Krein Space Representations and Radon–Nikodým Theorem for Local $\alpha$-Completely Positive Maps

Abstract

In this paper, we prove a Krein space J-representation theorem for local $\alpha$-completely positive maps on locally $C^*$-algebras. Using this representation, we construct a Krein space J-representation associated with a pair of two maps $(\varphi ,\Phi )$ where $\varphi$ is a local $\alpha$-completely positive map on a locally $C^*$-algebra and $\Phi$ is a $\varphi$-map. Also, we discuss the minimality of Krein space J-representations, and as an application, we establish the Radon–Nikodým theorem for local $\alpha$-completely positive maps.

1 Introduction

The notion of locally $C^*$-algebras was first systematically studied by Inoue [12] as a generalization of $C^*$-algebras. It is known that locally $C^*$-algebras are useful for the study of non-commutative algebraic topology, pseudo differential operators and quantum field theory. Motivated by operator spaces which is regarded as quantized normed spaces, Dosiev [6] introduced local operator spaces and operator systems and proved a Stinespring representation theorem for local completely positive maps on locally $C^*$-algebras. To replace ${\mathcal {B}}({\mathcal {H}})$ in this theorem, he introduced a quantized domain to be an upward filtered family of closed subspaces $\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ with the union ${\mathcal {D}}$ which is a dense subspace of a Hilbert space ${\mathcal {H}}$. Bhat et al. [5] presented a Stinespring representation theorem for unbounded operator valued local completely positive maps on Hilbert modules over locally $C^*$-algebras.

Due to the lack of positivity in a local quantum field theory, the Gelfand-Naimark-Segal (GNS) construction associated with Hermitian linear functionals instead of positive linear functionals has been of increasing interest. The underlying space of the GNS representation associated with Hermitian linear functional is a space with an indefinite metric, and so it becomes a Krein space [3]. Antoine and Ôta [1] provided a GNS representation in a Krein space for a not necessarily positive linear functional on a unital *-algebra satisfying a suitable condition. Constantinescu and Gheondea [4] studied Hermitian kernels invariant under the action of a semigroup with involution and characterized those kernels that realize the action given by bounded operators on a Krein space. Heo et al. [7] introduced a notion of the $\alpha$-complete positivity where complete positivity is inherent in Hermitian maps in terms of the map $\alpha$. The $\alpha$-complete positivity provides an indefinite inner product, and the interplay between these two is indeed the characteristic feature of Krein spaces among all indefinite metric spaces. Heo and Ji [8] established a Radon–Nikodým type theorem for $\alpha$-completely positive maps. Heo et al. [10] also gave the Krein module representation for $\alpha$-completely positive maps between locally $C^*$-algebras. For more information of $\alpha$-complete positivity, we refer [8,9,10,11].

This paper is organized as follows. In Sect. 2, we recall some basic definitions for locality of (positive) maps on locally $C^*$-algebras and some properties of operators in the *-algebra of noncommutative continuous functions on a quantized domain. In Sect. 3, we briefly review some notions of (local) $\alpha$-completely positive maps and construct a (minimal) Krein space J-representation associated with a local $\alpha$-completely positive map from a locally $C^*$-algebra into a *-algebra of noncommutative continuous functions. In Sect. 4, we obtain a (minimal) J-representation on a Krein module over a locally $C^*$-algebra associated with two local maps. In Sect. 5, we establish a Radon–Nikodým theorem in the sense of Arveson [2] for local $\alpha$-completely positive maps.

2 Preliminaries and Notations

Recall that a seminorm p on a unital *-algebra ${\mathcal {A}}$ is said to be sub-multiplicative if $p(1_{\mathcal {A}})= 1$ and $p(ab) \le p(a)p(b)$ for all $a,b \in {\mathcal {A}}$. A sub-multiplicative seminorm p on ${\mathcal {A}}$ is called a $C^*$-seminorm if $p(a^*)=p(a)$ and $p(a^*a)=p(a)^2$ for all $a \in {\mathcal {A}}$. If $S({\mathcal {A}})$ is the set of all continuous $C^*$-seminorms on ${\mathcal {A}}$, then $S({\mathcal {A}})$ can be considered as a directed set with the order $q \preceq p$ (if $q(a) \le p(a)$ for all $a \in {\mathcal {A}}$). Throughout this paper, $S({\mathcal {A}})$ denotes a directed set of all continuous $C^*$-seminorms on ${\mathcal {A}}$, unless specified otherwise.

Definition 2.1

([12]) A locally $C^*$-algebra ${\mathcal {A}}$ is a *-algebra which is complete with respect to the locally convex topology generated by a directed set $S({\mathcal {A}})$.

We note that a locally $C^*$-algebra ${\mathcal {A}}$ can be understood as the inverse limit of $C^*$-algebras as follows: for each $p\in S({\mathcal {A}})$, the kernel $\ker (p)=\{a \in {\mathcal {A}}: p(a)=0\}$ is a closed ideal in ${\mathcal {A}}$, so that the quotient space ${\mathcal {A}}_p={\mathcal {A}}/\ker (p)$ becomes a $C^*$-algebra with the norm induced by p. We denote by $\pi _p :=\pi _p^{{\mathcal {A}}}$ the canonical quotient map from ${\mathcal {A}}$ onto ${\mathcal {A}}_p$ and by $a_p=\pi _p(a)$ the image of a under $\pi _p$ in ${\mathcal {A}}_p$. For $p \succeq q$ in $S({\mathcal {A}})$, there is a canonical surjective map $\pi _{pq}: {\mathcal {A}}_p \rightarrow {\mathcal {A}}_q$ such that $\pi _{pq}(a_p)=a_q$ for every $a_p \in {\mathcal {A}}_p$. Then the set

$$\begin{aligned} \{{\mathcal {A}}_p,~\pi _{pq}: {\mathcal {A}}_p \rightarrow {\mathcal {A}}_q, ~q \preceq p\} \end{aligned}$$

becomes an inverse system of $C^*$-algebras and the inverse limit

$$\begin{aligned} \varprojlim _{p} {\mathcal {A}}_p = \bigg \{ \{a_p\} \in \prod _{p\in S({\mathcal {A}})} {\mathcal {A}}_p : \pi _{pq}(a_p)=a_q ~~\hbox { for}\ q\preceq p\bigg \} \end{aligned}$$

is identified with the locally $C^*$-algebra ${\mathcal {A}}$ via the identification $a \longleftrightarrow \{a_p\}_p$. For more detailed study of inverse limits of $C^*$-algebras, we refer to [13].

Example 2.2

• (1) Let $C(\Omega )$ be the set of all continuous complex-valued functions on a compactly generated space $\Omega$. If we equip $C(\Omega )$ with the topology of uniform convergence on compact subsets, then $C(\Omega )$ becomes a locally $C^*$-algebra. We also see that every $C^*$-algebra is a locally $C^*$-algebra.

• (2) Let ${\mathcal {A}}$ be a locally $C^*$-algebra. The unitization ${\mathcal {A}}^1$ of ${\mathcal {A}}$ is the vector space ${\mathcal {A}}\oplus \mathbb {C}$, topologized as the direct sum and the multiplication defined as for the unitization of $C^*$-algebras. Then ${\mathcal {A}}^1$ is a unital locally $C^*$-algebra since ${\mathcal {A}}^1=\varprojlim _p {\mathcal {A}}^1_p$ where ${\mathcal {A}}^1_p$ is the unitization of a $C^*$-algebra ${\mathcal {A}}_p$.

• (3) Let ${\mathcal {A}}$ be a locally $C^*$-algebra. For each $n\in \mathbb {N}$, we denote by $M_n({\mathcal {A}})$ the locally $C^*$-algebra of $n\times n$ matrices with entries in ${\mathcal {A}}$. More precisely, let ${\mathcal {A}}$ be a locally $C^*$-algebra with a directed set $S({\mathcal {A}})$. Then for each $p\in S({\mathcal {A}})$, $M_n({\mathcal {A}}_p)$ is the $C^*$-algebra of $n\times n$ matrices with entries in ${\mathcal {A}}_p$. The $C^*$-seminorm on the $C^*$-algebra $M_n({\mathcal {A}})$ is denoted by $p^n$. Then the locally $C^*$-algebra $M_n({\mathcal {A}})$ is the inverse limit of the $C^*$-algebras $M_n({\mathcal {A}}_p)$, i.e., $M_n({\mathcal {A}})=\varprojlim _p M_n({\mathcal {A}}_p)$. $\square$

Let ${\mathcal {A}}$ be a locally $C^*$-algebra. We say that an element $a\in {\mathcal {A}}$ is local self-adjoint if $a=a^*+ x$ for some $x \in {\mathcal {A}}$ such that $p(x)=0$, where $p \in S({\mathcal {A}})$. An element $a\in {\mathcal {A}}$ is called local positive if $a=b^*b +y$ for some $b, y \in {\mathcal {A}}$ such that $p(y)=0$, where $p \in S({\mathcal {A}})$. For $p\in S({\mathcal {A}})$, we define an order relation $\ge _p$ on ${\mathcal {A}}$ by

$$\begin{aligned} a \ge _p b ~~ \Longleftrightarrow ~~ a-b \ge _p 0 ~~ \Longleftrightarrow ~~ \pi _p(a-b) \ge 0 ~~\hbox { in}\ {\mathcal {A}}_p. \end{aligned}$$

Definition 2.3

([6]) Let ${\mathcal {A}}$ and ${\mathcal {B}}$ be locally $C^*$-algebras with directed sets $S({\mathcal {A}})$ and $S({\mathcal {B}})$, respectively. A linear map $\varphi : {\mathcal {A}}\rightarrow {\mathcal {B}}$ is called

• (i) local bounded if for any $p\in S({\mathcal {A}})$, there exist $q \in S({\mathcal {B}})$ and a constant $c_{p,q}>0$ such that $q(\varphi (a)) \le c_{p,q} \cdot p(a)$ for all $a \in {\mathcal {A}}$;

• (ii) local contractive if for any $p\in S({\mathcal {A}})$, there exists $q \in S({\mathcal {B}})$ such that $q(\varphi (a)) \le p(a)$ for all $a \in {\mathcal {A}}$;

• (iii) local positive if for any $p\in S({\mathcal {A}})$, there exists $q \in S({\mathcal {B}})$ such that $\varphi (a) \ge _q 0$ for all $a \ge _p 0$;

• (iv) local completely bounded if for any $p\in S({\mathcal {A}})$, there exist $q \in S({\mathcal {B}})$ and a constant $c_{p,q}>0$ such that $q^n([\varphi (a_{ij})]_{ij}) \le c_{p,q}\cdot p^n([a_{ij}]_{ij})$ for all $n\in \mathbb {N}$ and $[a_{ij}]_{ij} \in M_n({\mathcal {A}})$;

• (v) local completely contractive if for any $p\in S({\mathcal {A}})$, there exists $q \in S({\mathcal {B}})$ such that $q^n([\varphi (a_{ij})]_{ij}) \le p^n([a_{ij}]_{ij})$ for all $n\in \mathbb {N}$ and $[a_{ij}]_{ij} \in M_n({\mathcal {A}})$;

• (vi) local completely positive if for any $p\in S({\mathcal {A}})$, there exists $q \in S({\mathcal {B}})$ such that $[\varphi (a_{ij})]_{ij} \ge _q 0$ for all $n\in \mathbb {N}$ and $[a_{ij}]_{ij} \ge _p 0$ (as $[a_{ij}]_{ij} \in M_n({\mathcal {A}}_p)$).

We review quantized domains in a Hilbert space which was introduced in [6]. Let ${\mathcal {H}}$ be a Hilbert space and ${\mathcal {E}}=\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ (for a directed poset $\Lambda$) be an upward filtered family of closed subspaces in ${\mathcal {H}}$ whose union ${\mathcal {D}}=\bigcup _{\lambda \in \Lambda } {\mathcal {H}}_\lambda$ is dense in ${\mathcal {H}}$. Such a family ${\mathcal {E}}$ is called a quantized domain in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$.

Example 2.4

Let $\Lambda$ be a directed poset and ${\mathcal {H}}$ be a Hilbert space. If $\{P_\lambda \}_{\lambda \in \Lambda }$ is an upward directed family of orthogonal projections in ${\mathcal {B}}({\mathcal {H}})$, then ${\mathcal {H}}_\lambda =P_\lambda ({\mathcal {H}})$ is a closed subspace in ${\mathcal {H}}$ and $\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ is an upward filtered family of closed subspaces in ${\mathcal {H}}$. Thus, if ${\mathcal {D}}=\bigcup _{\lambda \in \Lambda } {\mathcal {H}}_\lambda$ is dense in ${\mathcal {H}}$, then ${\mathcal {E}}= \{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ becomes a quantized domain in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$. $\square$

In [6], Dosiev introduced a notion of locally $C^*$-algebras over a quantized domain to replace ${\mathcal {B}}({\mathcal {H}})$ in a local operator space theory and provided an intrinsic characterization of local operator spaces, extending the known results for operator spaces.

Definition 2.5

([5, 6]) Let ${\mathcal {E}}=\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ be a quantized domain in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$.

