\section{Introduction}

Locally \ $C^{\ast }$-algebras are generalizations of $C^{\ast }$-algebras,
the topology on a locally $C^{\ast }$-algebra is defined by an upward
directed family of $C^{\ast }$-seminorms. The concrete models for locally
C^{\ast }$-algebras are $\ast $-algebras of unbounded linear operators on a
Hilbert space. In the literature, the locally $C^{\ast }$-algebras are
studied under different names like pro-$C^{\ast }$-algebras (D. Voiculescu,
N.C. Philips), $LMC^{\ast }$-algebras (G. Lassner, K. Schm\"{u}dgen),
b^{\ast }$-algebras (C. Apostol) and multinormed $C^{\ast }$-algebras (A.
Dosiev). The term locally \ $C^{\ast }$-algebra is due to A. Inoue \cite{I}.

A locally $C^{\ast }$-algebra is a complete Hausdorff complex topological
\ast $-algebra $\mathcal{A}$ whose topology is determined by an upward
filtered family $\{p_{\lambda }\}_{\lambda \in \Lambda }\ $of $C^{\ast }
-seminorms defined on $\mathcal{A}$. An element $a\in \mathcal{A}$ is
positive if there exists $b\in \mathcal{A}$ such that $a=b^{\ast }b$ and it
is local positive if there exist $c,d\in \mathcal{A}$ and $\lambda \in
\Lambda $ such that $a=c^{\ast }c+d$ and $p_{\lambda }\left( d\right) =0$.
Thus, the notion of (local) completely positive maps appeared naturally
while studying linear maps between locally $C^{\ast }$-algebras. In \cit
{MJ3}, we showed that an unbounded map is local completely positive if and
only if it is continuous and completely positive. The structure of the
unbounded local completely positive and local completely contractive maps on
unital locally $C^{\ast }$-algebras is described in \cite{D1}. More exactly,
Dosiev proved a Stinespring type theorem for unbounded local $\mathcal{CCP}$
(completely contractive and completely positive) maps on unital locally
C^{\ast }$-algebras. Bhat, Ghatak and Kumar \cite{BGK} introduced a partial
order relation on the set of all unbounded local $\mathcal{CCP}$-maps on a
unital locally $C^{\ast }$-algebra $\mathcal{A}$ and described this relation
in terms of their minimal Stinespring dilations. In \cite{MJ1}, we
determined the structure of unbounded local $\mathcal{CCP}$-maps on unital
locally $C^{\ast }$-algebras which preserve the local orthogonality and in
\cite{MJ3}, we proved some factorization properties for unbounded local
positive maps. In \cite{MJ2}, we proved a local convex version of
Wittstock's extension theorem and a Stinespring type theorem for unbounded
local $\mathcal{CC}$ (completely contractive) maps. In this paper, we
consider the unbounded local $\mathcal{CCP}$-maps on maximal tensor product
of unital locally $C^{\ast }$-algebras and discuss on extremal points of
certain convex subsets in the set of such maps.

In \cite{HHP}, Haapasalo, Heinosaari and Pellonp\"{a}\"{a} considered
bounded $\mathcal{CP\ }$(completely positive) maps on tensor product of von
Neumann algebras and showed that if one of the marginal maps of such map is
an extremal point, then the marginal maps uniquely determine the map. We
extend the results of \cite{HHP} to unbounded local $\mathcal{CCP}$-maps on
maximal tensor product of unital locally $C^{\ast }$-algebras.

The paper is organized as follows. In Section $2$, we gather some basic
facts on locally $C^{\ast }$-algebras, concrete models for locally $C^{\ast }
$-algebras and unbounded local $\mathcal{CCP}$-maps that are needed for
understanding the main results of the paper. Section $3$ is devoted to the
characterization of the extremal elements in certain convex subsets of
\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))
$. The extremal points in $[0,\varphi ]$ are those maps of the form $\varphi
_{P}$, where $P$ is a projection in $\pi _{\varphi }\left( \mathcal{A
\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }})$
and $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi },\mathcal{E
^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ is a minimal
Stinespring dilation of $\varphi $ (Proposition \ref{0}). Also we show that
the extremal elements in $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast
}(\mathcal{D}_{\mathcal{E}}))$ are those maps $\varphi $ for which the
linear map $T\in \pi _{\varphi }\left( \mathcal{A}\right) ^{^{\prime }}\cap
C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }})\mapsto $ $V_{\varphi }^{\ast
}TV_{\varphi }\ $is injective, where $\left( \pi _{\varphi },V_{\varphi },\
\mathcal{H}^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E
^{\varphi }}\}\right) $ is a minimal Stinespring dilation of $\varphi $
(Proposition \ref{1}). In Section $4$, we show that two unital unbounded
local $\mathcal{CCP}$-maps on unital locally $C^{\ast }$-algebras $\mathcal{
}$ and $\mathcal{B}$, whose ranges commute, induce an unbounded local
\mathcal{CCP}$-map an the maximal tensor product $\mathcal{A\otimes }_{\max
\mathcal{B}$ of $\mathcal{A}$ and $\mathcal{B}$ (Proposition \ref{tmax}).
Section $5$ is devoted to marginals maps and joint maps. The structure of an
unbounded local $\mathcal{CCP}$-map on the maximal tensor product of locally
$C^{\ast }$-algebras in terms of a minimal Stinespring dilation of one of
its marginal map is given in Theorem \ref{Help}. As in the case of bounded
\mathcal{CP}$-maps, we show that if one of the marginal maps is an extremal
point, then the joint map is unique (Theorem \ref{C} $(1)$) and when both
marginals maps are extremal points, then the unique joint map is extremal
(Theorem \ref{C} $(2)$).

\section{Preliminaries}

\paragraph{\textbf{Locally }$C^{\ast }$\textbf{-algebras}}

A \textit{locally }$C^{\ast }$\textit{-algebra }is a complete Hausdorff
complex topological $\ast $-algebra $\mathcal{A}$ whose topology is
determined by an upward filtered family $\{p_{\lambda }\}_{\lambda \in
\Lambda }\ $of $C^{\ast }$-seminorms defined on $\mathcal{A}$.

An element $a\in \mathcal{A}$ is \textit{bounded} if $\sup \{p_{\lambda
}\left( a\right) ;\lambda \in \Lambda \}<\infty .$ Then $b\left( \mathcal{A
\right) =\{a\in \mathcal{A};\left\Vert a\right\Vert _{\infty }:=\sup
\{p_{\lambda }\left( a\right) ;\lambda \in \Lambda \}<\infty \}\ $is a
C^{\ast }$-algebra with respect to the $C^{\ast }$-norm $\left\Vert \cdot
\right\Vert _{\infty }$. Moreover, $b\left( \mathcal{A}\right) $ is dense in
$\mathcal{A}.$

We see that $\mathcal{A}$ can be realized as a projective limit of an
inverse family of $C^{\ast }$-algebras as follows: For each $\lambda \in
\Lambda $, let $\mathcal{I}_{\lambda }=\{a\in \mathcal{A};p_{\lambda }\left(
a\right) =0\}$. Clearly, $\mathcal{I}_{\lambda }$ is a closed two sided
\ast $-ideal in $\mathcal{A}$ and $\mathcal{A}_{\lambda }=\mathcal{A}
\mathcal{I}_{\lambda }$ is a $C^{\ast }$-algebra with respect to the norm
induced by $p_{\lambda }$. The canonical quotient $\ast $-morphism from
\mathcal{A\ }$to $\mathcal{A}_{\lambda }$ is denoted by $\pi _{\lambda }^
\mathcal{A}}$. For each $\lambda _{1},\lambda _{2}\in \Lambda $ with
\lambda _{1}\leq \lambda _{2}$, there is a canonical surjective $\ast
-morphism $\pi _{\lambda _{2}\lambda _{1}}^{\mathcal{A}}:$ $\mathcal{A
_{\lambda _{2}}\rightarrow \mathcal{A}_{\lambda _{1}}$ given by $\pi
_{\lambda _{2}\lambda _{1}}^{\mathcal{A}}\left( a+\mathcal{I}_{\lambda
_{2}}\right) =a+\mathcal{I}_{\lambda _{1}},$ $a\in \mathcal{A}$. Then, $\
\mathcal{A}_{\lambda },\pi _{\lambda _{2}\lambda _{1}}^{\mathcal{A}}\}$\
forms an inverse system of $C^{\ast }$-algebras, since $\pi _{\lambda _{1}}^
\mathcal{A}}=$ $\pi _{\lambda _{2}\lambda _{1}}^{\mathcal{A}}\circ \pi
_{\lambda _{2}}^{\mathcal{A}}$ whenever $\lambda _{1}\leq \lambda _{2}$. The
projective limi
\begin{equation*}
\lim\limits_{\underset{\lambda }{\leftarrow }}\mathcal{A}_{\lambda
}=\{\left( a_{\lambda }\right) _{\lambda \in \Lambda }\in
\tprod\limits_{\lambda \in \Lambda }\mathcal{A}_{\lambda };\pi _{\lambda
_{2}\lambda _{1}}^{\mathcal{A}}\left( a_{\lambda _{2}}\right) =a_{\lambda
_{1}}\text{ whenever }\lambda _{1}\leq \lambda _{2},\lambda _{1},\lambda
_{2}\in \Lambda \}
\end{equation*
of the inverse system of $C^{\ast }$-algebras $\{\mathcal{A}_{\lambda },\pi
_{\lambda _{2}\lambda _{1}}^{\mathcal{A}}\}$ is a locally $C^{\ast }
-algebra that can be identified with $\mathcal{A}$ by the map $a\mapsto
\left( \pi _{\lambda }^{\mathcal{A}}\left( a\right) \right) _{\lambda \in
\Lambda }.$

\paragraph{\textbf{Positive and local positive elements}}

Let $\mathcal{A}$ be a locally $C^{\ast }$-algebra with the topology defined
by the family of $C^{\ast }$-seminorms $\left\{ p_{\lambda }\right\}
_{\lambda \in \Lambda }.$

