For every *ϕ* ∈ S(X ⊗min *l*∞(*<sup>S</sup>*0))∩(V ⊗ *<sup>l</sup>*<sup>1</sup>(*<sup>S</sup>*0)) and Δ ∈ F, <sup>I</sup>(Δ)*ϕ* ◦– <sup>I</sup>(Δ*<sup>c</sup>*)*ϕ*. For every *ω* ∈ S(X ) ∩ V, *ω* = *ω* ⊗ *δs*0 , where *s*0 ∈ *S*0 satisfies *ω*(*<sup>X</sup>* ⊗ 1) = *ω*(*X*) and [I (Δ)*ω*](*<sup>Y</sup>* ⊗ <sup>1</sup>)=[I(Δ)*ω*](*Y*) for all *X*,*Y* ∈ X and Δ ∈ F. Therefore, I has the second subcentral lifting property.

We conjecture that every *CP instrument* has both subcentral lifting properties.