• (i) The algebra of all noncommutative continuous functions on ${\mathcal {E}}$ is defined by

  $$\begin{aligned} C_{\mathcal {E}}({\mathcal {D}}) = \{ T \in L({\mathcal {D}}): TP_\lambda =P_\lambda TP_\lambda \in {\mathcal {B}}({\mathcal {H}}) ~~\text {for all}~~ \lambda \in \Lambda \}, \end{aligned}$$

  where $L({\mathcal {D}})$ is the algebra of all linear operators on ${\mathcal {D}}$.

• (ii) The *-algebra of all noncommutative continuous functions on ${\mathcal {E}}$ is defined by

  $$\begin{aligned} C^*_{\mathcal {E}}({\mathcal {D}}) = \{ T \in C_{\mathcal {E}}({\mathcal {D}}): P_\lambda T \subseteq TP_\lambda ~~\text {for all}~~ \lambda \in \Lambda \}, \end{aligned}$$

  where $T_1 \subseteq T_2$ means that $\text {dom}(T_1) \subseteq \text {dom}(T_2)$ and $T_1 = T_2|_{\text {dom}(T_1)}$.

Throughout the paper, ${\mathcal {B}}({\mathcal {H}})$ and ${\mathcal {E}}=\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ denote the algebras of all bounded linear operators on ${\mathcal {H}}$ and a quantized domain in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$, respectively.

Remark 2.6

([6]) For a quantized domain ${\mathcal {E}}$, we can see the following properties:

• (1) Each $T \in C_{\mathcal {E}}({\mathcal {D}})$ is an unbounded operator on ${\mathcal {H}}$ with domain ${\mathcal {D}}$ such that $T({\mathcal {H}}_\lambda ) \subseteq {\mathcal {H}}_\lambda$ and $T|_{{\mathcal {H}}_\lambda } \in {\mathcal {B}}({\mathcal {H}}_\lambda )$ for all $\lambda \in \Lambda$.

• (2) Each $T \in C^*_{\mathcal {E}}({\mathcal {D}})$ has an unbounded dual $T^\star$ such that ${\mathcal {D}}\subseteq \text {dom}(T^\star )$, $T^\star ({\mathcal {D}}) \subseteq {\mathcal {D}}$, and $T^*:=T^\star |_{\mathcal {D}}\in C^*_{\mathcal {E}}({\mathcal {D}})$.

• (3) The map $T \longmapsto T^*$ is an involution on $C^*_{\mathcal {E}}({\mathcal {D}})$.

• (4) $p^{(n)}_\lambda (T):=\Vert T|_{{\mathcal {H}}_\lambda ^n}\Vert$ is a seminorm on $M_n(C_{\mathcal {E}}({\mathcal {D}}))$, where ${\mathcal {H}}_\lambda ^n$ is the direct sum of n-copies of a Hilbert space ${\mathcal {H}}_\lambda$.

• (5) $T\in C^*_{\mathcal {E}}({\mathcal {D}}) ~~\Longleftrightarrow ~~ T({\mathcal {H}}_\lambda ^\perp \cap {\mathcal {D}}) \subseteq {\mathcal {H}}_\lambda ^\perp \cap {\mathcal {D}}$ for all $\lambda$.

• (6) If $T\in C^*_{\mathcal {E}}({\mathcal {D}})$, then ${\mathcal {D}}\subseteq \text {dom}(T^\star )$ and $T^\star ({\mathcal {H}}_\lambda ) \subseteq {\mathcal {H}}_\lambda$ for all $\lambda$.

As a particular case of above (3), for each $\lambda \in \Lambda$, we define a seminorm $\Vert \cdot \Vert _\lambda$ on $C^*_{\mathcal {E}}({\mathcal {D}})$ by

$$\begin{aligned} \Vert T\Vert _\lambda := \Vert T|_{{\mathcal {H}}_\lambda }\Vert \quad \text {for all}~~ T \in C^*_{\mathcal {E}}({\mathcal {D}}). \end{aligned}$$

Then $\{\Vert \cdot \Vert _\lambda \}_{\lambda \in \Lambda }$ is a directed set of $C^*$-seminorms on $C^*_{\mathcal {E}}({\mathcal {D}})$, so that $C^*_{\mathcal {E}}({\mathcal {D}})$ is a locally $C^*$-algebra determined by $\{\Vert \cdot \Vert _\lambda \}_\lambda$.

Remark 2.7

([6]) If ${\mathcal {E}}=\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ is a quantized domain in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$, then ${\mathcal {E}}^n=\{{\mathcal {H}}_\lambda ^n\}_{\lambda \in \Lambda }$ is a quantized domain in ${\mathcal {H}}^n$ with its union space ${\mathcal {D}}^n$. Moreover, $C^*_{{\mathcal {E}}^n}(D^n)$ is a locally $C^*$-algebra which is isomorphic to $M_n(C^*_{\mathcal {E}}({\mathcal {D}}))$.

Definition 2.8

Let $[T_{ij}]_{1\le i,j\le n} \in M_n(C^*_{\mathcal {E}}({\mathcal {D}}))$ for given $n\in \mathbb {N}$. Then we define as follows:

• (i) $[T_{ij}]\ge _\mu 0$ if $[T_{ij}]|_{{\mathcal {H}}_\mu ^n}$ is positive in ${\mathcal {B}}({\mathcal {H}}_\mu ^n)$, and

• (ii) $[T_{ij}] =_\mu 0$ if $[T_{ij}]|_{{\mathcal {H}}_\mu ^n}=0$ in ${\mathcal {B}}({\mathcal {H}}_\mu ^n)$.

Let ${\mathcal {E}}=\{{\mathcal {H}}_\lambda \}_{\lambda \in \Lambda }$ and ${\mathcal {F}}=\{{\mathcal {K}}_\lambda \}_{\lambda \in \Lambda }$ be quantized domains in Hilbert spaces ${\mathcal {H}}$ and ${\mathcal {K}}$ with their union spaces ${\mathcal {D}}$ and ${\mathfrak {F}}$, respectively. Let $V: {\mathcal {D}}\rightarrow {\mathcal {K}}$ be a linear operator. Then we write $V({\mathcal {E}})\subset {\mathcal {F}}$ if for any $\lambda \in \Lambda$, $V({\mathcal {H}}_\lambda )\subset {\mathcal {K}}_\lambda$.

The following is a Stinespring type theorem for local completely positive and local completely contractive maps.

Theorem 2.9

[6] If $\phi$ is a local completely positive and local completely contractive map from a locally $C^*$-algebra ${\mathcal {A}}$ into $C^*_{\mathcal {E}}({\mathcal {D}})$, then there exist a quantized domain ${\mathcal {E}}^\phi$ in ${\mathcal {H}}^\phi$ with its union space ${\mathcal {D}}^\phi$, a contraction $V_\phi :{\mathcal {H}}\rightarrow {\mathcal {H}}^\phi$ and a unital local contractive *-homomorphism $\pi _\phi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ such that

$$\begin{aligned} V_\phi ({\mathcal {E}}) \subseteq {\mathcal {E}}^\phi , \quad \phi (a) \subseteq V_\phi ^* \pi _\phi (a) V_\phi , \ \ \forall a \in {\mathcal {A}}. \end{aligned}$$

Moreover, if $\phi (1_{\mathcal {A}}) = I_{\mathcal {D}}$, then $V_\phi$ is an isometry.

Dosiev [6] also proved that for a locally $C^*$-algebra ${\mathcal {A}}$, there exist a quantized domain ${\mathcal {E}}$ in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$ and a local isometrical *-homomorphism $\pi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$. Bhat at al. [5] introduced the minimality for Stinespring theorem and proved the uniqueness of minimality up to unitary equivalence.

3 Representation Theorem for Local $\alpha$-Completely Positive Maps

Let ${\mathcal {A}}$ be a unital $C^*$-algebra and $\alpha : {\mathcal {A}}\rightarrow {\mathcal {A}}$ be a bounded Hermitian map such that $\alpha (1_{\mathcal {A}})=1_{\mathcal {A}}$ and $\alpha ^2=\text {id}_{\mathcal {A}}$ where $1_{\mathcal {A}}$ is the unit element of ${\mathcal {A}}$ and $\text {id}_{\mathcal {A}}$ is the identity map on ${\mathcal {A}}$. We say that a Hermitian linear map $\phi$ of ${\mathcal {A}}$ into ${\mathcal {B}}({\mathcal {H}})$ is $\alpha$-completely positive (briefly, $\alpha$-CP) if

• (i) $\phi (ab)=\phi (\alpha (ab))=\phi (\alpha (a)\alpha (b))$ for all $a,b\in {\mathcal {A}}$,

• (ii) for any $n\ge 1$, $a_1,\ldots ,a_n\in {\mathcal {A}}$ and $\xi _1,\ldots ,\xi _n\in {\mathcal {H}}$,

  $$\begin{aligned} \sum _{i,j=1}^{n} \big \langle \xi _i, \phi (\alpha (a_i)^*a_j)\xi _j \big \rangle \ge 0, \end{aligned}$$

• (iii) for each $a,a_1,\ldots ,a_n \in {\mathcal {A}}$, there exists a constant $C(a) \ge 0$ such that

  $$\begin{aligned} \biggl (\phi (\alpha (aa_i)^*aa_j)\biggr ) \le C(a) \cdot \bigg (\phi (\alpha (a_i)^*a_j)\bigg ), \end{aligned}$$

  where a big parenthesis denotes an $n\times n$ matrix.

For more detailed study of $\alpha$-CP maps, we refer to [7,8,9].

Let ${\mathcal {A}}$ and ${\mathcal {B}}$ be locally $C^*$-algebras with seminorms $\{p_\lambda \}_{\lambda \in \Lambda }$ and $\{q_\omega \}_{\omega \in \Omega }$, respectively. To extend the notion of $\alpha$-complete positivity to a linear map between locally $C^*$-algebras, we define a strict continuity of a linear map.

Definition 3.1

A linear map $\varphi : {\mathcal {A}}\rightarrow {\mathcal {B}}$ is strictly continuous if for any $\omega \in \Omega$, there exist a $\lambda _\omega \in \Lambda$ and a constant $C_\omega > 0$ such that

$$\begin{aligned} q_\omega (\varphi (a)) \le C_\omega \cdot p_{\lambda _\omega }(a), \quad \forall a \in {\mathcal {A}}. \end{aligned}$$

Definition 3.2

Let ${\mathcal {A}}$ be a unital locally $C^*$-algebra and $\alpha : {\mathcal {A}}\rightarrow {\mathcal {A}}$ be a local bounded Hermitian map such that $\alpha (1_{\mathcal {A}})=1_{\mathcal {A}}$ and $\alpha ^2=\text {id}_{\mathcal {A}}$. A linear map $\varphi : {\mathcal {A}}\rightarrow {\mathcal {B}}$ is local $\alpha$-completely positive (briefly, local $\alpha$-CP) if

• (i) for all $a,b \in {\mathcal {A}}$, $\varphi (ab)=\varphi (\alpha (ab))=\varphi (\alpha (a)\alpha (b))$,

• (ii) for any $n\ge 1$ and $\omega \in \Omega$,

  $$\begin{aligned} \bigg (\varphi (\alpha (a_i)^*a_j) \bigg ) \ge _{q_\omega } 0, \ \ i.e.,~ \bigg (\pi _\omega ^{\mathcal {B}}\circ \varphi (\alpha (a_i)^*a_j) \bigg ) \ge 0 \end{aligned}$$

  for all $a_1,\ldots ,a_n\in {\mathcal {A}}$, where $\pi _\omega ^{\mathcal {B}}: {\mathcal {B}}\rightarrow {\mathcal {B}}_\omega$ is the canonical homomorphism,

• (iii) for any $n\ge 1$ and $\omega \in \Omega$, there exists a constant $C(a) \ge 0$ such that

  $$\begin{aligned} \biggl (\varphi (\alpha (aa_i)^*aa_j)\biggr ) \le _{q_\omega } C(a)\cdot \bigg (\varphi (\alpha (a_i)^*a_j)\bigg ) \end{aligned}$$

  for all $a,a_1,\ldots ,a_n \in {\mathcal {A}}$.

In the remaining of this section, ${\mathcal {A}}$ denotes a locally $C^*$-algebra with seminorms $\{p_\lambda \}_{\lambda \in \Lambda }$, unless specified otherwise.