An element $a\in \mathcal{A}$ is \textit{self-adjoint} if $a^{\ast }=a$ and
it is\textit{\ positive} if $a=b^{\ast }b$ for some $b\in \mathcal{A}.$

An element $a\in \mathcal{A}$ is called \textit{local self-adjoint} if
a=a^{\ast }+c$, where $c\in \mathcal{A}$ such that $p_{\lambda }\left(
c\right) =0$ for some $\lambda \in \Lambda $, and we call $a$ as $\lambda
-self-adjoint, and \textit{local positive} if $a=b^{\ast }b+c$, where
b,c\in $ $\mathcal{A}$ such that $p_{\lambda }\left( c\right) =0\ $ for some
$\lambda \in \Lambda $, we call $a$ as $\lambda $-positive and note $a\geq
_{\lambda }0$. We write, $a=_{\lambda }0$ whenever $p_{\lambda }\left(
a\right) =0$. Note that $a\in \mathcal{A}$ is local self-adjoint if and only
if there is $\lambda \in \Lambda $ such that $\pi _{\lambda }^{\mathcal{A
}\left( a\right) $ is self adjoint in $\mathcal{A}_{\lambda }$ and $a\in
\mathcal{A}$ is local positive if and only if there is $\lambda \in \Lambda $
such that $\pi _{\lambda }^{\mathcal{A}}\left( a\right) $ is positive in
\mathcal{A}_{\lambda }.$

\paragraph{\textbf{Quantized domains}}

Throughout the paper, $\mathcal{H}$ is a complex Hilbert space and $B
\mathcal{H})$ is the algebra of all bounded linear operators on the Hilbert
space $\mathcal{H}$.

Let $(\Upsilon ,\leq )$ be a directed poset. A \textit{quantized domain }in
a Hilbert space $\mathcal{H}$ is a triple $\{\mathcal{H},\mathcal{E}
\mathcal{D}_{\mathcal{E}}\}$, where $\mathcal{E}=\{\mathcal{H}_{\iota
};\iota \in \Upsilon \}$ is an upward filtered family of closed subspaces
such that the union space $\mathcal{D}_{\mathcal{E}}=\tbigcup\limits_{\iota
\in \Upsilon }\mathcal{H}_{\iota }$ is dense in $\mathcal{H\ }$\cite{D1}.

A quantized family $\mathcal{E}=\{\mathcal{H}_{\iota };\iota \in \Upsilon \}$
determines an upward filtered family $\{P_{\iota };\iota \in \Upsilon \}$ of
projections in $B(\mathcal{H})$, where $P_{\iota }$ is a projection onto
\mathcal{H}_{\iota }$.

We say that a quantized domain $\mathcal{F}=\{\mathcal{K}_{\iota };\iota \in
\Upsilon \}$ \ of $\mathcal{H}$ with its union space $\mathcal{D}_{\mathcal{
}}$ and $\mathcal{K}$ $=\overline{\tbigcup\limits_{\iota \in \Upsilon
\mathcal{K}_{\iota }}$ is a \textit{quantized subdomian} of $\mathcal{E}$,
if $\mathcal{K}_{\iota }\subseteq \mathcal{H}_{\iota }$ for all $\iota \in
\Upsilon $ and denote by $\mathcal{F}\subseteq $ $\mathcal{E}$.

Let $\mathcal{E}^{i}=\{\mathcal{H}_{\iota }^{i};\iota \in \Upsilon \}$ be a
quantized domain in a Hilbert space $\mathcal{H}^{i}$ for $i=1,2$. If $V
\mathcal{D}_{\mathcal{E}^{1}}\rightarrow \mathcal{H}^{2}$ is a linear
operator such that $V(\mathcal{H}_{\iota }^{1})\subseteq \mathcal{H}_{\iota
}^{2}$\ for all $\iota \in \Upsilon $, we write $V(\mathcal{E}^{1})\subseteq
\mathcal{E}^{2}$.

A linear operator $T$ $:$ dom$(T)$ $\subseteq $ $\mathcal{H}$ $\rightarrow $
$\mathcal{H}$ is said to be densely defined if dom$(T)$ is a dense subspace
of $\mathcal{H}$. The adjoint of $T$ is a linear map $T^{\bigstar }:$ dom
(T^{\bigstar })$ $\subseteq $ $\mathcal{H}\rightarrow \mathcal{H}$, wher
\begin{equation*}
\text{dom}(T^{\bigstar })=\{\xi \in \mathcal{H};\eta \rightarrow
\left\langle T\eta ,\xi \right\rangle \ \text{is continuous for every }\eta
\in \text{dom}(T)\}
\end{equation*
satisfying $\left\langle T\eta ,\xi \right\rangle =\left\langle \eta
,T^{\bigstar }\xi \right\rangle $ for all $\xi \in $dom$(T^{\bigstar })$ and
$\eta \in $dom$(T).$

Let $\mathcal{E}=\{\mathcal{H}_{\iota };\iota \in \Upsilon \}$ be a
quantized domain in a Hilbert space $\mathcal{H}$ and
\begin{equation*}
C(\mathcal{D}_{\mathcal{E}})=\{T\in \mathcal{L}(\mathcal{D}_{\mathcal{E
});TP_{\iota }=P_{\iota }TP_{\iota }\in B(\mathcal{H})\text{ for all }\iota
\in \Upsilon \}
\end{equation*
where $\mathcal{L}(\mathcal{D}_{\mathcal{E}})$ is the collection of all
linear operators on $\mathcal{D}_{\mathcal{E}}$. If $T\in \mathcal{L}
\mathcal{D}_{\mathcal{E}})$, then $T\in C(\mathcal{D}_{\mathcal{E}})$ if and
only if $T(\mathcal{H}_{\iota })\subseteq \mathcal{H}_{\mathcal{\iota }}$
and $\left. T\right\vert _{\mathcal{H}_{\iota }}\in B(\mathcal{H}_{\iota })$
for all $\iota \in \Upsilon $, and so $C(\mathcal{D}_{\mathcal{E}})$ is an
algebra. Let
\begin{equation*}
C^{\ast }(\mathcal{D}_{\mathcal{E}})=\{T\in C(\mathcal{D}_{\mathcal{E
});P_{\iota }T\subseteq TP_{\iota }\text{ for all }\iota \in \Upsilon \}.
\end{equation*
If $T\in C(\mathcal{D}_{\mathcal{E}})$, then $T\in C^{\ast }(\mathcal{D}_
\mathcal{E}})$ if and only if $T(\mathcal{H}_{\iota }^{\bot }\cap \mathcal{D
_{\mathcal{E}})\subseteq \mathcal{H}_{\iota }^{\bot }\cap \mathcal{D}_
\mathcal{E}}$ for all $\iota \in \Upsilon .\ $

If $T\in C^{\ast }(\mathcal{D}_{\mathcal{E}})$, then $\mathcal{D}_{\mathcal{
}}$ $\subseteq $ dom$(T^{\bigstar })$. Moreover, $T^{\bigstar }(\mathcal{H
_{\iota })\subseteq \mathcal{H}_{\iota }$ for all $\iota \in \Upsilon $.
Now, let $T^{\ast }=\left. T^{\bigstar }\right\vert _{\mathcal{D}_{\mathcal{
}}}$. It is easy to check that $T^{\ast }\in C^{\ast }(\mathcal{D}_{\mathcal
E}}),$ and so $C^{\ast }(\mathcal{D}_{\mathcal{E}})$ is a unital $\ast
-algebra.

For each $\iota \in \Upsilon ,$ the map $\left\Vert \cdot \right\Vert
_{\iota }:C^{\ast }(\mathcal{D}_{\mathcal{E}})\rightarrow \lbrack 0,\infty )
,
\begin{equation*}
\left\Vert T\right\Vert _{\iota }=\left\Vert \left. T\right\vert _{\mathcal{
}_{\iota }}\right\Vert =\sup \{\left\Vert T\left( \xi \right) \right\Vert
;\xi \in \mathcal{H}_{\iota },\left\Vert \xi \right\Vert \leq 1\}
\end{equation*
is a $C^{\ast }$-seminorm on $C^{\ast }(\mathcal{D}_{\mathcal{E}})$.
Moreover, $C^{\ast }(\mathcal{D}_{\mathcal{E}})$ is a locally $C^{\ast }
-algebra with respect to the topology determined by the family of $C^{\ast }
-seminorms $\{\left\Vert \cdot \right\Vert _{\iota }\}_{\iota \in \Upsilon }$
and $b(C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ is isomorphic with the
C^{\ast }$-algebra $\{T\in B\left( \mathcal{H}\right) ;P_{\iota }T=TP_{\iota
}\ $for all $\iota \in \Upsilon \}.$

If $\mathcal{E}=\{\mathcal{H}_{\iota };\iota \in \Upsilon \}$ is a quantized
domain in the Hilbert space $\mathcal{H}$, then $\mathcal{D}_{\mathcal{E}}$
can be regarded as a strict inductive limit of the direct family of Hilbert
spaces $\mathcal{E}=\{\mathcal{H}_{\iota };\iota \in \Upsilon \},$ $\mathcal
D}_{\mathcal{E}}=\lim\limits_{\rightarrow }\mathcal{H}_{\iota },$ and it is
called a locally Hilbert space (see \cite{I}). If $T\in C^{\ast }(\mathcal{D
_{\mathcal{E}})$, then $T$ is continuous \cite[Lemma 5.2 ]{I}.

For every locally $C^{\ast }$-algebra $\mathcal{A}$ there exist a quantized
domain $\mathcal{E}$ in a Hilbert space $\mathcal{H}$ and a local isometric
\ast $-homomorphism $\pi :\mathcal{A\rightarrow }C^{\ast }(\mathcal{D}_
\mathcal{E}})$ \cite[Theorem 7.2]{D1}. This result can be regarded as an
unbounded analog of the Ghelfand-Naimark theorem.

\paragraph{\textbf{Local completely contractive and local completely
positive maps }}

Let $\mathcal{A}$ and $\mathcal{B}$ be two locally $C^{\ast }$-algebras with
the topology defined by the families of $C^{\ast }$-seminorms $\left\{
p_{\lambda }\right\} _{\lambda \in \Lambda }$ and $\left\{ q_{\delta
}\right\} _{\delta \in \Delta }$, respectively. For each $n\in \mathbb{N},$
M_{n}(\mathcal{A})$ denotes the collection of all matrices of order $n$ with
elements in $\mathcal{A}$. Note that $M_{n}(\mathcal{A})$ is a locally
C^{\ast }$-algebra where the associated family of $C^{\ast }$-seminorms is
denoted by $\{p_{\lambda }^{n}\}_{\lambda \in \Lambda }.$

For each $n\in \mathbb{N}$, the $n$-amplification of the linear map $\varphi
:\mathcal{A}\rightarrow \mathcal{B}$ is the map $\varphi ^{\left( n\right)
}:M_{n}(\mathcal{A})$ $\rightarrow $ $M_{n}(\mathcal{B})$ defined by
\begin{equation*}
\varphi ^{\left( n\right) }\left( \left[ a_{ij}\right] _{i,j=1}^{n}\right)
\left[ \varphi \left( a_{ij}\right) \right] _{i,j=1}^{n}
\end{equation*
for all $\left[ a_{ij}\right] _{i,j=1}^{n}\in M_{n}(\mathcal{A})$ .

A linear map $\varphi :\mathcal{A}\rightarrow \mathcal{B}$ is called :

\begin{enumerate}
\item \textit{local contractive} if for each $\delta \in \Delta $, there
exists $\lambda \in \Lambda $ such tha
\begin{equation*}
q_{\delta }\left( \varphi \left( a\right) \right) \leq p_{\lambda }\left(
a\right) \text{ for all }a\in \mathcal{A};
\end{equation*}

\item \textit{local positive} if for each $\delta \in \Delta $, there exists
$\lambda \in \Lambda $\ such that $\varphi \left( a\right) \geq _{\delta }0$
whenever $a\geq _{\lambda }0$ and $\varphi \left( a\right) =_{\delta }0$\ \
\ whenever $a=_{\lambda }0;$

\item \textit{local completely contractive }(\textit{local }$\mathcal{CC}
\textit{) } \textit{\ }if for each $\delta \in \Delta $, there exists
\lambda \in \Lambda $ such that
\begin{equation*}
q_{\delta }^{n}\left( \varphi ^{\left( n\right) }\left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) \right) \leq p_{\lambda }^{n}\left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) \text{ }
\end{equation*
for all $\left[ a_{ij}\right] _{i,j=1}^{n}\in M_{n}(\mathcal{A})\ $and for
all $n\in \mathbb{N};$

\item \textit{local completely positive }(\textit{local }$\mathcal{CP}$
\textit{\ }if for each $\delta \in \Delta $, there exists $\lambda \in
\Lambda $ such that $\varphi ^{\left( n\right) }\left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) \geq _{\delta }0\ $whenever $\left[ a_{ij}\right]
_{i,j=1}^{n}\geq _{\lambda }0$ and $\varphi ^{\left( n\right) }\left( \left[
a_{ij}\right] _{i,j=1}^{n}\right) =_{\delta }0\ \ $whenever $\left[ a_{ij
\right] _{i,j=1}^{n}=_{\lambda }0,\ $for all $n\in \mathbb{N}.$
\end{enumerate}

Let $\mathcal{A}\ $be a unital locally $C^{\ast }$-algebra and $\{\mathcal{H
,\mathcal{E},\mathcal{D}_{\mathcal{E}}\}$ be a quantized domain in a Hilbert
space $\mathcal{H\ }$with $\mathcal{E=\{H}_{\iota }\mathcal{\}}_{\iota \in
\Upsilon }$. The set of all local\textit{\ }$\mathcal{CCP}$\textit{-}maps
from\textit{\ }$\mathcal{A}$ to $C^{\ast }(\mathcal{D}_{\mathcal{E}})
\textit{\ }is denoted by $\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}})).$

\begin{theorem}
\label{s} \cite[Theorem 5.1]{D1} Let $\varphi \in \mathcal{CPCC}_{\text{loc
}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$. Then there exist a
quantized domain $\{\mathcal{H}^{\varphi },\mathcal{E}^{\varphi },\mathcal{D
_{\mathcal{E}^{\varphi }}\}$, where $\mathcal{E}^{\varphi }=\{\mathcal{H
_{\iota }^{\varphi };\iota \in \Upsilon \}$ is an upward filtered family of
closed subspaces of $\mathcal{H}^{\varphi }$, a contraction $V_{\varphi }
\mathcal{H}\rightarrow \mathcal{H}^{\varphi }$ and a unital local
contractive $\ast $-morphism $\pi _{\varphi }:\mathcal{A\rightarrow }C^{\ast
}(\mathcal{D}_{\mathcal{E}^{\varphi }})$ such that

\begin{enumerate}
\item $V_{\varphi }\left( \mathcal{E}\right) \subseteq \mathcal{E}^{\varphi
};$

\item $\varphi \left( a\right) \subseteq V_{\varphi }^{\ast }\pi _{\varphi
}\left( a\right) V_{\varphi };$

\item $\mathcal{H}_{\iota }^{\varphi }=\left[ \pi _{\varphi }\left( \mathcal
A}\right) V_{\varphi }\mathcal{H}_{\iota }\right] $ for all $\iota \in
\Upsilon \ $\cite[Proposition 3.3]{BGK}.

Moreover, if $\varphi \left( 1_{\mathcal{A}}\right) =$id$_{\mathcal{D}_
\mathcal{E}}}$, then $V_{\varphi }$ is an isometry.
\end{enumerate}
\end{theorem}

The triple $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi }
\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $
constructed in Theorem \ref{s} is called a minimal Stinespring dilation of
\varphi $ or a minimal Stinespring representation associated to $\varphi $.
Moreover, the minimal Stinespring dilation of $\varphi $ is unique up to
unitary equivalence in the following sense, if $\left( \pi _{\varphi
},V_{\varphi },\{\mathcal{H}^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_
\mathcal{E}^{\varphi }}\}\right) $ and$\left( \widetilde{\pi }_{\varphi }
\widetilde{V}_{\varphi },\{\widetilde{\mathcal{H}}^{\varphi },\widetilde
\mathcal{E}}^{\varphi },\widetilde{\mathcal{D}}_{\widetilde{\mathcal{E}
^{\varphi }}\}\right) $\ are two minimal Stinespring dilations of $\varphi
, then there exists a unitary operator $U_{\varphi }:\mathcal{H}^{\varphi
}\rightarrow \widetilde{\mathcal{H}}^{\varphi }$ such that $U_{\varphi
}V_{\varphi }=\widetilde{V}_{\varphi }$ and $U_{\varphi }\pi _{\varphi
}\left( a\right) \subseteq \widetilde{\pi }_{\varphi }\left( a\right)
U_{\varphi }$ for all $a\in \mathcal{A\ }$\cite[Theorem 3.4]{BGK}.

If $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D
_{\mathcal{E}})),$ then $\varphi \left( b(\mathcal{A})\right) \subseteq
b(C^{\ast }(\mathcal{D}_{\mathcal{E}}))$. Moreover, there is a $\mathcal{CCP}
$-map $\left. \varphi \right\vert _{b(\mathcal{A})}:b(\mathcal{A
)\rightarrow B(\mathcal{H})$ such that $\left. \left. \varphi \right\vert
_{b(\mathcal{A})}\left( a\right) \right\vert _{\mathcal{D}_{\mathcal{E
}}=\varphi \left( a\right) \ $for all $a\in b(\mathcal{A})\ $\cite[Remark 3.
]{MJ3}, and if $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi }
\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ is a
minimal Stinespring dilation of $\varphi $, then $\left( \left. \pi
_{\varphi }\right\vert _{b(\mathcal{A})},V_{\varphi },\mathcal{H}^{\varphi
}\right) $, where $\left. \left. \pi _{\varphi }\right\vert _{b(\mathcal{A
)}\left( a\right) \right\vert _{\mathcal{D}_{\widetilde{\mathcal{E}
^{\varphi }}}=\pi _{\varphi }\left( a\right) \ $for all $a\in b(\mathcal{A})
, is a minimal Stinespring dilation of $\left. \varphi \right\vert _{b
\mathcal{A})}$ \cite[Remark 3.8]{MJ3}.

\section{Extremal points}

Let $\mathcal{A}$ be a unital locally $C^{\ast }$-algebra with the topology
defined by the family of $C^{\ast }$-seminorms $\left\{ p_{\lambda }\right\}
_{\lambda \in \Lambda }$and $\{\mathcal{H},\mathcal{E},\mathcal{D}_{\mathcal
E}}\}$ be a quantized domain in a Hilbert space $\mathcal{H\ }$with
\mathcal{E=\{H}_{\iota }\mathcal{\}}_{\iota \in \Upsilon }.$

For a local contractive $\ast $-momorphism $\pi :\mathcal{A\rightarrow
C^{\ast }(\mathcal{D}_{\mathcal{E}}),
\begin{equation*}
\pi \left( \mathcal{A}\right) ^{^{\prime }}=\{T\in B\left( \mathcal{H
\right) ;T\pi \left( a\right) \subseteq \pi \left( a\right) T\ \ \text{for\
all\ }a\in \mathcal{A}\}.
\end{equation*}

Let $\varphi ,\psi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$. We say that $\psi $ is \textit{dominated
\textbf{\ }by $\varphi $, and note by $\varphi \geq \psi $, if $\varphi
-\psi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_
\mathcal{E}}))\ $\cite[Definition 4.1]{BGK}$.$

Let $\varphi ,\psi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ such that $\psi $ is dominated by $\varphi $.
By Radon Nikodym type theorem \cite[Theorem 4.5]{BGK}, if $\left( \pi
_{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi },\mathcal{E}^{\varphi }
\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ is a minimal Stinespring
dilation of $\varphi ,$ then there is a unique element $T\in \pi _{\varphi
}\left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal
E}^{\varphi }})$ such that
\begin{equation*}
\psi \left( a\right) =\varphi _{T}\left( a\right) =\left. V_{\varphi }^{\ast
}T\pi _{\varphi }\left( a\right) V_{\varphi }\right\vert _{\mathcal{D}_
\mathcal{E}}}
\end{equation*
for all $a\in \mathcal{A}.$

We say that $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ is \textit{pure} if whenever $\psi \in \mathcal
CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ and
\varphi \geq \psi $, there is a positive number $t$ such that $\psi
=t\varphi $.