For simple notations, we denote by $\pi _\lambda :=\pi _{p_\lambda }: {\mathcal {A}}\rightarrow {\mathcal {A}}_\lambda :=A_{p_\lambda }= {\mathcal {A}}/\ker p_\lambda$ the canonical *-homomorphism and for a quantized domain ${\mathcal {E}}=\{{\mathcal {H}}_\mu \}_{\mu \in \Omega }$ in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$, the directed set of $C^*$-seminorms on the locally $C^*$-algebra $C^*_{\mathcal {E}}({\mathcal {D}})$ is denoted by $\{q_\mu \}_{\mu \in \Omega }$, i.e. $q_{\mu }=\Vert \cdot \Vert _\mu$ for $\mu \in \Omega$. Then for each $n\in \mathbb {N}$, put

$$\begin{aligned} q^n_\mu (T):= \left\| P_\mu ^{\oplus ^n}TP_\mu ^{\oplus ^n}\right\| \quad \hbox { for}\ T \in C^*_{{\mathcal {E}}^n}({\mathcal {D}}^n). \end{aligned}$$

Let $\alpha :{\mathcal {A}}\rightarrow {\mathcal {A}}$ be a local bounded Hermitian map with $\alpha ^2=\text {id}_{\mathcal {A}}$ and $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ be a local $\alpha$-CP map. We define a sesquilinear form on an algebraic tensor product ${\mathcal {A}}\otimes {\mathcal {D}}$ by

$$\begin{aligned} \bigg \langle \sum _i a_i \otimes \xi _i, \sum _j b_j \otimes \eta _j \bigg \rangle = \sum _{i,j} \langle \xi _i \big | \varphi (\alpha (a_i)^*b_j) \eta _j \rangle \end{aligned}$$

where $\langle \cdot | \cdot \rangle$ denotes an inner product on ${\mathcal {H}}$. Let ${\mathcal {N}}_\varphi := \{u \in {\mathcal {A}}\otimes {\mathcal {D}}: \langle u,u\rangle =0\}$. Then the sesquilinear form $\langle \cdot ,\cdot \rangle$ induces an inner product, still denoted by $\langle \cdot ,\cdot \rangle$, on the quotient space ${\mathcal {A}}\otimes {\mathcal {D}}/{\mathcal {N}}_\varphi$. We denote by ${\mathcal {K}}$ the completion of ${\mathcal {A}}\otimes {\mathcal {D}}/{\mathcal {N}}_\varphi$ with respect to the norm induced by the inner product $\langle \cdot ,\cdot \rangle$.

Now we define an indefinite inner product $[\cdot ,\cdot ]$ on the quotient space ${\mathcal {A}}\otimes {\mathcal {D}}/{\mathcal {N}}_\varphi$ by

$$\begin{aligned} \bigg [ \sum _i a_i \otimes \xi _i + {\mathcal {N}}_\varphi , \sum _j b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg ] = \sum _{i,j} \langle \xi _i \big | \varphi (a_i^*b_j) \eta _j \rangle . \end{aligned}$$

By Cauchy-Schwarz inequality, we have that

$$\begin{aligned}&\bigg |\bigg [ \sum _i a_i \otimes \xi _i + {\mathcal {N}}_\varphi , \sum _j b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg ] \bigg |= \bigg | \sum _{i,j} \langle \xi _i | \varphi (a_i^*b_j) \eta _j \rangle \bigg | \\&\quad = \bigg | \sum _{i,j} \langle \alpha (a_i) \otimes \xi _i + {\mathcal {N}}_\varphi , b_j \otimes \eta _j + {\mathcal {N}}_\varphi \rangle \bigg | \\&\quad \le \bigg \Vert \sum _i \alpha (a_i) \otimes \xi _i + {\mathcal {N}}_\varphi \bigg \Vert \bigg \Vert \sum _j b_j \otimes \xi _i + {\mathcal {N}}_\varphi \bigg \Vert \\&\quad =\bigg \Vert \sum _i a_i \otimes \xi _i + {\mathcal {N}}_\varphi \bigg \Vert \bigg \Vert \sum _j b_j \otimes \xi _i + {\mathcal {N}}_\varphi \bigg \Vert \end{aligned}$$

We see that $[u,v]=\langle (\alpha \otimes I)(u),v\rangle$ for all $u,v \in {\mathcal {A}}\otimes {\mathcal {D}}/{\mathcal {N}}_\varphi$. Put $J:=\alpha \otimes I$ on ${\mathcal {A}}\otimes {\mathcal {D}}/{\mathcal {N}}_\varphi$ and extend to ${\mathcal {K}}$. Then it follows from continuity of $\langle \cdot ,\cdot \rangle$ and $\alpha$ that

$$\begin{aligned}{}[u,v]=\langle Ju,v\rangle \quad \hbox { for all}\ u,v \in {\mathcal {K}}. \end{aligned}$$

Moreover, we have that $J^2 = \text {id}_{\mathcal {K}}$ and $J=J^*=J^{-1}$ since $\alpha ^2=\text {id}_{\mathcal {A}}$ and $\alpha$ is Hermitian. Therefore, J is a fundamental symmetry on ${\mathcal {K}}$, so that the pair $({\mathcal {K}},J)$ becomes a Krein space.

For each $a\in {\mathcal {A}}$, we consider the linear map $\pi (a): {\mathcal {A}}\otimes {\mathcal {D}}\rightarrow {\mathcal {A}}\otimes {\mathcal {D}}$ given by $\pi (a)= L_a \otimes \text {id}_{\mathcal {D}}$ where $L_a$ is a left multiplication map on ${\mathcal {A}}$. Let $u=\sum _{i=1}^n a_i \otimes \xi _i \in {\mathcal {A}}\otimes {\mathcal {D}}$ be given. Then since $\{{\mathcal {H}}_\mu \}$ is a directed family of closed subspaces in ${\mathcal {H}}$, there is an index $\mu \in \Omega$ such that $\{\xi _i: 1\le i \le n\} \subseteq {\mathcal {H}}_\mu$, i.e., $\xi =(\xi _i) \in {\mathcal {H}}_\mu ^n$. We note that

$$\begin{aligned} \Big [\varphi \left( \alpha (aa_i)^*aa_j\right) \Big ]&\le _\mu&C(a)\Big [\varphi (\alpha (a_i)^*a_j \Big ] \Longleftrightarrow \Big [\pi _\mu \circ \varphi (\alpha (aa_i)^*aa_j) \Big ] \\\le & {} C(a)\Big [\pi _\mu \circ \varphi (\alpha (a_i)^*a_j)\Big ] \end{aligned}$$

where $\pi _\mu \circ \varphi (c) = \varphi (c) P_\mu$. By condition (iii) of Definition 3.2, we obtain that

$$\begin{aligned} \langle \pi (a)u, \pi (a)u \rangle&= \bigg \langle \sum _{i=1}^n aa_i \otimes \xi _i, \sum _{j=1}^n aa_j \otimes \xi _j \bigg \rangle \\&= \sum _{i,j}^n \langle \xi _i \big | \varphi (\alpha (aa_i)^*aa_j) \xi _j \rangle \\&\le C_\mu (a) \Big \langle \xi \Big | \Big [\varphi (\alpha (a_i)^*a_j)\Big ]\xi \Big \rangle _n = C_\mu (a)\langle u,u \rangle \end{aligned}$$

where $\xi =(\xi _i) \in {\mathcal {H}}_\mu ^n$ and $\langle \cdot \big | \cdot \rangle _n$ is the inner product on ${\mathcal {H}}_\mu ^n$.

Since $\pi (a) ({\mathcal {N}}_\varphi ) \subseteq {\mathcal {N}}_\varphi$, $\pi (a)$ determines a linear map, denoted by $\pi (a)$, on ${\mathcal {A}}\otimes {\mathcal {D}}/ {\mathcal {N}}_\varphi$, so that $\pi (a)$ is a densely defined unbounded linear operator on $({\mathcal {K}},J)$. Moreover, we obtain that

$$\begin{aligned} \bigg [ \pi (a) \bigg ( \sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi \bigg ),&\sum _{j=1}^m b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg ] \\&= \bigg [ \sum _{i=1}^n aa_i \otimes \xi _i + {\mathcal {N}}_\varphi , \sum _{j=1}^m b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg ] \\&= \sum _{i,j}^{n, m} \Big \langle \xi _i \Big | \varphi (a_i^*a^*b_j) \eta _j \Big \rangle \\&= \bigg [ \sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi , \pi (a^*) \bigg (\sum _{j=1}^m b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg ) \bigg ], \end{aligned}$$

which implies that $\pi (a)^\# = \pi (a^*)$ where ${}^\#$ denotes the J-adjoint with respect to the indefinite inner product $[\cdot ,\cdot ]$. Therefore, $\pi$ is a J-homomorphism.

We define the set

$$\begin{aligned} {\mathcal {M}}_\mu :=\bigg \{\sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi : \xi _i \in {\mathcal {H}}_\mu , ~a_i \in {\mathcal {A}}\bigg \}. \end{aligned}$$

Then ${\mathcal {M}}_\mu$ is the range of a subspace ${\mathcal {A}}\otimes {\mathcal {H}}_\mu$ via the quotient map ${\mathcal {A}}\otimes {\mathcal {D}}\rightarrow {\mathcal {A}}\otimes {\mathcal {D}}/ {\mathcal {N}}_\varphi$ and is a subspace of ${\mathcal {K}}$ which is invariant under $\pi (a)$. We denote by ${\mathcal {K}}_\mu$ the closure of ${\mathcal {M}}_\mu$ in ${\mathcal {K}}$. Since $\Vert \pi (a)u\Vert ^2 \le C_\mu (a)\Vert u\Vert ^2$ and ${\mathcal {M}}_\mu$ is dense in ${\mathcal {K}}_\mu$, $\pi (a)$ extends to a bounded linear operator on ${\mathcal {K}}_\mu$, which we denote by $\pi (a)_\mu$. Then we obtain that $\Vert \pi (a)_\mu \Vert _{{\mathcal {B}}({\mathcal {K}}_\mu )} \le C_\mu (a)^{1/2}$ and that if $\mu \preceq \nu$, then ${\mathcal {K}}_\mu \subseteq {\mathcal {K}}_\nu$ and $\pi (a)_\nu |_{{\mathcal {K}}_\mu } = \pi (a)_\mu$.

Let ${\mathcal {F}}= \{{\mathcal {K}}_\mu \}$ and ${\mathfrak {F}} = \cup _\mu {\mathcal {K}}_\mu$. Then ${\mathcal {F}}$ is a quantized domain in ${\mathcal {K}}$ with its union space ${\mathfrak {F}}$. For each $a\in {\mathcal {A}}$, $\pi (a)$ is a unbounded operator on ${\mathcal {K}}$ such that

$$\begin{aligned} \text {dom}(\pi (a)) = {\mathfrak {F}}, \quad \pi (a)|_{{\mathcal {K}}_\mu } = \pi (a)_\mu \quad \hbox { for all}\ \mu \in \Omega . \end{aligned}$$

Moreover, each $\pi (a)$ leaves each subspace ${\mathcal {K}}_\mu$ invariant. Since $\Vert \pi (a)|_{{\mathcal {K}}_\mu }\Vert \le C_\mu (a)^{1/2}$ for each $\mu$, we see that $\pi (a)$ belongs to the algebra $C_{\mathcal {F}}({\mathfrak {F}})$ and $\pi : {\mathcal {A}}\rightarrow C_{\mathcal {F}}({\mathfrak {F}})$ is a unital homomorphism.

Proposition 3.3

The range of $\pi$ is contained in a locally $C^*$-algebra $C_{\mathcal {F}}^*({\mathfrak {F}})$. Furthermore, $\pi : {\mathcal {A}}\rightarrow C_{\mathcal {F}}^*({\mathfrak {F}})$ is a unital J-homomorphism.

Proof

For any $u, v \in {\mathcal {M}}_\mu$, we can write as follows;

$$\begin{aligned} u= \sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi , \quad v= \sum _{j=1}^m b_j \otimes \eta _j + {\mathcal {N}}_\varphi \end{aligned}$$

where $\xi _i, \eta _j \in {\mathcal {H}}_\mu$ and $a_i, b_j \in {\mathcal {A}}$. For $a \in {\mathcal {A}}$, we have that

$$\begin{aligned}&\bigg \langle \pi (a) \bigg ( \sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi \bigg ),\sum _{j=1}^m b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg \rangle \\&\quad = \sum _{i,j=1}^{n, m} \Big \langle \xi _i \Big | \varphi (\alpha (aa_i)^*b_j) \eta _j \Big \rangle = \sum _{i,j=1}^{n, m} \Big \langle \xi _i \Big | \varphi (\alpha (a_i)^*\alpha (a^*\alpha (b_j))) \eta _j \Big \rangle \\&\quad = \bigg \langle \sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi , \sum _{j=1}^m (\alpha \otimes I)\pi (a^*)(\alpha \otimes I)(b_j \otimes \eta _j + {\mathcal {N}}_\varphi ) \bigg ) \bigg \rangle \\&\quad =\bigg \langle \sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi , J\pi (a^*)J\bigg (\sum _{j=1}^m b_j \otimes \eta _j + {\mathcal {N}}_\varphi \bigg ) \bigg \rangle . \end{aligned}$$

This means that $\pi (a)^* = J \pi (a^*)J$ on ${\mathcal {M}}_\mu$, so that $\pi (a)^\# = J\pi (a)^*J=\pi (a^*)$ on ${\mathcal {M}}_\mu$. By continuity, we have that $\langle \pi (a)x,y\rangle = \langle x, J\pi (a^*)J y\rangle$ for all $x,y\in {\mathfrak {F}}$ and $\pi (a)^\# =\pi (a^*)$ on ${\mathfrak {F}}$.