A map $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ is pure if and only if $\pi _{\varphi }\left(
\mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E
^{\varphi }})$ $=\{\alpha $id$_{\mathcal{D}_{\mathcal{E}^{\varphi }}};\alpha
\in \mathbb{C}\},$ where $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H
^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi
}}\}\right) $ is a minimal Stinespring dilation of $\varphi $\ \cite{MJ3}.

The set $\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_
\mathcal{E}}))$ is convex. An element $\varphi \in \mathcal{CPCC}_{\text{loc
}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ is \textit{extremal} in
$\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E
}))$ if $\varphi $ can not be written as $\varphi =t\varphi _{1}+\left(
1-t\right) \varphi _{2}$ for $\varphi _{1},\varphi _{2}\in \mathcal{CPCC}_
\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}})),$ $\varphi
_{1}\neq \varphi _{2}$ and $0<t<1.\ $Clearly, $\varphi $ is extremal in
\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E
})) $ if and only if $\varphi $ is pure.

The following proposition is a local convex version of Corollary 1.4.4 \cit
{AW}.

\begin{proposition}
\label{0} Let $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))\ $and $\left( \pi _{\varphi },V_{\varphi },\
\mathcal{H}^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E
^{\varphi }}\}\right) $ be a minimal Stinespring dilation of $\varphi $.
Then the extremal points in $[0,\varphi ]$ are those maps of the form
\varphi _{P}$, where $P$ is a projection in $\pi _{\varphi }\left( \mathcal{
}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }}).$
\end{proposition}

\begin{proof}
By \cite[Corollary 4.6]{BGK}, the map
\begin{equation*}
\zeta :\{T\in \pi _{\varphi }\left( \mathcal{A}\right) ^{^{\prime }}\cap
C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }}),0\leq T\leq id_{\mathcal{H
}\}\rightarrow \lbrack 0,\varphi ]\text{, defined by }\zeta \left( T\right)
=\varphi _{T}
\end{equation*
is an order isomorphism preserving convexity structure. Consequently,
\varphi _{T}$ is an extremal point in $[0,\varphi ]$ if and only if $T$ is
an extremal point in $\{T\in \pi _{\varphi }\left( \mathcal{A}\right)
^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }}),$ $0\leq
T\leq $id$_{\mathcal{H}}\}$. Clearly, $\{T\in \pi _{\varphi }\left( \mathcal
A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }}),$
$0\leq T\leq $id$_{\mathcal{H}}\}\subseteq \{T\in \left( \left. \pi
_{\varphi }\right\vert _{b(\mathcal{A)}}\left( b(\mathcal{A)}\right) \right)
^{^{\prime }},$ $0\leq T\leq $id$_{\mathcal{H}}\}$. The extreme points in
\{T\in \left( \left. \pi _{\varphi }\right\vert _{b(\mathcal{A)}}\left( b
\mathcal{A)}\right) \right) ^{^{\prime }},$ $0\leq T\leq $id$_{\mathcal{H
}\} $ are the projections in $\left( \left. \pi _{\varphi }\right\vert _{b
\mathcal{A)}}\left( b(\mathcal{A)}\right) \right) ^{^{\prime }}\ $\cite
Corollary 1.4.4]{AW}. If $P\in $ $\{T\in \pi _{\varphi }\left( \mathcal{A
\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }}),$
0\leq T\leq $id$_{\mathcal{H}}\}$ is a projection, then $P$ is an extreme
point in $\{T\in \left( \left. \pi _{\varphi }\right\vert _{b(\mathcal{A)
}\left( b(\mathcal{A)}\right) \right) ^{^{\prime }},$ $0\leq T\leq $id$_
\mathcal{H}}\}$, and so, it is an extreme point in $\{T\in \pi _{\varphi
}\left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal
E}^{\varphi }}),$ $0\leq T\leq $id$_{\mathcal{H}}\}.$

Conversely, let $S$ be an extreme point in $\{T\in \pi _{\varphi }\left(
\mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E
^{\varphi }}),$ $0\leq T\leq $id$_{\mathcal{H}}\}$. Suppose that $S$ is not
a projection, then $S$ is not an extreme point in $\{T\in \left( \left. \pi
_{\varphi }\right\vert _{b(\mathcal{A)}}\left( b(\mathcal{A)}\right) \right)
^{^{\prime }},$ $0\leq T\leq $id$_{\mathcal{H}}\}$, and so there exist
S_{1},S_{2}\in \{T\in \left( \left. \pi _{\varphi }\right\vert _{b(\mathcal
A)}}\left( b(\mathcal{A)}\right) \right) ^{^{\prime }},$ $0\leq T\leq $id$_
\mathcal{H}}\},\ S_{1}\neq S_{2}\ $such that $\left( \left. \varphi
\right\vert _{b(\mathcal{A})}\right) _{S}=\frac{1}{2}\varphi _{S_{1}}+\frac{
}{2}\varphi _{S_{2}}$. Since $\varphi _{S_{i}}\leq 2\left( \left. \varphi
\right\vert _{b(\mathcal{A})}\right) _{S},i=1,2$, and $\left( \left. \varphi
\right\vert _{b(\mathcal{A})}\right) _{S}$ is continuous with respect to the
families of $C^{\ast }$-seminorms $\{\left. p_{\lambda }\right\vert _{b
\mathcal{A})}\}_{\lambda \in \Lambda }$ and $\{\left\Vert \cdot \right\Vert
_{\lambda }\}_{\lambda \in \Lambda },$ it follows that $\varphi _{S_{1}}$
and $\varphi _{S_{2}}$ are continuous too. Therefore, $\varphi _{S_{i}}$
extends to a local $\mathcal{CCP}$-map from $\mathcal{A}$ to $C^{\ast }
\mathcal{D}_{\mathcal{E}})$, denoted by $\psi _{i},i=1,2.$ Moreover,
\varphi _{S}=\frac{1}{2}\psi _{1}+\frac{1}{2}\psi _{2}$. Therefore, there
exist $\psi _{1},\psi _{2}\in \mathcal{CPCC}_{\text{loc}}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$, $\psi _{1}\neq \psi _{2},$ such
that $\varphi _{S}=\frac{1}{2}\psi _{1}+\frac{1}{2}\psi _{2}$, a
contradiction. Consequently, $S$ is a projection.
\end{proof}

Let $P\in C^{\ast }(\mathcal{D}_{\mathcal{E}})$. The se
\begin{equation*}
\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E
}))=\{\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal
D}_{\mathcal{E}}));\varphi \left( 1_{\mathcal{A}}\right) =P\}
\end{equation*
is convex.

\begin{proposition}
\label{1}Let $\varphi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast
}(\mathcal{D}_{\mathcal{E}}))\ $and $\left( \pi _{\varphi },V_{\varphi },\
\mathcal{H}^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E
^{\varphi }}\}\right) $ be a minimal Stinespring dilation of $\varphi $.
Then $\varphi $ is extremal in $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ if and only if for any $T\in \pi
_{\varphi }\left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D
_{\mathcal{E}^{\varphi }})$, the condition $V_{\varphi }^{\ast }TV_{\varphi
}=0$ implies $T=0.$
\end{proposition}

\begin{proof}
The proof is similar to the case of $C^{\ast }$-algebras \cite{AW}. Suppose
that $\varphi $ is extremal in $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$. Let $T\in \pi _{\varphi }\left(
\mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E
^{\varphi }})$ such that $V_{\varphi }^{\ast }TV_{\varphi }=0$. We can
suppose that $0\leq T\leq $id$_{\mathcal{H}}.$ Then $\varphi _{\left( \text
id}_{\mathcal{H}}-T\right) },$ $\varphi _{\left( \text{id}_{\mathcal{H
}+T\right) }\in $ $\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$. \ Since $V_{\varphi }^{\ast }TV_{\varphi }=0,$
it follows that $\varphi _{\left( \text{id}_{\mathcal{H}}-T\right) }\left(
1_{\mathcal{A}}\right) =$ $\varphi _{\left( \text{id}_{\mathcal{H}}+T\right)
}\left( 1_{\mathcal{A}}\right) =P,$ and so $\varphi _{\left( \text{id}_
\mathcal{H}}-T\right) },$ $\varphi _{\left( \text{id}_{\mathcal{H}}+T\right)
}\in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D}_
\mathcal{E}}))$. On the other hand, $\frac{1}{2}\varphi _{\left( \text{id}_
\mathcal{H}}-T\right) }+\frac{1}{2}\varphi _{\left( \text{id}_{\mathcal{H
}+T\right) }=\varphi ,$ whence, since $\varphi $ is extremal in $\mathcal
CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$, we
deduce that $\varphi _{\left( \text{id}_{\mathcal{H}}-T\right) }=\varphi
_{\left( \text{id}_{\mathcal{H}}+T\right) }=\varphi .$ Therefore, id$_
\mathcal{H}}-T=$id$_{\mathcal{H}}+T=$id$_{\mathcal{H}},$ \cite[Theorem 4.5
{BGK}, and then $T=0.$

Conversely,\ suppose that $\varphi $ is not extremal in $\mathcal{CPCC}_
\text{loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$. Then,
there exist $\psi _{1},\psi _{2}\in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}})),$ $\psi _{1}\neq \psi _{2}\ $such
that $\varphi =\frac{1}{2}\psi _{1}+\frac{1}{2}\psi _{2}$.\ Since $\frac{1}{
}\psi _{i}\leq \varphi ,i=1,2$, by Radon Nikodym type theorem \cite[Theorem
4.5]{BGK}, there exists $T_{i}\in \pi _{\varphi }\left( \mathcal{A}\right)
^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi }})$ such that
$\psi _{i}\left( a\right) =2\left. V_{\varphi }^{\ast }T_{i}\pi _{\varphi
}\left( a\right) V_{\varphi }\right\vert _{\mathcal{D}_{\mathcal{E}}}$ for
all $a\in \mathcal{A},i=1,2$.$\ $Then,$\ P=2V_{\varphi }^{\ast
}T_{i}V_{\varphi },i=1,2,$ whence $0=V_{\varphi }^{\ast }\left(
T_{1}-T_{2}\right) V_{\varphi },$ and consequently $T_{1}=T_{2}$. This
implies that $\psi _{1}=\psi _{2}$, a contradiction.
\end{proof}

\begin{corollary}
Let $\varphi \in \mathcal{CPCC}_{\text{loc,}1}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$. If $\varphi $ is a local contractive $\ast
-morphism, then $\varphi $ is extremal in $\mathcal{CPCC}_{\text{loc,}1}
\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}})).$
\end{corollary}

\section{Tensor product of unbounded local $\mathcal{CCP}$-maps}

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital locally $C^{\ast }
-algebras with the topologies defined by the family of $C^{\ast }$-seminorms
$\left\{ p_{\lambda }\right\} _{\lambda \in \Lambda }$ and $\left\{
q_{\delta }\right\} _{\delta \in \Delta }$, respectively, and $\{\mathcal{H}
\mathcal{E},\mathcal{D}_{\mathcal{E}}\}$ be a quantized domain in a Hilbert
space $\mathcal{H\ }$with $\mathcal{E=\{H}_{\iota }\mathcal{\}}_{\iota \in
\Upsilon }.$

A $\ast $-representation of $\mathcal{A}$ on a Hilbert space $\mathcal{H}$
is a continuous $\ast $-morphism $\varphi :$ $\mathcal{A}$ $\rightarrow B
\mathcal{H})$. For each $\lambda \in \Lambda ,\mathcal{R}_{\lambda }
\mathcal{A})=\{\varphi \ $is a $\ast $-representation of $\mathcal{A
;\left\Vert \varphi \left( a\right) \right\Vert \leq p_{\lambda }(a)$ for
all $a\in \mathcal{A}\}$. The \textit{maximal tensor product} of the locally
$C^{\ast }$-algebras $\mathcal{A}$ and $\mathcal{B}$ is the locally $C^{\ast
}$-algebra $\mathcal{A}\otimes _{\max }\mathcal{B}$ obtained by the
completion of the algebraic tensor product $\mathcal{A}\otimes _{\text{alg}
\mathcal{B}$ with respect to the topology given by the family of $C^{\ast }
-seminorms $\{\upsilon _{\left( \lambda ,\delta \right) }\}_{(\lambda
,\delta )\in \Lambda \times \Delta }$, where

$\upsilon _{\left( \lambda ,\delta \right) }\left(
\tsum\limits_{k=1}^{n}a_{k}\otimes b_{k}\right) =\sup \{\left\Vert \left(
\varphi \left( \tsum\limits_{k=1}^{n}a_{k}\right) \psi \left(
\tsum\limits_{k=1}^{n}b_{k}\right) \right) \right\Vert ;\varphi \in \mathcal
R}_{\lambda }(\mathcal{A}),\psi \in \mathcal{R}_{\delta }(\mathcal{B})$

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ with commuting ranges $\}.$

Moreover, for each $\left( \lambda ,\delta \right) \in \Lambda \times \Delta
$, the $C^{\ast }$-algebras $\left( \mathcal{A}\otimes _{\max }\mathcal{B
\right) _{(\lambda ,\delta )}$ and $\mathcal{A}_{\lambda }\otimes _{\max
\mathcal{B}_{\delta }$ are isomorphic. We refer the reader to \cite{F} for
further information about locally $C^{\ast }$-algebras and tensor products
of locally $C^{\ast }$-algebras.

\begin{remark}
\label{H}Let $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ and $\psi \in \mathcal{CPCC}_{\text{loc}}
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such that $\varphi \left(
\mathcal{A}\right) $\ and $\psi \left( \mathcal{B}\right) $ commute. By \cit
[Lemma 2.8]{MJ1}, for each $\iota \in \Upsilon $, there exist $\varphi
_{\lambda }:\mathcal{A}_{\lambda }\rightarrow B\left( \mathcal{H}_{\iota
}\right) ,\varphi _{\lambda }\left( \pi _{\lambda }^{\mathcal{A}}\left(
a\right) \right) =\left. \varphi \left( a\right) \right\vert _{\mathcal{H
_{\iota }}$ and $\psi _{\delta }:\mathcal{B}_{\delta }\rightarrow B\left(
\mathcal{H}_{\iota }\right) ,\psi _{\delta }\left( \pi _{\delta }^{\mathcal{
}}\left( b\right) \right) =\left. \psi \left( b\right) \right\vert _
\mathcal{H}_{\iota }}$. Moreover, $\varphi _{\lambda }$ and $\psi _{\delta }$
are completely positive maps such that $\varphi _{\lambda }\left( \mathcal{A
_{\lambda }\right) \ $and $\psi _{\delta }\left( \mathcal{B}_{\delta
}\right) \ $commute.
\end{remark}

\begin{lemma}
\label{max}\ Let $\varphi \in \mathcal{CPCC}_{\text{loc,}1}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ and $\psi \in \mathcal{CPCC}_{\text
loc,}1}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such that
\varphi \left( \mathcal{A}\right) $\ and $\psi \left( \mathcal{B}\right) $
commute. Then there exist a quantized domain $\{\mathcal{K},\mathcal{F}
\mathcal{D}_{\mathcal{F}}\}$,\ two local contractive $\ast $-morphisms $\pi
_{1}:\mathcal{A}\rightarrow $ $C^{\ast }(\mathcal{D}_{\mathcal{F}})$ and
\pi _{2}:\mathcal{B}\rightarrow $ $C^{\ast }(\mathcal{D}_{\mathcal{F}})$,
and an isometry $V\in B\left( \mathcal{H},\mathcal{K}\right) $ such that:

\begin{enumerate}
\item $V\left( \mathcal{E}\right) \subseteq \mathcal{F};$

\item $\pi _{1}\left( a\right) \pi _{2}\left( b\right) =\pi _{2}\left(
b\right) \pi _{1}\left( a\right) $ for all $a\in \mathcal{A}$ and for all
b\in \mathcal{B}$;

\item $\varphi \left( a\right) \subseteq V^{\ast }\pi _{1}\left( a\right) V\
$for all $a\in \mathcal{A};$

\item $\psi \left( b\right) \subseteq V^{\ast }\pi _{2}\left( b\right) V\
for all $b\in \mathcal{B}.$
\end{enumerate}
\end{lemma}

\begin{proof}
Let $\left\langle \cdot ,\cdot \right\rangle $ be the sesquilinear form on
\mathcal{A}\otimes _{\text{alg}}\mathcal{B}\otimes _{\text{alg}}\mathcal{D}_
\mathcal{E}}\ $defined by
\begin{equation*}
\left\langle \tsum\limits_{j=1}^{m}c_{j}\otimes d_{j}\otimes \eta
_{j},\tsum\limits_{i=1}^{n}a_{i}\otimes b_{i}\otimes \xi _{i}\right\rangle
=\tsum\limits_{j=1}^{m}\tsum\limits_{i=1}^{n}\left\langle \eta _{j},\varphi
\left( c_{j}^{\ast }a_{i}\right) \psi \left( d_{j}^{\ast }b_{i}\right) \xi
_{i}\right\rangle
\end{equation*
for all $a_{i},c_{j}\in \mathcal{A},$ for all $b_{i},d_{j}\in \mathcal{B}$
and for all $\xi _{i},\eta _{j}\in \mathcal{D}_{\mathcal{E}}$.$\ $In the
same manner as in the normed case\ (see \cite[II.9.7.2]{B} and \cite
Proposition 4.23]{T}) \ and using Remark \ref{H}, we show that the
sesquilinear form $\left\langle \cdot ,\cdot \right\rangle $ is a positive
semidefined form on $\mathcal{A}\otimes _{\text{alg}}\mathcal{B}\otimes _
\text{alg}}\mathcal{D}_{\mathcal{E}}$ and the quotient space $\left(
\mathcal{A}\otimes _{\text{alg}}\mathcal{B}\otimes _{\text{alg}}\mathcal{D}_
\mathcal{E}}\right) /\mathcal{N},\ $where $\mathcal{N=\{\zeta \in A}\otimes
_{\text{alg}}\mathcal{B}\otimes _{\text{alg}}\mathcal{D}_{\mathcal{E
};\left\langle \zeta ,\zeta \right\rangle =0\}$ is a pre-Hilbert space.

Let $\mathcal{K}$ be the completion of the pre-Hilbert space $\left(
\mathcal{A}\otimes _{\text{alg}}\mathcal{B}\otimes _{\text{alg}}\mathcal{D}_
\mathcal{E}}\right) /\mathcal{N}$. For each $\iota \in \Upsilon ,$ let
K_{\iota }=$span$\{\tsum\limits_{i=1}^{n}a_{i}\otimes b_{i}\otimes \xi _{i}
\mathcal{N};\xi _{i}\in \mathcal{H}_{\iota },i=1,...,n\}$ be a subspace in
\mathcal{K}$, and let $\mathcal{K}_{\iota }$ be the closure of $K_{\iota }$
in $\mathcal{K}$. Then $\{\mathcal{K},\mathcal{F},\mathcal{D}_{\mathcal{F
}\} $ is a quantized domain in the Hilbert space $\mathcal{K\ }$with
\mathcal{F=\{K}_{\iota }\mathcal{\}}_{\iota \in \Upsilon }$.