We observe that $(\alpha \otimes I)({\mathcal {N}}_\varphi ) \subseteq {\mathcal {N}}_\varphi$ and $(\alpha \otimes I)({\mathcal {M}}_\mu ) \subseteq {\mathcal {M}}_\mu$. Since $\pi (a^*)P_\mu = P_\mu \pi (a^*)P_\mu$, $\pi (a)^\# \in C_{\mathcal {F}}({\mathfrak {F}})$. For any $u=\sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi \in {\mathcal {M}}_\mu$, we have that

$$\begin{aligned} \pi (a)^*(u) = J\pi (a^*)J \bigg (\sum _{i=1}^n a_i \otimes \xi _i + {\mathcal {N}}_\varphi \bigg ) = \sum _{i=1}^n \alpha (a^*\alpha (a_i)) \otimes \xi _i + {\mathcal {N}}_\varphi , \end{aligned}$$

so that $\pi (a)^*({\mathcal {M}}_\mu ) \subseteq {\mathcal {M}}_\mu$. By [6, Proposition 3.1], we see that $\pi (a) \in C_{\mathcal {F}}^*({\mathfrak {F}})$. Hence $\pi : {\mathcal {A}}\rightarrow C_{\mathcal {F}}^*({\mathfrak {F}})$ is a unital J-homomorphism. $\square$

Theorem 3.4

Let ${\mathcal {A}}$ be a unital locally $C^*$-algebra with and ${\mathcal {E}}= \{{\mathcal {H}}_\mu \}_{\mu \in \Omega }$ be a quantized domain in ${\mathcal {H}}$ with its union space ${\mathcal {D}}$. If $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ is a local contractive and local $\alpha$-CP map, then there exist a quantized domain ${\mathcal {F}}= \{{\mathcal {K}}_\mu \}_{\mu \in \Omega }$ in a Krein space $({\mathcal {K}},J)$ with its union space ${\mathfrak {F}}$, a contraction $V:{\mathcal {H}}\rightarrow {\mathcal {K}}$ and a unital J-representation $\pi : {\mathcal {A}}\rightarrow C_{\mathcal {F}}^*({\mathfrak {F}})$ such that for all $a,b\in {\mathcal {A}}$,

• (i) $V({\mathcal {E}}) \subseteq {\mathcal {F}}$,

• (ii) $\varphi (a) \subseteq V^*\pi (a)V$,

• (iii) $V^*\pi (a)^*\pi (b)V = V^*\pi (\alpha (a)^*b)V$,

• (iv) $\varphi (a^*) \subseteq V^*\pi (a)^\#V$.

Moreover, if $\varphi (1_{\mathcal {A}})=I_{{\mathcal {D}}}$, then V is an isometry.

Proof

A Krein space $({\mathcal {K}},J)$, a quantized domain ${\mathcal {F}}= \{{\mathcal {K}}_\mu \}_{\mu \in \Omega }$ in ${\mathcal {K}}$ with its union space ${\mathfrak {F}}$ and a unital J-representation $\pi : {\mathcal {A}}\rightarrow C_{\mathcal {F}}^*({\mathfrak {F}})$ have been already constructed in Proposition 3.3 and its previous discussions.

We now consider a linear map $V:{\mathcal {D}}\rightarrow {\mathcal {K}}$ defined by $V\xi = 1_{\mathcal {A}}\otimes \xi + {\mathcal {N}}_\varphi$ where $\xi \in {\mathcal {D}}=\bigcup _\mu {\mathcal {H}}_\mu$. For each $\mu \in \Omega$ and $\xi \in {\mathcal {H}}_\mu$, we have that

$$\begin{aligned} V\xi = 1_{\mathcal {A}}\otimes \xi + {\mathcal {N}}_\varphi \in {\mathcal {M}}_\mu \subseteq {\mathcal {K}}_\mu \subseteq {\mathfrak {F}}, \end{aligned}$$

so that $V({\mathcal {H}}_\mu ) \subseteq {\mathcal {K}}_\mu$. Then for any $\xi \in {\mathcal {H}}_\mu$, we have

$$\begin{aligned} \Vert V\xi \Vert ^2=\langle 1_{\mathcal {A}}\otimes \xi + {\mathcal {N}}_\varphi , 1_{\mathcal {A}}\otimes \xi + {\mathcal {N}}_\varphi \rangle = \langle \xi | \varphi (1_{\mathcal {A}}) \xi \rangle , \end{aligned}$$

from which we see that if $\varphi (1_{\mathcal {A}})=I_{{\mathcal {D}}}$, then V is an isometry. Since for any $\mu \in \Omega$, the local contractivity of $\varphi$ gives the inequality $q_\mu \circ \varphi \le p_\lambda$ for some $\lambda \in \Lambda$,

$$\begin{aligned} \Vert V\xi \Vert ^2&= \langle \xi | \varphi (1_{\mathcal {A}}) \xi \rangle \le \Vert \xi \Vert \Vert \varphi (1_{\mathcal {A}}) \xi \Vert \\&\le \Vert \varphi (1_{\mathcal {A}})|_{{\mathcal {H}}_\mu }\Vert \Vert \xi \Vert ^2 = q_\mu (\varphi (1_{\mathcal {A}}))\cdot \Vert \xi \Vert ^2 \le p_\lambda (1_{\mathcal {A}}) \cdot \Vert \xi \Vert ^2 \\&= \Vert \xi \Vert ^2 \end{aligned}$$

for all $\xi \in {\mathcal {H}}_{\mu }$. Hence V is a contraction, so that V has a unique extension on $\overline{{\mathcal {D}}}={\mathcal {H}}$.

Let $a \in {\mathcal {A}}$ and $\xi ,\eta \in {\mathcal {D}}$. Then $\xi ,\eta \in {\mathcal {H}}_\mu$ for some $\mu$, so we have

$$\begin{aligned} \langle \xi | V^*\pi (a)V \eta \rangle&= \langle 1_{\mathcal {A}}\otimes \xi + N_\varphi , \pi (a)(1_{\mathcal {A}}\otimes \eta + N_\varphi ) \rangle \\&= \langle \xi | \varphi (a) \eta \rangle . \end{aligned}$$

This implies that $\varphi (a)\subseteq V^*\pi (a)V$ and $\varphi (a)P_\mu = V^*\pi (a)VP_\mu$ for all $\mu \in \Omega$. Moreover, we also obtain that

$$\begin{aligned} \langle \xi | V^*\pi (a)^*\pi (b)V \eta \rangle = \langle a \otimes \xi + N_\varphi ,b \otimes \eta + N_\varphi \rangle = \langle \xi | \varphi (\alpha (a)^*b) \eta \rangle , \end{aligned}$$

so that $V^*\pi (a)^*\pi (b)V = \varphi (\alpha (a)^*b) = V^*\pi (\alpha (a)^*b)V$. Since

$$\begin{aligned} \langle \xi | V^*\pi (a)^\#V \eta \rangle = \langle \xi | \varphi (\alpha (a)^*) \eta \rangle =\langle \xi | \varphi (a^*) \eta \rangle , \end{aligned}$$

we obtain that $\varphi (a^*)\subseteq V^*\pi (a)^\#V$. $\square$

Corollary 3.5

Let ${\mathcal {A}}$, ${\mathcal {E}}$ and $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ be as in Theorem 3.4. If, in addition, $\varphi$ satisfies the following condition:

(iii$'$):

for any $n\ge 1$ and $\omega \in \Omega$, there exists a $\lambda \in \Lambda$ such that for any $a\in {\mathcal {A}}_{\lambda }$,

$$\begin{aligned} \Big (\varphi (\alpha (aa_i)^*aa_j)\Big ) \le _{q_\omega } p_{\lambda }(a)\cdot \Big (\varphi (\alpha (a_i)^*a_j)\Big ) \end{aligned}$$

for all $a_1,\ldots ,a_n \in {\mathcal {A}}$,

then the unital J-representation $\pi : {\mathcal {A}}\rightarrow C_{\mathcal {F}}^*({\mathfrak {F}})$ constructed in Theorem 3.4 is local contractive.

Proof

The proof is immediate from the construction of the J-representation. $\square$

In the same way as in [5], we can define the minimality for a Krein space J-representation associated with a quantized domain as follows. Let ${\mathcal {E}}$, $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ and $(\pi , V, ({\mathcal {K}},J;{\mathcal {F}}))$ be as in Theorem 3.4. The Krein space J-representation $(\pi ,V,({\mathcal {K}},J;{\mathcal {F}}))$ of $\varphi$ is called minimal if ${\mathcal {K}}_\mu = \big [\pi ({\mathcal {A}}) V {\mathcal {H}}_\mu \big ]$ for every $\mu \in \Omega$ where [S] is the closed linear span of S. From the notions of a locally $C^*$-algebra and a quantized domain, it seems to be more appropriate than the usual definition of minimality, ${\mathcal {K}}= \big [\pi ({\mathcal {A}}) V {\mathcal {H}}\big ]$ (see [5]).

In the following proposition, we prove the existence of the minimal Krein space J-representation of a local $\alpha$-CP map $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$.

Proposition 3.6

Let ${\mathcal {A}}$, ${\mathcal {E}}$ and $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ be as in Theorem 3.4. If $(\pi _\varphi ,V_\varphi ,({\mathcal {H}}^\varphi ,J_\varphi ;{\mathcal {E}}^\varphi ))$ is a Krein space J-representation for $\varphi$, then there exists a minimal Krein space J-representation $(\widetilde{\pi }_\varphi ,\widetilde{V}_\varphi ,({\mathcal {H}}^\varphi ,\widetilde{J}_\varphi ;\widetilde{{\mathcal {E}}}^\varphi ))$ such that $\widetilde{{\mathcal {E}}}^\varphi \subseteq {\mathcal {E}}^\varphi$ and $\widetilde{{\mathcal {H}}}^\varphi = [\widetilde{\pi }_\varphi ({\mathcal {A}})\widetilde{V}_\varphi {\mathcal {D}}]$.

Proof

The proof is a simple modification of the proof of Proposition 3.3 of [5]. We give a sketch of the proof for the readers’ convenience. For each $\mu \in \Omega$, we let $\widetilde{{\mathcal {H}}}_\mu ^\varphi = [\widetilde{\pi }_\varphi ({\mathcal {A}})\widetilde{V}_\varphi {\mathcal {H}}_{\mu }]$. Since $\pi _\varphi (a)|_{{\mathcal {H}}_\mu } \in {\mathcal {B}}({\mathcal {H}}_\mu )$ and $V_\varphi ({\mathcal {H}}_\mu ) \subseteq {\mathcal {H}}_\mu ^\varphi$, the space $\widetilde{{\mathcal {H}}}_\mu ^\varphi$ is a closed subspace of ${\mathcal {H}}_\mu ^\varphi$. Moreover, $\widetilde{{\mathcal {H}}}_\mu ^\varphi \subseteq \widetilde{{\mathcal {H}}}_\nu ^\varphi$ for $\mu \le \nu$, so that $\widetilde{{\mathcal {E}}}^\varphi := \{\widetilde{{\mathcal {H}}}_\mu ^\varphi \}_{\mu \in \Omega }$ is an upward filtered family of Hilbert spaces such that $\widetilde{{\mathcal {H}}}_\mu ^\varphi \subseteq {\mathcal {H}}_\mu ^\varphi$. Put

$$\begin{aligned} \widetilde{{\mathcal {D}}}^\varphi := \bigcup _\mu \widetilde{{\mathcal {H}}}_\mu ^\varphi \quad \text {and} \quad \widetilde{{\mathcal {H}}}^\varphi := \overline{\widetilde{{\mathcal {D}}}^\varphi }. \end{aligned}$$

Then the family $\widetilde{{\mathcal {E}}}^\varphi$ is a quantized domain in $\widetilde{{\mathcal {H}}}^\varphi$ with its union space $\widetilde{{\mathcal {D}}}^\varphi$. Since $\widetilde{{\mathcal {H}}}_\mu ^\varphi$ reduces every operator in $\pi _\varphi ({\mathcal {A}})$, we have that $\pi _\varphi (a)|_{\widetilde{{\mathcal {H}}}_\mu ^\varphi } \in {\mathcal {B}}(\widetilde{{\mathcal {H}}}_\mu ^\varphi )$. Thus, the map $\widetilde{\pi }_\varphi :{\mathcal {A}}\rightarrow C^*_{\widetilde{{\mathcal {E}}}^\varphi }(\widetilde{{\mathcal {D}}}^\varphi )$ defined by

$$\begin{aligned} \widetilde{\pi }_\varphi (a) :=\pi _\varphi (a)|_{\widetilde{{\mathcal {D}}}^\varphi }, \quad a\in {\mathcal {A}}, \end{aligned}$$

is a well-defined unital homomorphism.