For each $a\in \mathcal{A}$, consider the linear map $\pi _{1}\left(
a\right) :$ $\mathcal{A}\otimes _{\text{alg}}\mathcal{B}\otimes _{\text{alg}
\mathcal{D}_{\mathcal{E}}\rightarrow $ $\mathcal{A}\otimes _{\text{alg}
\mathcal{B}\otimes _{\text{alg}}\mathcal{D}_{\mathcal{E}}$,$\ $defined by
\begin{equation*}
\pi _{1}\left( a\right) \left( c\otimes b\otimes \xi \right) =ac\otimes
b\otimes \xi \text{.}
\end{equation*
Let $\zeta =\tsum\limits_{i=1}^{n}a_{i}\otimes b_{i}\otimes \xi _{i}$. There
is $\iota \in \Upsilon $ such that $\xi _{i}\in \mathcal{H}_{\iota
},i=1,...,n.\ $By Remark \ref{H}, there exist two completely positive maps
\varphi _{\lambda _{\iota }}:\mathcal{A}_{\lambda _{\iota }}\rightarrow
B\left( \mathcal{H}_{\iota }\right) ,\varphi _{\lambda _{\iota }}\left( \pi
_{\lambda _{\iota }}^{\mathcal{A}}\left( a\right) \right) =\left. \varphi
\left( a\right) \right\vert _{\mathcal{H}_{\iota }}$ and $\psi _{\delta
_{\iota }}:\mathcal{B}_{\delta _{\iota }}\rightarrow B\left( \mathcal{H
_{\iota }\right) ,\psi _{\delta _{\iota }}\left( \pi _{\delta _{\iota }}^
\mathcal{B}}\left( b\right) \right) =\left. \psi \left( b\right) \right\vert
_{\mathcal{H}_{\iota }}$, such that $\varphi _{\lambda _{\iota }}\left(
\mathcal{A}_{\lambda _{\iota }}\right) \ $and $\psi _{\delta _{\iota
}}\left( \mathcal{B}_{\delta _{\iota }}\right) \ $commute. The
\begin{eqnarray*}
&&\left\langle \pi _{1}\left( a\right) \zeta ,\pi _{1}\left( a\right) \zeta
\right\rangle  \\
&=&\tsum\limits_{i,j=1}^{n}\left\langle \xi _{i},\varphi \left( a_{i}^{\ast
}a^{\ast }aa_{j}\right) \psi \left( b_{i}^{\ast }b_{j}\right) \xi
_{j}\right\rangle  \\
&=&\tsum\limits_{i,j=1}^{n}\left\langle \xi _{i},\varphi _{\lambda _{\iota
}}\left( \pi _{\lambda _{\iota }}^{\mathcal{A}}\left( a_{i}\right) ^{\ast
}\pi _{\lambda _{\iota }}^{\mathcal{A}}\left( a\right) ^{\ast }\pi _{\lambda
_{\iota }}^{\mathcal{A}}\left( a\right) \pi _{\lambda _{\iota }}^{\mathcal{A
}\left( a_{j}\right) \right) \psi _{\delta _{\iota }}\left( \pi _{\delta
_{\iota }}^{\mathcal{B}}\left( b_{i}\right) ^{\ast }\pi _{\delta _{\iota }}^
\mathcal{B}}\left( b_{j}\right) \right) \xi _{j}\right\rangle  \\
&\leq &\left\Vert \pi _{\lambda _{\iota }}^{\mathcal{A}}\left( a\right)
\right\Vert ^{2}\tsum\limits_{i,j=1}^{n}\left\langle \xi _{i},\varphi
_{\lambda _{\iota }}\left( \pi _{\lambda _{\iota }}^{\mathcal{A}}\left(
a_{i}\right) ^{\ast }\pi _{\lambda _{\iota }}^{\mathcal{A}}\left(
a_{j}\right) \right) \psi _{\delta _{\iota }}\left( \pi _{\delta _{\iota }}^
\mathcal{B}}\left( b_{i}\right) ^{\ast }\pi _{\delta _{\iota }}^{\mathcal{B
}\left( b_{j}\right) \right) \xi _{j}\right\rangle  \\
&=&p_{\lambda _{\iota }}\left( a\right)
^{2}\tsum\limits_{i,j=1}^{n}\left\langle \xi _{i},\varphi \left( a_{i}^{\ast
}a_{j}\right) \psi \left( b_{i}^{\ast }b_{j}\right) \xi _{j}\right\rangle  \\
&=&p_{\lambda _{\iota }}\left( a\right) ^{2}\left\langle \zeta ,\zeta
\right\rangle .
\end{eqnarray*
From the above relation, we conclude that $\pi _{1}\left( a\right) \left(
\mathcal{N}\right) \subseteq \mathcal{N}$. Therefore $\pi _{1}\left(
a\right) $ determines a linear operator on\ \ $\left( \mathcal{A}\otimes _
\text{alg}}\mathcal{B}\otimes _{\text{alg}}\mathcal{D}_{\mathcal{E}}\right)
\mathcal{N}$, denoted by $\pi _{1}\left( a\right) $ too. So, $\pi _{1}\left(
a\right) $ is a densely defined linear operator on $\mathcal{K}$. Moreover,
\left. \pi _{1}\left( a\right) \right\vert _{\mathcal{K}_{\iota }}\in B
\mathcal{K}_{\iota })$ and $\left\Vert \left. \pi _{1}\left( a\right)
\right\vert _{\mathcal{K}_{\iota }}\right\Vert \leq p_{\lambda _{\iota
}}\left( a\right) $. From
\begin{eqnarray*}
&&\left\langle \pi _{1}\left( a\right) \left(
\tsum\limits_{i=1}^{n}a_{i}\otimes b_{i}\otimes \xi _{i}+\mathcal{N}\right)
,\left( \tsum\limits_{j=1}^{m}c_{j}\otimes d_{j}\otimes \eta _{j}+\mathcal{N
\right) \right\rangle  \\
&=&\tsum\limits_{i=1}^{n}\tsum\limits_{j=1}^{m}\left\langle \xi _{i},\varphi
\left( a_{i}^{\ast }a^{\ast }c_{j}\right) \psi \left( b_{i}^{\ast
}d_{j}\right) \eta _{j}\right\rangle  \\
&=&\left\langle \left( \tsum\limits_{i=1}^{n}a_{i}\otimes b_{i}\otimes \xi
_{i}+\mathcal{N}\right) ,\pi _{1}\left( a^{\ast }\right) \left(
\tsum\limits_{j=1}^{m}c_{j}\otimes d_{j}\otimes \eta _{j}+\mathcal{N}\right)
\right\rangle
\end{eqnarray*
for all $a_{i},c_{j}\in \mathcal{A},b_{i},d_{j}\in \mathcal{B}$ and $\xi
_{i},\eta _{j}\in \mathcal{H}_{\iota }$, we conclude that $\left. \pi
_{1}\left( a^{\ast }\right) \right\vert _{\mathcal{K}_{\iota }}=\left. \pi
_{1}\left( a\right) \right\vert _{\mathcal{K}_{\iota }}^{\ast }$. Therefore,
$\pi _{1}\left( a\right) \in C^{\ast }\left( \mathcal{D}_{\mathcal{F
}\right) $ and $\pi _{1}\left( a^{\ast }\right) =\pi _{1}\left( a\right)
^{\ast }$. Thus, we obtained a unital $\ast $-morphism $\pi _{1}:\mathcal{A
\rightarrow C^{\ast }\left( \mathcal{D}_{\mathcal{F}}\right) $ which is a
local contractive map, since for each $\iota \in \Upsilon ,$ there is
\lambda _{\iota }\in \Lambda $ such that $\left\Vert \left. \pi _{1}\left(
a\right) \right\vert _{\mathcal{H}_{\iota }}\right\Vert \leq p_{\lambda
_{\iota }}\left( a\right) $ for all $a\in \mathcal{A}.$

In the same way, we obtain a local contractive unital $\ast $-morphism $\pi
_{2}:\mathcal{B}\rightarrow C^{\ast }\left( \mathcal{D}_{\mathcal{F}}\right)
$, such that, for each $b\in \mathcal{B},$
\begin{equation*}
\pi _{2}\left( b\right) \left( c\otimes d\otimes \xi +\mathcal{N}\right)
=c\otimes bd\otimes \xi +\mathcal{N}\text{.}
\end{equation*
From
\begin{equation*}
\pi _{1}\left( a\right) \pi _{2}\left( b\right) \left( c\otimes d\otimes \xi
+\mathcal{N}\right) =ac\otimes bd\otimes \xi +\mathcal{N=}\pi _{2}\left(
b\right) \pi _{1}\left( a\right) \left( c\otimes d\otimes \xi +\mathcal{N
\right)
\end{equation*
for all $a,c\in \mathcal{A},b,d\in \mathcal{B}$ and $\xi \in \mathcal{D}_
\mathcal{E}},$ we deduce that $\pi _{1}\left( a\right) \pi _{2}\left(
b\right) =\pi _{2}\left( b\right) \pi _{1}\left( a\right) $ for all $a,c\in
\mathcal{A}$ and $b,d\in \mathcal{B}.$

Consider a linear map $V:\mathcal{D}_{\mathcal{E}}\rightarrow \mathcal{D}_
\mathcal{F}}$ defined by $V\xi =1_{\mathcal{A}}\otimes 1_{\mathcal{B
}\otimes \xi +\mathcal{N}.$ Clearly, $V\left( \mathcal{H}_{\iota }\right)
\subseteq \mathcal{K}_{\iota }$ for all $\iota \in \Upsilon $. For each
\iota \in \Upsilon $, we have
\begin{equation*}
\left\langle V\xi ,V\xi \right\rangle =\left\langle \xi ,\varphi \left( 1_
\mathcal{A}}\right) \psi \left( 1_{\mathcal{B}}\right) \xi \right\rangle
=\left\langle \xi ,\xi \right\rangle
\end{equation*
for all $\xi \in \mathcal{H}_{\iota }$. Therefore, $V$ extends to an
isometry from $\mathcal{H}$ to $\mathcal{K}$, denoted by $V$ too.

Let $a\in \mathcal{A},$ and $b\in \mathcal{B}$. From
\begin{eqnarray*}
\left\langle V^{\ast }\pi _{1}\left( a\right) V\xi ,\eta \right\rangle
&=&\left\langle \pi _{1}\left( a\right) V\xi ,V\eta \right\rangle
=\left\langle a\otimes 1_{\mathcal{B}}\otimes \xi ,1_{\mathcal{A}}\otimes 1_
\mathcal{B}}\otimes \eta \right\rangle  \\
&=&\left\langle \xi ,\varphi \left( a^{\ast }\right) \psi \left( 1_{\mathcal
B}}\right) \eta \right\rangle =\left\langle \varphi \left( a\right) \xi
,\eta \right\rangle
\end{eqnarray*
and
\begin{eqnarray*}
\left\langle V^{\ast }\pi _{2}\left( b\right) V\xi ,\eta \right\rangle
&=&\left\langle \pi _{2}\left( b\right) V\xi ,V\eta \right\rangle
=\left\langle 1_{\mathcal{A}}\otimes b\otimes \xi ,1_{\mathcal{A}}\otimes 1_
\mathcal{B}}\otimes \eta \right\rangle  \\
&=&\left\langle \xi ,\varphi \left( 1_{\mathcal{A}}\right) \psi \left(
b^{\ast }\right) \eta \right\rangle =\left\langle \psi \left( b\right) \xi
,\eta \right\rangle .
\end{eqnarray*
for all $\xi ,\eta \in \mathcal{D}_{\mathcal{E}},$ we deduce that $\varphi
\left( a\right) \subseteq V^{\ast }\pi _{1}\left( a\right) V$ and $\psi
\left( b\right) \subseteq V^{\ast }\pi _{2}\left( b\right) V$, and the lemma
is proved.
\end{proof}

\begin{proposition}
\label{tmax}Let $\varphi \in \mathcal{CPCC}_{\text{loc,}1}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ and $\psi \in \mathcal{CPCC}_{\text
loc,}1}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such that
\varphi \left( \mathcal{A}\right) $\ and $\psi \left( \mathcal{B}\right) $
commute. Then there exists a unique unital local \ $\mathcal{CCP}$-map,
\varphi \otimes \psi :\mathcal{A\otimes }_{\max }\mathcal{B}\rightarrow
C^{\ast }(\mathcal{D}_{\mathcal{E}})$, such that
\begin{equation*}
\left( \varphi \otimes \psi \right) \left( a\otimes b\right) =\varphi \left(
a\right) \psi \left( b\right)
\end{equation*
for all $a\in \mathcal{A}$ and for all $b\in \mathcal{B}$.
\end{proposition}

\begin{proof}
By Lemma \ref{max}, there exist a quantized domain $\{\mathcal{K},\mathcal{F
,\mathcal{D}_{\mathcal{F}}\}$,\ two local contractive $\ast $ -morphisms
\pi _{1}:\mathcal{A}\rightarrow $ $C^{\ast }(\mathcal{D}_{\mathcal{F}})$ and
$\pi _{2}:\mathcal{B}\rightarrow $ $C^{\ast }(\mathcal{D}_{\mathcal{F}})$,
and an isometry $V\in B\left( \mathcal{H},\mathcal{K}\right) $ such that:
V\left( \mathcal{E}\right) \subseteq \mathcal{F};\varphi \left( a\right)
\subseteq V^{\ast }\pi _{1}\left( a\right) V\ $for all $a\in \mathcal{A
;\psi \left( b\right) \subseteq V^{\ast }\pi _{2}\left( b\right) V\ $for all
$b\in \mathcal{B};\pi _{1}\left( a\right) \pi _{2}\left( b\right) =\pi
_{2}\left( b\right) \pi _{1}\left( a\right) $ for all $a\in \mathcal{A}$ and
for all $b\in \mathcal{B}$.

Since $\pi _{1}\left( a\right) \pi _{2}\left( b\right) =\pi _{2}\left(
b\right) \pi _{1}\left( a\right) $ for all $a\in \mathcal{A}$ and for all
b\in \mathcal{B}$, by \cite[Proposition 3.3]{Ph}, there exists a unique
local contractive $\ast $-morphism $\pi :\mathcal{A\otimes }_{\max }\mathcal
B}\rightarrow C^{\ast }(\mathcal{D}_{\mathcal{F}})$, such that
\begin{equation*}
\pi \left( a\otimes b\right) =\pi _{1}\left( a\right) \pi _{2}\left(
b\right)
\end{equation*
for all $a\in \mathcal{A}$ and for all $b\in \mathcal{B}$. Consider the
linear map $\rho :\mathcal{A\otimes }_{\max }\mathcal{B}\rightarrow C^{\ast
}(\mathcal{D}_{\mathcal{F}})$, defined by
\begin{equation*}
\rho \left( x\right) =\left. V^{\ast }\pi \left( x\right) V\right\vert _
\mathcal{D}_{\mathcal{E}}}.
\end{equation*
Clearly, $\rho $ is a unital local $\mathcal{CCP}$-map and
\begin{equation*}
\rho \left( a\otimes b\right) \xi =V^{\ast }\pi _{1}\left( a\right) \pi
_{2}\left( b\right) V\xi =V^{\ast }\pi _{1}\left( a\right) VV^{\ast }\pi
_{2}\left( b\right) V\xi =\varphi \left( a\right) \psi \left( b\right) \xi
\end{equation*
for all $\xi \in \mathcal{D}_{\mathcal{E}},$ for $a\in \mathcal{A}$ and for
b\in \mathcal{B}$.

If $\widetilde{\rho }\in \mathcal{CPCC}_{\text{loc,}1}(\mathcal{A\otimes
_{\max }\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ is another map
such tha
\begin{equation*}
\widetilde{\rho }\left( a\otimes b\right) =\varphi \left( a\right) \psi
\left( b\right)
\end{equation*
for all $a\in \mathcal{A}$ and for all $b\in \mathcal{B}$. Then $\left.
\widetilde{\rho }\right\vert _{\mathcal{A\otimes }_{\text{alg}}\mathcal{B}}=$
$\left. \rho \right\vert _{\mathcal{A\otimes }_{\text{alg}}\mathcal{B}}$ and
since $\mathcal{A\otimes }_{\text{alg}}\mathcal{B}$ is dense in $\mathcal
A\otimes }_{\max }\mathcal{B}$ , $\widetilde{\rho }$ and $\rho $ are
continuos (see \cite[Proposition 3.3]{MJ3}),  it follows that $\widetilde
\rho }=\rho .$
\end{proof}

\begin{remark}
Let $\varphi \in \mathcal{CPCC}_{\text{loc,}1}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ and $\psi \in \mathcal{CPCC}_{\text{loc,}1}
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such that $\varphi \left(
\mathcal{A}\right) $\ and $\psi \left( \mathcal{B}\right) $ commute. Then
the triple $\left( \pi ,V,\{\mathcal{K},\mathcal{F},\mathcal{D}_{\mathcal{F
}\}\right) $ constructed in the proof of Proposition \ref{tmax} is a minimal
dilation of $\varphi \otimes \psi .$

Indeed, we have:

\begin{enumerate}
\item $V\in B(\mathcal{H},\mathcal{K}),\ V$ is an isometry and $V\left(
\mathcal{E}\right) \subseteq \mathcal{F};$

\item $\left( \varphi \otimes \psi \right) \left( x\right) \subseteq V^{\ast
}\pi \left( x\right) V$, for all $x\in \mathcal{A\otimes }_{\max }\mathcal{B
;$

\item For each $\iota \in \Upsilon ,
\begin{eqnarray*}
\left[ \pi \left( \mathcal{A\otimes }_{\max }\mathcal{B}\right) V\mathcal{H
_{\iota }\right] &=&\overline{\text{span}}\{\pi \left( a\otimes b\right)
\mathcal{\xi };a\in \mathcal{A},b\in \mathcal{B},\mathcal{\xi \in H}_{\iota
}\} \\
&=&\overline{\text{span}}\{\pi _{1}\left( a\right) \pi _{2}\left( b\right)
\mathcal{\xi };a\in \mathcal{A},b\in \mathcal{B},\mathcal{\xi \in H}_{\iota
}\} \\
&=&\overline{\text{span}}\{a\otimes b\otimes \xi +\mathcal{N};a\in \mathcal{
},b\in \mathcal{B},\mathcal{\xi \in H}_{\iota }\}=\mathcal{K}_{\iota }
\end{eqnarray*
and by \cite[Theorem 4.5]{BGK}, the triple $\left( \pi ,V,\{\mathcal{K}
\mathcal{F},\mathcal{D}_{\mathcal{F}}\}\right) $ constructed in the proof of
Proposition \ref{tmax} is a minimal dilation of $\varphi \otimes \psi .$
\end{enumerate}
\end{remark}

\section{Marginals of a local $\mathcal{CCP}$ -map}

Let $\mathcal{A}$ and $\mathcal{B}$ be two unital locally $C^{\ast }
-algebras with the topologies defined by the families of $C^{\ast }
-seminorms $\left\{ p_{\lambda }\right\} _{\lambda \in \Lambda }$ and
\left\{ q_{\delta }\right\} _{\delta \in \Delta },$ respectively, and $\
\mathcal{H},\mathcal{E},\mathcal{D}_{\mathcal{E}}\}$ be a quantized domain
in a Hilbert space $\mathcal{H\ }$with $\mathcal{E=\{H}_{\iota }\mathcal{\}
_{\iota \in \Upsilon }.$