Let $\widetilde{V}_\varphi = V_\varphi$. For $\xi , \eta \in {\mathcal {D}}$, there is a $\mu \in \Omega$ such that $\xi , \eta \in {\mathcal {H}}_\mu$. Then we have

$$\begin{aligned} \Big \langle \xi \Big | \widetilde{V}_\varphi ^*\widetilde{\pi }_\varphi (a) \widetilde{V}_\varphi \eta \Big \rangle _{{\mathcal {H}}_\mu } = \big \langle \xi \big | \varphi (a) \eta \big \rangle _{{\mathcal {H}}_\mu }. \end{aligned}$$

By putting $\widetilde{J}_\varphi :=J_\varphi$, we get

$$\begin{aligned} \widetilde{\pi }_\varphi (a)^* \widetilde{\pi }_\varphi (a) = \widetilde{\pi }_\varphi (\alpha (a)^*b) \quad \text {and} \quad \widetilde{\pi }_\varphi (a)^* = \widetilde{J}_\varphi \widetilde{\pi }_\varphi (a^*) \widetilde{J}_\varphi . \end{aligned}$$

Moreover, we have that

$$\begin{aligned} \widetilde{{\mathcal {H}}}^\varphi = \overline{\widetilde{{\mathcal {D}}}^\varphi } = \overline{\bigcup _\mu \big [\widetilde{\pi }_\varphi ({\mathcal {A}})\widetilde{V}_\varphi {\mathcal {H}}_\mu \big ]} = \big [\widetilde{\pi }_\varphi ({\mathcal {A}})\widetilde{V}_\varphi {\mathcal {D}}\big ], \end{aligned}$$

which completes the proof. $\square$

Remark 3.7

By applying the arguments used in the proof of Theorem 3.4 of [5], we can prove the uniqueness up to unitary equivalence of a minimal Krein space J-representation of a local $\alpha$-CP map $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$.

4 Representation on Krein $C^*$-Modules Associated to Pairs of Two Maps

In this section, we denote by ${\mathcal {E}}=\{{\mathcal {H}}_\mu \}_{\mu \in \Omega }$ and ${\mathcal {F}}=\{{\mathcal {K}}_\mu \}_{\mu \in \Omega }$ quantized domains in Hilbert spaces ${\mathcal {H}}$ and ${\mathcal {K}}$ with their union spaces ${\mathcal {D}}$ and ${\mathfrak {F}}$, respectively. We recall that $\{{\mathcal {H}}_\mu \}_{\mu \in \Omega }$ and $\{{\mathcal {K}}_\mu \}_{\mu \in \Omega }$ are families of closed subspaces in ${\mathcal {H}}$ and ${\mathcal {K}}$, respectively.

Definition 4.1

[5] A class of noncommutative functions between quantized domains ${\mathcal {E}}$ and ${\mathcal {F}}$ is defined by

$$\begin{aligned} C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}}) =\{T\in L({\mathcal {D}},{\mathfrak {F}}):&~T({\mathcal {H}}_\mu )\subseteq {\mathcal {K}}_\mu , ~T({\mathcal {H}}_\mu ^\perp \cap {\mathcal {D}})\subseteq {\mathcal {K}}_\mu ^\perp \cap {\mathfrak {F}} \\&\qquad \qquad \qquad \text {and} \quad T|_{{\mathcal {H}}_\mu }\in {\mathcal {B}}({\mathcal {H}}_\mu ,{\mathcal {K}}_\mu ) ~\forall \mu \}. \end{aligned}$$

We can prove the following statements (see [5]):

• (1) The set $C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$ is a complex vector space.

• (2) The map $C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}}) \times C^*_{{\mathcal {E}}}({\mathcal {D}}) \rightarrow C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$, $(T, S) \mapsto TS$, is a module map.

• (3) The map $C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}}) \times C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}}) \rightarrow C^*_{{\mathcal {E}}}({\mathcal {D}})$ given by $\langle T,S\rangle = T^*S$ is a $C^*_{{\mathcal {E}}}({\mathcal {D}})$-valued inner product.

• (4) The locally convex topology on $C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$ induced from $\varprojlim _\mu {\mathcal {B}}({\mathcal {H}}_\mu ,{\mathcal {K}}_\mu )$ is the topology generated by the family of seminorms $\{q_\mu : ~\mu \in \Omega \}$ given by

  $$\begin{aligned} q_\mu (T) = \Vert \langle T,T\rangle \Vert _{{\mathcal {B}}({\mathcal {H}}_\mu )}^{1/2} = \Vert T^*T|_{{\mathcal {H}}_\mu }\Vert ^{1/2} = \Vert T|_{{\mathcal {H}}_\mu }\Vert _{{\mathcal {B}}({\mathcal {H}}_\mu ,{\mathcal {K}}_\mu )}. \end{aligned}$$

• (5) $C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$ is a Hilbert module over the locally $C^*$-algebra $C^*_{{\mathcal {E}}}({\mathcal {D}})$.

Definition 4.2

[5] Let ${\mathcal {E}}=\{{\mathcal {H}}_\mu \}_{\mu \in \Omega }$ and ${\mathcal {F}}=\{{\mathcal {K}}_\mu \}_{\mu \in \Omega }$ be as above and let E be a Hilbert module over a locally $C^*$-algebra ${\mathcal {A}}$. Suppose that $\varphi : {\mathcal {A}}\rightarrow C^*_{{\mathcal {E}}}({\mathcal {D}})$ and $\Phi : E \rightarrow C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$ are linear maps.

• (i) $\Phi$ is a $\varphi$-map if $\langle \Phi (x),\Phi (y)\rangle = \varphi (\langle x,y\rangle )$ for all $x,y \in E$.

• (ii) $\Phi$ is a $\varphi$-morphism if it is a $\varphi$-map and $\varphi$ is a morphism.

In the remaining of this section, we denote by E a Hilbert module over a locally $C^*$-algebra ${\mathcal {A}}$, unless specified otherwise. If $\Phi : E \rightarrow C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$ is a linear map, then we have that

$$\begin{aligned} \overline{~\bigcup _\mu [\Phi (E){\mathcal {H}}_\mu ]~} = [\Phi (E){\mathcal {D}}]. \end{aligned}$$

Suppose that $\varphi : {\mathcal {A}}\rightarrow C^*_{{\mathcal {E}}}({\mathcal {D}})$ is a local contractive and local $\alpha$-CP map and $\Phi : E \rightarrow C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$ is a $\varphi$-map. By Theorem 3.4, there exist

• (a) a quantized domain ${\mathcal {E}}^\varphi = \{{\mathcal {H}}_\mu ^\varphi : \mu \in \Omega \}$ in a Hilbert space ${\mathcal {H}}^\varphi$ with its union space ${\mathcal {D}}^\varphi$,

• (b) a contraction $V_\varphi : {\mathcal {H}}\rightarrow {\mathcal {H}}^\varphi$,

• (c) a unital J-representation $\pi _\varphi : {\mathcal {A}}\rightarrow C^*_{{\mathcal {E}}^\varphi }({\mathcal {D}}^\varphi )$

such that

• (i) $V_\varphi ({\mathcal {E}}) \subseteq {\mathcal {E}}^\varphi$,

• (ii) $\varphi (a) \subseteq V_\varphi ^*\pi _\varphi (a)V_\varphi$,

• (iii) $V_\varphi ^*\pi _\varphi (a)^*\pi _\varphi (b)V_\varphi = V_\varphi ^*\pi _\varphi (\alpha (a)^*b)V_\varphi$,

• (iv) $\varphi (a^*) \subseteq V_\varphi ^*\pi _\varphi (a)^\#V_\varphi$.

With loss of generality, we may assume that $\pi _\varphi$ is the minimal representation for $\varphi$, so that

$$\begin{aligned} {\mathcal {H}}_\mu ^\varphi = [\pi _\varphi ({\mathcal {A}})V_\varphi {\mathcal {H}}_\mu ] \quad \text {and} \quad {\mathcal {H}}^\varphi = [\pi _\varphi ({\mathcal {A}})V_\varphi {\mathcal {D}}]. \end{aligned}$$

We define the four sets as follows:

$$\begin{aligned} {\mathcal {K}}_\mu ^\Phi =[\Phi (E){\mathcal {H}}_\mu ], \quad {\mathfrak {F}}^\Phi = \bigcup _{\mu \in \Omega }{\mathcal {K}}_\mu ^\Phi , \quad {\mathcal {K}}^\Phi = \overline{{\mathfrak {F}}^\Phi } , \quad {\mathcal {F}}^\Phi = \{{\mathcal {K}}_\mu ^\Phi \}_{\mu \in \Omega }. \end{aligned}$$

Then ${\mathcal {F}}^\Phi$ is an upward filtered family, which is a quantized domain in a Hilbert space ${\mathcal {K}}^\Phi$ with its union space ${\mathfrak {F}}^\Phi$. By construction, we have that ${\mathcal {K}}_\mu ^\Phi \subseteq {\mathcal {K}}_\mu$ for all $\mu \in \Omega$ and ${\mathcal {K}}^\Phi \subseteq {\mathcal {K}}$.

For each $x \in E$, we define a linear map $\Pi _\Phi (x): {\mathcal {D}}_\varphi \rightarrow {\mathfrak {F}}^\Phi$ by

$$\begin{aligned} \Pi _\Phi (x) \bigg ( \sum _{i=1}^n \pi _\varphi (a_i) V_\varphi \xi _i \bigg ) = \sum _{i=1}^n \Phi (xa_i)\xi _i. \end{aligned}$$

For every $\mu \in \Omega$ and $x\in E$, we have that

$$\begin{aligned}&\bigg \Vert \Pi _\Phi (x) \bigg ( \sum _{i=1}^n \pi _\varphi (a_i) V_\varphi \xi _i \bigg ) \bigg \Vert _{{\mathcal {K}}_\mu ^\Phi }^2 = \sum _{i,j=1}^n \big \langle \Phi (xa_i)\xi _i, \Phi (xa_j)\xi _j \big \rangle _{{\mathcal {K}}_\mu ^\Phi } \\&\quad = \sum _{i,j=1}^n \big \langle \xi _i | \varphi (\langle xa_i,xa_j\rangle )\xi _j \big \rangle _{{\mathcal {H}}_\mu } \\&\quad = \sum _{i,j=1}^n \big \langle \xi _i | V_\varphi ^* \pi _\varphi (a_i^*\langle x,x\rangle a_j)V_\varphi \xi _j \big \rangle _{{\mathcal {H}}_\mu } \\&\quad = \sum _{i,j=1}^n \big \langle \pi _\varphi (a_i^*)^*V_\varphi \xi _i,\pi _\varphi (\langle x,x\rangle ) \pi _\varphi (a_j)V_\varphi \xi _j \big \rangle _{{\mathcal {H}}_\mu ^\varphi } \\&\quad = \bigg \langle \sum _{i=1}^n J_\varphi \pi _\varphi (a_i) J_\varphi V_\varphi \xi _i, \pi _\varphi (\langle x,x\rangle ) \bigg ( \sum _{i=1}^n \pi _\varphi (a_i) V_\varphi \xi _i \bigg ) \bigg \rangle _{{\mathcal {H}}_\mu ^\varphi } \\&\quad \le \left\| \pi _\varphi (\langle x,x\rangle )|_{{\mathcal {H}}_\mu ^\varphi }\right\| \cdot \bigg \Vert \sum _{i=1}^n \pi _\varphi (a_i) V_\varphi \xi _i\bigg \Vert _{{\mathcal {H}}_\mu ^\varphi }^2 \end{aligned}$$

where the inequality follows from the equality $J_\varphi V_\varphi \xi _i = V_\varphi \xi _i$. Thus, $\Pi _\Phi (x)$ is bounded on ${\mathcal {K}}_\mu ^\Phi$, so that $\Pi _\Phi (x) \in C^*_{{\mathcal {E}}^\varphi ,{\mathcal {F}}^\Phi }({\mathcal {D}}^\varphi ,{\mathfrak {F}}^\Phi )$.

Lemma 4.3

The map $\Pi _\Phi$ is a $J_\varphi \circ \pi _\varphi$-morphism.

Proof

For any $a_i \in {\mathcal {A}}$ and $\xi _i \in {\mathcal {H}}_\mu$, we have

$$\begin{aligned}&\bigg \langle \Pi _\Phi (x) \bigg (\sum _{i=1}^n \pi _\varphi (a_i) V_\varphi \xi _i \bigg ), \sum _{j=1}^m \Phi (x_j) \eta _j \bigg \rangle _{{\mathcal {K}}_\mu ^\Phi } = \bigg \langle \bigg (\sum _{i=1}^n \Phi (xa_i)\xi _i \bigg ),\sum _{j=1}^m \Phi (x_j) \eta _j \bigg \rangle _{{\mathcal {K}}_\mu ^\Phi } \\&\quad = \sum _{i,j=1}^{n,m} \big \langle \xi _i | V_\varphi ^* \pi _\varphi (a_i^* \langle x, x_j \rangle )V_\varphi \eta _j \big \rangle _{{\mathcal {H}}_\mu } \\&\quad = \sum _{i,j=1}^{n,m} \big \langle \pi _\varphi (a_i^*)^* V_\varphi \xi _i| \pi _\varphi (\langle x, x_j \rangle )V_\varphi \eta _j \big \rangle _{{\mathcal {H}}_\mu ^\varphi } \\&\quad = \bigg \langle \sum _{i=1}^{n} \pi _\varphi (a_i) V_\varphi \xi _i \bigg | \sum _{j=1}^{m} J_\varphi \pi _\varphi (\langle x, x_j \rangle )V_\varphi \eta _j \bigg \rangle _{{\mathcal {H}}_\mu ^\varphi } \end{aligned}$$

Thus, we obtain that for each $x\in E$,

$$\begin{aligned} \Pi _\Phi (x)^*\bigg (\sum _{j=1}^m \Phi (x_j) \eta _j \bigg ) = \sum _{j=1}^m J_\varphi \pi _\varphi (\langle x, x_j \rangle )V_\varphi \eta _j, \end{aligned}$$

so that

$$\begin{aligned} \Pi _\Phi (x)^*\Pi _\Phi (y)\bigg (\sum _{i=1}^n \pi _\varphi (a_i) V_\varphi \xi _i \bigg )&= \Pi _\Phi (x)^*\bigg (\sum _{i=1}^n \Phi (ya_i)\xi _i \bigg ) \\&= \sum _{i=1}^n J_\varphi \pi _\varphi (\langle x,ya_i\rangle ) V_\varphi \xi _i \\&= J_\varphi \pi _\varphi (\langle x,y\rangle ) \sum _{i=1}^n \pi (a_i) V_\varphi \xi _i, \end{aligned}$$

so that $\Pi _\Phi (x)^*\Pi _\Phi (y) = J_\varphi \pi _\varphi (\langle x,y\rangle )$. This means that $\Pi _\Phi$ is a $J_\varphi \circ \pi _\varphi$-morphism on the dense set $\{\pi _\varphi ({\mathcal {A}}) V_\varphi {\mathcal {H}}_\mu \}$ of ${\mathcal {H}}_\mu ^\varphi$ for every $\mu$. By density of ${\mathcal {D}}$ in ${\mathcal {H}}$, $\Pi _\Phi$ is a $J_\varphi \circ \pi _\varphi$-morphism. $\square$

It follows from the construction that ${\mathcal {K}}^\Phi$ is a closed subspace of ${\mathcal {K}}$. If $Q_\Phi$ is the orthogonal projection of ${\mathcal {K}}$ onto ${\mathcal {K}}^\Phi$, then $Q_\Phi Q_\Phi ^* = I_{{\mathcal {K}}^\Phi }$ is the identity operator of ${\mathcal {K}}^\Phi$. Let $x\in E$ and $\xi \in {\mathcal {D}}$. For some $\mu \in \Omega$, we have $\xi \in {\mathcal {H}}_\mu$ and

$$\begin{aligned} Q_\Phi ^* \Pi _\Phi (x) V_\varphi \xi = \Pi _\Phi (x)\big (\pi _\varphi (1_{\mathcal {A}}) V_\varphi \xi \big ) = \Phi (x) \xi , \end{aligned}$$

so that $\Phi (x) \subseteq Q_\Phi ^* \Pi _\Phi (x) V_\varphi$.

Definition 4.4

Let $\varphi : {\mathcal {A}}\rightarrow C^*_{\mathcal {E}}({\mathcal {D}})$ be a local contractive and local $\alpha$-CP map and $\Phi$ be a $\varphi$-map from E into $C^*_{{\mathcal {E}},{\mathcal {F}}}({\mathcal {D}},{\mathfrak {F}})$.

• (1) We say that the pair $((\pi _\varphi ,V_\varphi ,({\mathcal {H}}_\varphi ,J_\varphi ;{\mathcal {E}}^\varphi )), (\Pi _\Phi , Q_\Phi ,({\mathcal {K}}^\Phi ;{\mathcal {F}}^\Phi ))$ is a Krein space J-representation for two maps $(\varphi ,\Phi )$.

• (2) We say that a Krein space J-representation $((\pi _\varphi ,V_\varphi ,({\mathcal {H}}_\varphi ,J_\varphi ;{\mathcal {E}}^\varphi )), (\Pi _\Phi , Q_\Phi ,({\mathcal {K}}^\Phi ;{\mathcal {F}}^\Phi ))$ for two maps $(\varphi ,\Phi )$ is minimal if

  • (i) ${\mathcal {E}}^\varphi \subseteq {\mathcal {E}}$ and ${\mathcal {H}}_\mu ^\varphi = \big [\pi _\varphi ({\mathcal {A}})V_\varphi {\mathcal {H}}_\mu \big ]$ for all $\mu$,

  • (ii) ${\mathcal {F}}^\Phi \subseteq {\mathcal {F}}$ and ${\mathcal {K}}_\mu ^\Phi = \big [\Pi _\Phi (E) V_\varphi {\mathcal {H}}_\mu \big ]$ for all $\mu$.

Theorem 4.5

Let $\varphi$ and $\Phi$ be as in Definition 4.4. If $((\pi _\varphi ,V_\varphi ,({\mathcal {H}}_\varphi ,J_\varphi ;{\mathcal {E}}^\varphi )), (\Pi _\Phi , Q_\Phi ,({\mathcal {K}}^\Phi ;{\mathcal {F}}^\Phi ))$ is a Krein space J-representation for two maps $(\varphi ,\Phi )$, then there exist a minimal Krein space J-representation $((\widetilde{\pi }_\varphi ,\widetilde{V}_\varphi ,(\widetilde{{\mathcal {H}}}_\varphi ,\widetilde{J}_\varphi ;\widetilde{{\mathcal {E}}}^\varphi )), (\widetilde{\Pi }_\Phi , \widetilde{Q}_\Phi ,(\widetilde{{\mathcal {K}}}^\Phi ;\widetilde{{\mathcal {F}}}^\Phi ))$ such that

• (i) $\widetilde{{\mathcal {E}}}^\varphi \subseteq {\mathcal {E}}^\varphi$ and $\widetilde{{\mathcal {F}}}^\Phi \subseteq {\mathcal {F}}^\Phi$,

• (ii) $\widetilde{{\mathcal {H}}}_\mu ^\varphi = [\widetilde{\pi }_\varphi ({\mathcal {A}})\widetilde{V}_\varphi {\mathcal {D}}]$ and $\widetilde{{\mathcal {K}}}^\Phi = [\widetilde{\Pi }_\Phi (E) \widetilde{V}_\varphi {\mathcal {D}}]$.

Proof

By Proposition 3.6, there exists, up to unitary equivalence, a unique minimal Krein space J-representation $(\widetilde{\pi }_\varphi ,\widetilde{V}_\varphi ,(\widetilde{{\mathcal {H}}}_\varphi ,\widetilde{J}_\varphi ;\widetilde{{\mathcal {E}}}^\varphi ))$ such that $\widetilde{{\mathcal {E}}}^\varphi \subseteq {\mathcal {E}}^\varphi$ and $\widetilde{{\mathcal {H}}}^\varphi = [\widetilde{\pi }_\varphi ({\mathcal {A}})\widetilde{V}_\varphi {\mathcal {D}}]$. We note that $\widetilde{V}_\varphi = V_\varphi$ and $\widetilde{\pi }_\varphi (a) = \pi _\varphi (a) |_{\widetilde{{\mathcal {D}}}^\varphi }$. We define the spaces by

$$\begin{aligned} \widetilde{{\mathcal {K}}}_\mu ^\Phi = [\widetilde{\Pi }_\Phi (E) \widetilde{V}_\varphi {\mathcal {H}}_\mu ] , \qquad \widetilde{{\mathfrak {F}}}^\Phi = \bigcup _\mu \widetilde{{\mathcal {K}}}_\mu ^\Phi . \end{aligned}$$

Then we have $\widetilde{{\mathcal {K}}}_\mu ^\Phi \subseteq {\mathcal {K}}_\mu ^\Phi$ and $\widetilde{{\mathcal {K}}}_\mu ^\Phi \subseteq \widetilde{{\mathcal {K}}}_\nu ^\Phi$ for $\mu \le \nu$. We also see that $\widetilde{{\mathcal {F}}}^\Phi :=\{\widetilde{{\mathcal {K}}}_\mu ^\Phi : \mu \in \Omega \}$ is a quantized domain, i.e., an upward filtered family in $\widetilde{{\mathcal {K}}}^\Phi = \overline{\widetilde{{\mathcal {F}}}^\Phi }$ and $\widetilde{{\mathcal {F}}}^\Phi \subseteq {\mathcal {F}}^\Phi$.

For any $x \in E$, we define a map $\widetilde{\Pi }_\Phi (x): \widetilde{{\mathcal {D}}}^\varphi \rightarrow \widetilde{{\mathfrak {F}}}^\Phi$ by

$$\begin{aligned} \widetilde{\Pi }_\Phi (x)= \Pi _\Phi (x)|_{\widetilde{{\mathcal {D}}}^\varphi }. \end{aligned}$$

Then we have $\widetilde{\Pi }_\Phi (x)|_{\widetilde{{\mathcal {H}}}_\mu ^\varphi } \in {\mathcal {B}}(\widetilde{{\mathcal {H}}}_\mu ^\varphi ,\widetilde{{\mathcal {K}}}_\mu ^\Phi )$, so that $\widetilde{\Pi }_\Phi : E \rightarrow C^*_{\widetilde{{\mathcal {E}}}^\varphi ,\widetilde{{\mathcal {F}}}^\Phi }(\widetilde{{\mathcal {D}}}^\varphi ,\widetilde{{\mathfrak {F}}}^\Phi )$ is well-defined. Since $\widetilde{\Pi }_\Phi (x)^*\widetilde{\Pi }_\Phi (y) = J_\varphi \circ \widetilde{\pi }_\varphi (\langle x,y\rangle )$, $\widetilde{\Pi }_\Phi$ is a $J_\varphi \circ \widetilde{\pi }_\varphi$-morphism. Let $\widetilde{Q}_\Phi$ be the orthogonal projection of ${\mathcal {K}}$ onto $\widetilde{{\mathcal {K}}}_\Phi$. Then, for any $x\in E$ and $\xi \in {\mathcal {D}}$, we have

$$\begin{aligned} \widetilde{Q}_\Phi ^* \widetilde{\Pi }_\Phi (x) \widetilde{V}_\varphi \xi = \widetilde{\Pi }_\Phi (x) \widetilde{\pi }_\varphi (1_{\mathcal {A}})V_\varphi \xi =\Phi (x) \xi , \end{aligned}$$

so that $\Phi (x) \subseteq \widetilde{Q}_\Phi ^* \widetilde{\Pi }_\Phi (x) \widetilde{V}_\varphi$. Therefore, $((\widetilde{\pi }_\varphi ,\widetilde{V}_\varphi ,(\widetilde{{\mathcal {H}}}_\varphi ,\widetilde{J}_\varphi ;\widetilde{{\mathcal {E}}}^\varphi )), (\widetilde{\Pi }_\Phi , \widetilde{Q}_\Phi ,(\widetilde{{\mathcal {K}}}^\Phi ;\widetilde{{\mathcal {F}}}^\Phi ))$ is a minimal Krein space J-representation. $\square$

Remark 4.6

By applying the arguments used in the proof of Theorem 5.6 of [5], we can prove the uniqueness up to unitary equivalence of a minimal Krein space J-representation for two maps $(\varphi ,\Phi )$.

5 Radon–Nikodým Theorem for Local $\alpha$-CP Maps

In this section we establish the Radon–Nikodým theorem for local $\alpha$-CP maps.

For notational convenience, the subspace of a vector space X generated by a subset S of X is denoted by $\mathrm{span}(S)$, i.e. $\mathrm{span}(S)$ is the space of all (finite) linear combinations of elements of the set S.

Lemma 5.1

Let $\left( \pi _\varphi ,V_\varphi ,({\mathcal {H}}^\varphi ,J_\varphi ,{\mathcal {E}}^\varphi =\{{\mathcal {H}}_{\mu }^\varphi \}_{\mu \in \Omega })\right)$ be a minimal Krein space J-representation for a local contractive and local $\alpha$-CP map $\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$. Then for each $\mu \in \Omega$, we have

$$\begin{aligned} \left[ \pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \right] =\left( {\mathcal {H}}_{\mu }^\varphi \right) ^\perp . \end{aligned}$$

Proof

The proof is a modification of the proof of Lemma 4.2 of [5]. For any $\mu \in \Omega$, we choose arbitrary elements $u\in \mathrm{span}(\{\pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \})$ and $v\in \mathrm{span}(\{\pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }\right) \})$. We can write as follows;

$$\begin{aligned} u =\sum _{i=1}^n \pi _\varphi (a_i)V_\varphi (\xi _i), \quad v =\sum _{j=1}^m \pi _\varphi (b_j)V_\varphi (\eta _j) \end{aligned}$$

where $a_i, b_j \in {\mathcal {A}}$, $\xi _i\in {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}$ and $b_j\in {\mathcal {A}}$ and $\eta _j\in {\mathcal {H}}_{\mu }$. Then we obtain that

$$\begin{aligned} \left\langle u,v\right\rangle&=\sum _{i=1}^n\sum _{j=1}^m \langle \xi _i|V_\varphi ^*\pi _\varphi (a_i)^*\pi _\varphi (b_j)V_\varphi \eta _j\rangle \\&=\sum _{i=1}^n\sum _{j=1}^m \langle \xi _i|V_\varphi ^*J_\varphi \pi _\varphi (a_i^*)J_\varphi \pi _\varphi (b_j)V_\varphi \eta _j\rangle \\&=\sum _{i=1}^n\sum _{j=1}^m \langle \xi _i|V_\varphi ^*\pi _\varphi (\alpha (a_i^*\alpha (b_j)))V_\varphi \eta _j\rangle \\&=\sum _{i=1}^n\sum _{j=1}^m \langle \xi _i|\varphi (\alpha (a_i^*\alpha (b_j)))\eta _j\rangle \\&=\sum _{i=1}^n\sum _{j=1}^m \langle \xi _i|\varphi (a_i^*\alpha (b_j))\eta _j\rangle =0, \end{aligned}$$

where the sixth equality follows from the fact that $\varphi (a)\eta \in {\mathcal {H}}_{\mu }$ for all $a\in {\mathcal {A}}$ and $\eta \in {\mathcal {H}}_{\mu }$. Therefore, we have

$$\begin{aligned} \mathrm{span}\Big \{\pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \Big \} \subset \Big (\mathrm{span}\{\pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }\right) \}\Big )^\perp , \end{aligned}$$

and so by the minimality of the Krein space J-representation, we have

$$\begin{aligned} \left[ \pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \right] \subset \left( {\mathcal {H}}_{\mu }^\varphi \right) ^\perp . \end{aligned}$$

Suppose that $w\in \left[ \pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \right] ^\perp \cap \left( {\mathcal {H}}_{\mu }^\varphi \right) ^\perp$. Let $\zeta \in {\mathcal {D}}$ be given. Then there exist vectors $\xi \in {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}$ and $\eta \in {\mathcal {H}}_{\mu }$ such that $\zeta =\xi +\eta$, and so for any $a\in {\mathcal {A}}$, we have

$$\begin{aligned} \langle w,\pi _\varphi (a)V_\varphi \zeta \rangle =\langle w,\pi _\varphi (a)V_\varphi \xi \rangle +\langle w,\pi _\varphi (a)V_\varphi \eta \rangle =0, \end{aligned}$$

which implies that since ${\mathcal {H}}^\varphi =[\pi _\varphi ({\mathcal {A}})V_\varphi {\mathcal {D}}]$, $\langle w,y\rangle =0$ for all $y\in {\mathcal {H}}^\varphi$, and so $w=0$. Therefore, we have $\left[ \pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \right] ^\perp \cap \left( {\mathcal {H}}_{\mu }^\varphi \right) ^\perp =\{0\}$, and hence $\left[ \pi _\varphi ({\mathcal {A}})V_\varphi \left( {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}\right) \right] =\left( {\mathcal {H}}_{\mu }^\varphi \right) ^\perp$. $\square$

For a local contractive and local $\alpha$-CP map $\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$, let $(\pi _\varphi ,V_\varphi ,({\mathcal {H}}^\varphi ,J_\varphi ,{\mathcal {E}}^\varphi =\{{\mathcal {H}}_{\mu }^\varphi \}))$ be its minimal Krein space J-representation. We put

$$\begin{aligned} \pi _\varphi ({\mathcal {A}})'=\{T\in {\mathcal {B}}({\mathcal {H}}^{\varphi }):T\pi _\varphi (a)\subset \pi _\varphi (a) T\text { for all }a\in {\mathcal {A}}\}, \end{aligned}$$

which is called the commutant of $\pi _\varphi ({\mathcal {A}})$ in ${\mathcal {B}}({\mathcal {H}}^{\varphi })$. Then for each $T\in \pi _\varphi ({\mathcal {A}})'\cap C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$ with $J_\varphi T\subset TJ_\varphi$, and $a\in {\mathcal {A}}$, the map $\varphi _T(a)$ defined on ${\mathcal {D}}$ by

$$\begin{aligned} \varphi _T(a):=V_\varphi ^*T\pi _\varphi (a)V_\varphi |_{{\mathcal {D}}} \end{aligned}$$

becomes an element of $C_{{\mathcal {E}}}^*({\mathcal {D}})$. In fact, for each fixed $\mu \in \Omega$, any $\xi \in {\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}$ and $\eta \in {\mathcal {H}}_{\mu }$, by applying Lemma 5.1, we see that $V_\varphi \xi =\pi _\varphi (1_{\mathcal {A}})V_\varphi (\xi )\in ({\mathcal {H}}_{\mu }^\varphi )^\perp$ and since $\pi _\varphi (a)\in C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$, $T\pi _\varphi (a)V_\varphi \eta \in {\mathcal {H}}_{\mu }^\varphi$, and hence we obtain that

$$\begin{aligned} \langle \xi |\varphi _T(a)\eta \rangle&=\langle \xi |V_\varphi ^*T\pi _\varphi (a)V_\varphi \eta \rangle =\langle V_\varphi \xi ,T\pi _\varphi (a)V_\varphi \eta \rangle =0. \end{aligned}$$

Therefore, ${\mathcal {H}}_{\mu }$ and ${\mathcal {H}}_{\mu }^\perp \cap {\mathcal {D}}$ are invariant under $\varphi _T(a)$ and $\varphi _T(a)P_{\mu }=P_{\mu }\varphi _T(a)P_{\mu }\in {\mathcal {B}}({\mathcal {H}})$, and hence by (5) of Remark 2.6, we see that $\varphi _T(a)\in C_{{\mathcal {E}}}^*({\mathcal {D}})$. We now define a map:

$$\begin{aligned} \varphi _T:{\mathcal {A}}\ni a\mapsto \varphi _T(a)=V_\varphi ^*T\pi _\varphi (a)V_\varphi |_{{\mathcal {D}}}\in C_{{\mathcal {E}}}^*({\mathcal {D}}). \end{aligned}$$

(1)

Then the correspondence $T \mapsto \varphi _T$ is linear and injective. Indeed, if $\varphi _T =0$, then for any $a,b \in {\mathcal {A}}$ and $\xi ,\eta \in {\mathcal {D}}$, we obtain that

$$\begin{aligned} \begin{aligned} \langle T\pi _\varphi (a)V_\varphi \xi ,\pi _\varphi (b)V_\varphi \eta \rangle&= \langle V_\varphi ^*\pi _\varphi (b)^*T\pi _\varphi (a)V_\varphi \xi |\eta \rangle \\&= \langle V_\varphi ^*J_\varphi \pi _\varphi (b^*)J_\varphi T\pi _\varphi (a)V_\varphi \xi |\eta \rangle \\&= \langle V_\varphi ^*TJ_\varphi \pi _\varphi (b^*)J_\varphi \pi _\varphi (a)V_\varphi \xi |\eta \rangle \\&= \langle V_\varphi ^*T\pi _\varphi (\alpha (b^*\alpha (a)))V_\varphi \xi |\eta \rangle \\&= \langle \varphi _T(\alpha (b^*\alpha (a)))\xi |\eta \rangle \\&= 0. \end{aligned} \end{aligned}$$

Since the set $\{\pi _\varphi (a)V_\varphi \xi : a\in {\mathcal {A}}, ~\xi \in {\mathcal {D}}\}$ is dense in ${\mathcal {H}}^\varphi$, we have $T=0$. Furthermore, $\varphi _{I_{{\mathcal {H}}^\varphi }}=\varphi$, where $I_{{\mathcal {H}}^\varphi }$ is the identity operator on ${\mathcal {H}}^\varphi$.

Lemma 5.2

Let $\left( \pi _\varphi ,V_\varphi ,({\mathcal {H}}^\varphi ,J_\varphi ;{\mathcal {E}}^\varphi =\{{\mathcal {H}}_{\mu }^\varphi \}_{\mu \in \Omega })\right)$ be a minimal Krein space J-representation of a local contractive and local $\alpha$-CP map $\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$. Let $T\in \pi _\varphi ({\mathcal {A}})'\cap C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$ be given such that $J_\varphi T\subseteq TJ_\varphi$. If the map $\varphi _T$ given by (1) is local $\alpha$-CP, then T is local positive, i.e. for any $\mu \in \Omega$, $T|_{{\mathcal {H}}_{\mu }^\varphi }$ is positive as in ${\mathcal {B}}({\mathcal {H}}_{\mu }^\varphi )$.

Proof

Let $\mu \in \Omega$ be given. For any $n \in \mathbb {N}$, $a_1,\ldots ,a_n \in {\mathcal {A}}$ and $\xi _1,\ldots ,\xi _n \in {\mathcal {H}}_{\mu }$, we obtain that

$$\begin{aligned} \begin{aligned} \left\langle T\sum _{i=1}^n \pi _\varphi (a_i)V_\varphi \xi _i,\sum _{i=1}^n \pi _\varphi (a_i)V_\varphi \xi _i \right\rangle&= \sum _{i,j=1}^n \langle V_\varphi ^*TJ_\varphi \pi _\varphi (a_j^*)J_\varphi \pi _\varphi (a_i)V_\varphi \xi _i|\xi _j \rangle \\&= \sum _{i,j=1}^n \langle V_\varphi ^*T\pi _\varphi (\alpha (a_j^*\alpha (a_i))V_\varphi \xi _i|\xi _j \rangle \\&= \sum _{i,j=1}^n \langle \varphi _T(\alpha (a_j)^*a_i)\xi _i|\xi _j \rangle \ge 0, \end{aligned} \end{aligned}$$

which means that $T \ge 0$ because $\pi ({\mathcal {A}})V{\mathcal {H}}_{\mu }$ is dense in ${\mathcal {H}}_{\mu }^\varphi$. $\square$

Let $\psi ,\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ be local contractive and local $\alpha$-CP maps. Then $\psi$ is said to be dominated by $\varphi$, denoted by $\psi \le \varphi$, if the difference $\varphi -\psi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ is a local contractive and local $\alpha$-CP map.

Proposition 5.3

Let $\left( \pi _\varphi ,V_\varphi ,({\mathcal {H}}^\varphi ,J_\varphi ,{\mathcal {E}}^\varphi =\{{\mathcal {H}}_{\mu }^\varphi \})\right)$ be a minimal Krein space J-representation of a local contractive and local $\alpha$-CP map $\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$. If $T\in \pi _\varphi ({\mathcal {A}})'\cap C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$ is a local positive operator such that $J_\varphi T\subseteq TJ_\varphi$, then $\varphi _T:{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ defined by $\varphi _T(a)=V_\varphi ^*T\pi _\varphi (a)V_\varphi |_{{\mathcal {D}}}$ is a local $\alpha$-CP map from ${\mathcal {A}}$ into $C_{{\mathcal {E}}}^*({\mathcal {D}})$. Furthermore, if $0\le _{\mu } T\le _{\mu } I$ for all $\mu \in \Omega$, then $\varphi _T\le \varphi$.

Proof

The first part of the proof is a slight modification of the proof of [8, Proposition 3.1]. We note that for any $a\in {\mathcal {A}}$, $\pi _\varphi (a)^*=J_\varphi \pi _\varphi (a^*)J_\varphi$ and $\pi _\varphi (\alpha (a))V_\varphi =J_\varphi \pi _\varphi (a)V_\varphi$. Therefore, for any $a,b \in {\mathcal {A}}$ and $\xi , \eta \in {\mathcal {D}}$, we obtain that

$$\begin{aligned} \bigl \langle \varphi _T(\alpha (a)\alpha (b))\xi |\eta \bigr \rangle&= \bigl \langle T\pi _\varphi (\alpha (b))V_\varphi \xi ,J_\varphi \pi _\varphi (\alpha (a^*))J_\varphi V_\varphi \eta \bigr \rangle \\&= \bigl \langle J_\varphi T\pi _\varphi (b)V_\varphi \xi , \pi _\varphi (a^*)J_\varphi V_\varphi \eta \bigr \rangle \\&= \bigl \langle V_\varphi ^*T \pi _\varphi (ab)V_\varphi \xi | \eta \bigr \rangle \\&= \bigl \langle \varphi _T(ab)\xi | \eta \bigr \rangle . \end{aligned}$$

Hence we have that $\varphi _T(\alpha (a)\alpha (b))=\varphi _T(ab)$ for any $a,b \in {\mathcal {A}}$, so that $\varphi _T(\alpha (ab))=\varphi _T(ab)$. Let $n\in \mathbb {N}$, $a_1,\ldots ,a_n \in {\mathcal {A}}$ and $\xi _1,\ldots ,\xi _n \in {\mathcal {H}}_{\mu }$ for $\mu \in \Omega$. Then we have that

$$\begin{aligned} \sum _{i,j=1}^n \langle \varphi _T(\alpha (a_j)^*a_i)\xi _i|\xi _j\rangle =\sum _{i,j=1}^n \langle \pi _\varphi (\alpha (a_j)^*a_i) T^{1/2}V_\varphi \xi _i, T^{1/2}V_\varphi \xi _j\rangle \ge 0. \end{aligned}$$

Let $a,a_1,\ldots ,a_n \in {\mathcal {A}}$. Since $\varphi$ is local $\alpha$-CP, for any $\mu \in \Omega$, there exists a positive constant C(a) such that

$$\begin{aligned} \Big (\varphi (\alpha (aa_j)^*aa_i)\Big ) \le _{\mu } C(a)\Big (\varphi (\alpha (a_j)^*a_i)\Big ), \end{aligned}$$

where a big parenthesis denotes an $n\times n$ matrix. Equivalently, for any $\xi _1,\ldots ,\xi _n \in {\mathcal {H}}_{\mu }$

$$\begin{aligned} 0 \le \sum _{i,j=1}^n \langle \pi _\varphi (\alpha (aa_j)^*aa_i)V_\varphi \xi _i, V_\varphi \xi _j\rangle \le C(a)\sum _{i,j=1}^n \langle \pi _\varphi (\alpha (a_j)^*a_i)V_\varphi \xi _i, V_\varphi \xi _j\rangle . \end{aligned}$$

Since T commutes with $\pi _\varphi (a)$ and is positive, it follows that

$$\begin{aligned} \begin{aligned}&\sum _{i,j=1}^n \langle T\pi _\varphi (\alpha (aa_j)^*aa_i) V_\varphi \xi _i, V_\varphi \xi _j\rangle \\&\quad = \sum _{i,j=1}^n \langle \pi _\varphi (\alpha (aa_j)^*aa_i) T^{1/2}V_\varphi \xi _i, T^{1/2}V_\varphi \xi _j\rangle \\&\quad \le C(a)\sum _{i,j=1}^n \langle \pi _\varphi (\alpha (a_j)^*a_i) T^{1/2}V_\varphi \xi _i, T^{1/2}V_\varphi \xi _j\rangle \\&\quad = C(a)\sum _{i,j=1}^n \langle T\pi _\varphi (\alpha (a_j)^*a_i) V_\varphi \xi _i,V_\varphi \xi _j\rangle , \end{aligned} \end{aligned}$$

which means that $\bigl (\varphi _T(\alpha (aa_j)^*aa_i)\bigr ) \le _{\mu } C(a)\bigl (\varphi _T(\alpha (a_j)^*a_i)\bigr )$ for any $\mu \in \Omega$. Therefore, the map $\varphi _T$ is local $\alpha$-CP.

We now suppose that $0\le _{\mu }T\le _{\mu } I$ for all $\mu \in \Omega$. Let $n\in \mathbb {N}$, $a_1,\ldots ,a_n \in {\mathcal {A}}$ and $\xi _1,\ldots ,\xi _n \in {\mathcal {H}}_{\mu }$ for $\mu \in \Omega$. Then since $I-T\ge _{\mu }0$ on ${\mathcal {H}}_{\mu }^\varphi$, we have that

$$\begin{aligned}&\sum _{i,j=1}^n \langle \left[ \varphi (\alpha (a_j)^*a_i)-\varphi _T(\alpha (a_j)^*a_i)\right] \xi _i|\xi _j\rangle \\&\quad =\sum _{i,j=1}^n \langle (I-T)\pi _\varphi (\alpha (a_j)^*a_i) V_\varphi \xi _i,V_\varphi \xi _j\rangle \ge 0, \end{aligned}$$

and for $a\in {\mathcal {A}}$, we obtain that

$$\begin{aligned} \begin{aligned}&\sum _{i,j=1}^n \langle (I-T)\pi _\varphi (\alpha (aa_j)^*aa_i) V_\varphi \xi _i, V_\varphi \xi _j\rangle \\&\qquad \le C(a)\sum _{i,j=1}^n \langle \pi _\varphi (\alpha (a_j)^*a_i) (I-T)^{1/2}V_\varphi \xi _i, (I-)T^{1/2}V_\varphi \xi _j\rangle \\&\qquad = C(a)\sum _{i,j=1}^n \langle (I-T)\pi _\varphi (\alpha (a_j)^*a_i) V_\varphi \xi _i,V_\varphi \xi _j\rangle . \end{aligned} \end{aligned}$$

Therefore, $\varphi -\varphi _T:{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ is a local contractive and local $\alpha$-CP map, and so $\varphi _T\le \varphi$. $\square$

Lemma 5.4

Let $\psi ,\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ be local contractive and local $\alpha$-CP maps such that $\psi \le \varphi$. Let $\big (\pi _\psi ,V_\psi ,({\mathcal {H}}^\psi ,J_\psi ;{\mathcal {E}}^\psi =\{{\mathcal {H}}_{\mu }^\psi \}_{\mu \in \Omega })\big )$ and $\big (\pi _\varphi ,V_\varphi ,({\mathcal {H}}^\varphi ,J_\varphi ;{\mathcal {E}}^\varphi =\{{\mathcal {H}}_{\mu }^\varphi \}_{\mu \in \Omega })\big )$ be minimal Krein space J-representations of $\psi$ and $\varphi$, respectively. There exists a unique contraction $S \in {\mathcal {B}}({\mathcal {H}}^\varphi ,{\mathcal {H}}^\psi )$ such that

• (i) for any $\mu \in \Omega$, $S|_{{\mathcal {H}}_{\mu }^\varphi }$ is a contraction as in ${\mathcal {B}}({\mathcal {H}}_{\mu }^\varphi ,{\mathcal {H}}_{\mu }^\psi )$,

• (ii) $SV_\varphi =V_\psi$ and $SJ_\varphi =J_\psi S$,

• (iii) $S\pi _\varphi (a)\subseteq \pi _\psi (a)S$ for any $a \in {\mathcal {A}}$.

• (iv) $S^*J_\psi \pi _\psi (a)J_\psi \subseteq J_\varphi \pi _\varphi (a)J_\varphi S^*$ for any $a \in {\mathcal {A}}$.

Proof

We denote by ${\mathcal {D}}^\psi =\bigcup _{\mu \in \Omega }{\mathcal {H}}_{\mu }^\psi$ and ${\mathcal {D}}^\varphi =\bigcup _{\mu \in \Omega }{\mathcal {H}}_{\mu }^\varphi$ the union spaces of quantized domains ${\mathcal {E}}^\psi$ and ${\mathcal {E}}^\varphi$ in ${\mathcal {H}}^\psi$ and ${\mathcal {H}}^\varphi$, respectively. Let $\mu \in \Omega$ be given. For each $a_1,\ldots ,a_n \in {\mathcal {A}}$ and $\xi _1,\ldots ,\xi _n \in H_\mu$, we have that

$$\begin{aligned} \begin{aligned} \left\| \sum _{i=1}^n \pi _\psi (a_i)V_\psi \xi _i\right\| _{{\mathcal {H}}_{\mu }^\psi }^2&= \sum _{i,j=1}^n \langle V_\psi ^*\pi _\psi (\alpha (a_j)^*a_i)V_\psi \xi _i| \xi _j\rangle \\&= \sum _{i,j=1}^n \langle \psi (\alpha (a_j)^*a_i)\xi _i| \xi _j\rangle \\&\le \sum _{i,j=1}^n \langle \varphi (\alpha (a_j)^*a_i)\xi _i| \xi _j\rangle =\left\| \sum _{i=1}^n \pi _\varphi (a_i)V_\varphi \xi _i\right\| _{{\mathcal {H}}_{\mu }^\varphi }^2 . \end{aligned} \end{aligned}$$

Therefore, there exist a unique contraction $S\in {\mathcal {B}}({\mathcal {H}}_\mu ^\varphi ,{\mathcal {H}}_{\mu }^\psi )$ which satisfies the equation

$$\begin{aligned} S\biggl (\sum _{i=1}^n \pi _\varphi (a_i)V_\varphi \xi _i \biggr ) = \sum _{i=1}^n \pi _\psi (a_i)V_\psi \xi _i, \end{aligned}$$

and we extend S to ${\mathcal {D}}^\varphi$ and its extension is denoted by S again. Then S becomes a contraction in ${\mathcal {B}}({\mathcal {H}}^\varphi ,{\mathcal {H}}^\psi )$. Also, we have that $S(\pi _\varphi (a)V_\varphi \xi )=\pi _\psi (a)V_\psi \xi$ for any $a \in {\mathcal {A}}$ and $\xi \in {\mathcal {D}}$, so we get $SV_\varphi =V_\psi$ by taking $a=1_{\mathcal {A}}$. Moreover, for any $a \in {\mathcal {A}}$ and $\xi \in {\mathcal {D}}$

$$\begin{aligned} \begin{aligned} J_\psi S\bigl (\pi _\varphi (a)V_\varphi \xi \bigr )&= J_\psi \pi _\psi (a)V_\psi \xi =\pi _\psi (\alpha (a))V_\psi \xi \\&= S \bigl (\pi _\varphi (\alpha (a))V_\varphi \xi \bigr ) = SJ_\varphi \bigl (\pi _\varphi (a)V_\varphi \xi \bigr ), \end{aligned} \end{aligned}$$

which implies that $J_\varphi S=SJ_\psi$. Let $a,b \in {\mathcal {A}}$ and $\xi \in {\mathcal {D}}$. We have that

$$\begin{aligned} S\pi _\varphi (a)(\pi _\varphi (b)V_\varphi \xi )= S(\pi _\varphi (ab)V_\varphi \xi )=\pi _\psi (ab)V_\psi \xi =\pi _\psi (a)S(\pi _\varphi (b)V_\varphi \xi ). \end{aligned}$$

Hence we have that $S\pi _\varphi (a)\subseteq \pi _\psi (a)S$ for any $a \in {\mathcal {A}}$. The property stated in (iv) is the dual property of (iii). The proof of the uniqueness of S is straightforward. $\square$

Theorem 5.5

Let $\psi ,\varphi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ be local contractive and local $\alpha$-CP maps. Then $\psi \le \varphi$ if and only if there exists $T\in \pi _\varphi ({\mathcal {A}})'\cap C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$ with $0\le T\le I$ and $J_\varphi T=TJ_\varphi$ such that

$$\begin{aligned} \psi (a)=V_\varphi ^*T\pi _\varphi (a)V_\varphi |_{{\mathcal {D}}} \end{aligned}$$

for all $a\in {\mathcal {A}}$.

Proof

We first assume that $T\in \pi _\varphi ({\mathcal {A}})'\cap C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$ with $0\le T\le I$ and $J_\varphi T=TJ_\varphi$. Let $\psi :{\mathcal {A}}\rightarrow C_{{\mathcal {E}}}^*({\mathcal {D}})$ be defined by $\psi (a)=V_\varphi ^*T\pi _\varphi (a)V_\varphi |_{{\mathcal {D}}}$ for all $a\in {\mathcal {A}}$. Then by Proposition 5.3, we see that $\psi \le \varphi$.

Conversely, suppose that $\psi \le \varphi$. Let $\big (\pi _\psi ,V_\psi ,({\mathcal {H}}^\psi ,J_\psi ,{\mathcal {E}}^\psi =\{{\mathcal {H}}_{\mu }^\psi \})\big )$ be a minimal Krein space J-representation of $\psi$. Then by Lemma 5.4, there exists a unique contraction $S \in {\mathcal {B}}({\mathcal {H}}^\varphi ,{\mathcal {H}}^\psi )$ such that (i), (ii), (iii) and (iv) of Lemma 5.4 hold. Put $T=S^*S$. Since S is a contraction in ${\mathcal {B}}({\mathcal {H}}^\varphi ,{\mathcal {H}}^\psi )$, we see that $0\le T\le I$ and from (ii) of Lemma 5.4, we see that

$$\begin{aligned} J_\varphi T =J_\varphi S^*S =S^*J_\psi S =S^*SJ_\varphi =TJ_\varphi . \end{aligned}$$

From the property stated in (i) of Lemma 5.4, we see that $T\in C_{{\mathcal {E}}^\varphi }^*({\mathcal {D}}^\varphi )$. It also follows from (iii) and (iv) of Lemma 5.4 that for any $a\in A$,

$$\begin{aligned} T\pi _\varphi (a)&=S^*S\pi _\varphi (a)\\&\subseteq S^*\pi _\psi (a)S=(S^*J_\psi )(J_\psi \pi _\psi (a)J_\psi )(J_\psi S) =J_\varphi S^*(J_\psi \pi _\psi (a)J_\psi )S J_\varphi \\&\subseteq J_\varphi (J_\varphi \pi _\varphi (a)J_\varphi ) S^*S J_\varphi =\pi _\varphi (a)J_\varphi T J_\varphi =\pi _\varphi (a)T, \end{aligned}$$

which implies that $T\in \pi _\varphi ({\mathcal {A}})'$. For any $\xi , \eta \in {\mathcal {D}}$ and $a \in {\mathcal {A}}$, we obtain that

$$\begin{aligned} \begin{aligned} \langle \varphi _{T}(a) \xi |\eta \rangle&= \langle V_\varphi ^*T\pi _\varphi (a)V_\varphi \xi |\eta \rangle = \langle V_\varphi ^*S^*S\pi _\varphi (a)V_\varphi \xi |\eta \rangle \\&= \langle S\pi _\varphi (a)V_\varphi \xi ,SV_\varphi \eta \rangle = \langle \pi _\psi (a)SV_\varphi \xi ,SV_\varphi \eta \rangle \\&= \langle \pi _\psi (a)V_\psi \xi ,V_\psi \eta \rangle = \langle V_\psi ^* \pi _\psi (a)V_\psi \xi |\eta \rangle \\&= \langle \psi (a)\xi |\eta \rangle , \end{aligned} \end{aligned}$$

which implies that $\varphi _{T}(a)= \psi (a)$ for any $a\in {\mathcal {A}}$. This completes the proof. $\square$