Let $\varphi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$.$\ $Clearly, the maps
\varphi _{\left( 1\right) }:\mathcal{A\rightarrow }C^{\ast }(\mathcal{D}_
\mathcal{E}})$ given by
\begin{equation*}
\varphi _{\left( 1\right) }\left( a\right) =\varphi \left( a\otimes 1_
\mathcal{B}}\right)
\end{equation*
and $\varphi _{\left( 2\right) }:\mathcal{B\rightarrow }C^{\ast }(\mathcal{D
_{\mathcal{E}})$ given b
\begin{equation*}
\varphi _{\left( 2\right) }\left( b\right) =\varphi \left( 1_{\mathcal{A
}\otimes b\right)
\end{equation*
are elements in $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))\ $and $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{E}})$, respectively.

\begin{definition}
Let $\varphi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$.$\ $The maps $\varphi
_{\left( 1\right) }$ and $\varphi _{\left( 2\right) }$ defined the above are
call the marginals of $\varphi .$
\end{definition}

\begin{definition}
Let $\varphi _{1}\in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))\ $and $\varphi _{2}\in \mathcal{CPCC}_{\text{loc
}P}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}})$. If there is $\psi \in
\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max }\mathcal{B},C^{\ast
}(\mathcal{D}_{\mathcal{E}}))$ such that $\psi _{\left( 1\right) }=\varphi
_{1}$ and $\psi _{\left( 2\right) }=\varphi _{2}$, we say that $\varphi _{1}$
and $\varphi _{2}$ are compatible and $\psi $ is a joint map for $\varphi
_{1}$ and $\varphi _{2}$.
\end{definition}

\begin{remark}
Let $\varphi \in \mathcal{CPCC}_{\text{loc,}1\text{ }}(\mathcal{A},C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ and $\psi \in \mathcal{CPCC}_{\text{loc,}1}
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such that $\varphi \left(
a\right) \psi \left( b\right) =\psi \left( b\right) \varphi \left( a\right) $
for all $a\in \mathcal{A}$ and for all $b\in \mathcal{B}$. Then $\varphi $
and $\psi $ are compatible and $\varphi \otimes \psi $ (see Proposition \re
{tmax}) is a unique joint map for $\varphi $ and $\psi .$
\end{remark}

The structure of an unbounded local $\mathcal{CCP}$-map in terms of a
minimal Stinespring dilation of one of its marginal maps is given in the
following theorem.

\begin{theorem}
\label{Help} Let $\varphi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal
A\otimes }_{\max }\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ and
(\pi _{\varphi _{\left( 1\right) }},V_{\varphi _{\left( 1\right) }},$ $\
\mathcal{H}^{\varphi _{\left( 1\right) }},\mathcal{E}^{\varphi _{\left(
1\right) }},$ $\mathcal{D}_{\mathcal{E}^{\varphi _{\left( 1\right) }}}\})$
be a minimal Stinespring dilation of $\varphi _{\left( 1\right) }$. Then,
there exists a unique $E\mathcal{\in CPCC}_{\text{loc,}1}(\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi _{\left( 1\right) }}}))$\ such
that $E\left( b\right) \pi _{\varphi _{\left( 1\right) }}\left( a\right)
=\pi _{\varphi _{\left( 1\right) }}\left( a\right) E\left( b\right) $ for
all $b\in \mathcal{B}$ and for all $a\in \mathcal{A}$, and
\begin{equation*}
\varphi \left( a\otimes b\right) \subseteq V_{\varphi _{\left( 1\right)
}}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a\right) E\left( b\right)
V_{\varphi _{\left( 1\right) }}
\end{equation*
for all $a\in \mathcal{A}$ and $b\in \mathcal{B}$.
\end{theorem}

\begin{proof}
Let $b\in b(\mathcal{B)}$. Consider the map $\varphi _{b}:\mathcal
A\rightarrow }C^{\ast }(\mathcal{D}_{\mathcal{E}})\ $defined b
\begin{equation*}
\varphi _{b}\left( a\right) =\varphi \left( a\otimes b\right) .
\end{equation*
Suppose that $0\leq b\leq 1_{\mathcal{B}}$. We will show that $\varphi
_{b}\in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_
\mathcal{E}}))$ and $\varphi _{\left( 1\right) }\geq \varphi _{b}$.

Since $\varphi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}})),$ for each $\iota \in
\Upsilon $, there is $\left( \lambda ,\delta \right) \in \Lambda \times
\Delta $ such that $\varphi ^{\left( n\right) }\left( \left[ d_{ij}\right]
_{i,j=1}^{n}\right) \geq _{\iota }0$ whenever $\left[ d_{ij}\right]
_{i,j=1}^{n}\geq _{\left( \lambda ,\delta \right) }0$ and $\varphi ^{\left(
n\right) }\left( \left[ d_{ij}\right] _{i,j=1}^{n}\right) =_{\iota }0$
whenever $\left[ d_{ij}\right] _{i,j=1}^{n}=_{\left( \lambda ,\delta \right)
}0$ for all $n$. Let $n$ be a positive integer and $\left[ a_{ij}\right]
_{i,j=1}^{n}\in M_{n}\left( \mathcal{A}\right) .$ If $\left[ a_{ij}\right]
_{i,j=1}^{n}\geq _{\lambda }0$, then $\left[ a_{ij}\otimes b\right]
_{i,j=1}^{n}\geq _{\left( \lambda ,\delta \right) }0$, an
\begin{equation*}
\varphi _{b}^{\left( n\right) }\left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) =\left[ \varphi _{b}\left( a_{ij}\right) \right]
_{i,j=1}^{n}=\left[ \varphi \left( a_{ij}\otimes b\right) \right]
_{i,j=1}^{n}=\varphi ^{\left( n\right) }\left( \left[ a_{ij}\otimes b\right]
_{i,j=1}^{n}\right) \geq _{\iota }0.
\end{equation*
If $\left[ a_{ij}\right] _{i,j=1}^{n}=_{\lambda }0,$ then $\left[
a_{ij}\otimes b\right] _{i,j=1}^{n}=_{\left( \lambda ,\delta \right) }0,$ an
\begin{equation*}
\varphi _{b}^{\left( n\right) }\left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) =\varphi ^{\left( n\right) }\left( \left[ a_{ij}\otimes
\right] _{i,j=1}^{n}\right) =_{\iota }0.
\end{equation*
Thus, we showed that $\varphi _{b}\in \mathcal{CPCC}_{\text{loc}}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$. Moreover, from
\begin{eqnarray*}
&&\varphi _{\left( 1\right) }^{\left( n\right) }\left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) -\varphi _{b}^{\left( n\right) }\left( \left[ a_{ij
\right] _{i,j=1}^{n}\right) \\
&=&\varphi ^{\left( n\right) }\left( \left[ a_{ij}\otimes 1_{\mathcal{B}
\right] _{i,j=1}^{n}\right) -\varphi ^{\left( n\right) }\left( \left[
a_{ij}\otimes b\right] _{i,j=1}^{n}\right) \\
&=&\varphi ^{\left( n\right) }\left( \left[ a_{ij}\otimes \left( 1_{\mathcal
B}}-b\right) \right] _{i,j=1}^{n}\right)
\end{eqnarray*
and taking into account that $\left[ a_{ij}\otimes \left( 1_{\mathcal{B
}-b\right) \right] _{i,j=1}^{n}\geq _{\left( \lambda ,\delta \right) }0$ if
\left[ a_{ij}\right] _{i,j=1}^{n}\geq _{\lambda }0$, and $\left[
a_{ij}\otimes \left( 1_{\mathcal{B}}-b\right) \right] _{i,j=1}^{n}=_{\left(
\lambda ,\delta \right) }0$ if $\left[ a_{ij}\right] _{i,j=1}^{n}=_{\lambda
}0$, we conclude that $\varphi _{\left( 1\right) }\geq \varphi _{b}$.
Consequently, there exists a unique $E\left( b\right) \in \pi _{\varphi
_{\left( 1\right) }}\left( \mathcal{A}\right) ^{\prime }\cap C^{\ast }
\mathcal{D}_{\mathcal{E}^{\varphi _{\left( 1\right) }}}),$ $0\leq $ $E\left(
b\right) \leq $id$_{\mathcal{H}^{\varphi _{\left( 1\right) }}},\ $such that
\begin{equation*}
\varphi _{b}\left( a\right) \subseteq V_{\varphi _{\left( 1\right) }}^{\ast
}\pi _{\varphi _{\left( 1\right) }}\left( a\right) E\left( b\right)
V_{\varphi _{\left( 1\right) }}
\end{equation*
for all $a\in \mathcal{A}$. In this way, we obtained a map $E:b(\mathcal{B}
\mathcal{\rightarrow }C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi _{\left(
1\right) }}})\ $such that
\begin{equation*}
\varphi \left( a\otimes b\right) \subseteq V_{\varphi _{\left( 1\right)
}}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a\right) E\left( b\right)
V_{\varphi _{\left( 1\right) }}
\end{equation*
for all $a\in \mathcal{A}$ and $b\in b(\mathcal{B)}$. Moreover, $E\left(
b\right) \pi _{\varphi _{\left( 1\right) }}\left( a\right) =\pi _{\varphi
_{\left( 1\right) }}\left( a\right) E\left( b\right) $ for all $b\in b
\mathcal{B)}$ and for all $a\in \mathcal{A}$, and since $\varphi _{1_
\mathcal{B}}}=\varphi _{\left( 1\right) }$, $E\left( 1_{\mathcal{B}}\right)
= $id$_{\mathcal{D}_{\mathcal{E}^{\varphi _{\left( 1\right) }}}}.$

To show that $E:b(\mathcal{B})\mathcal{\rightarrow }C^{\ast }(\mathcal{D}_
\mathcal{E}^{\varphi _{\left( 1\right) }}})$ is local completely positive,
let $n$ be a positive integer$\ $and $\xi _{i}\in \mathcal{H}_{\iota
}^{\varphi _{\left( 1\right) }},i=1,...,n$. Since $\left( \pi _{\varphi
_{\left( 1\right) }},V_{\varphi _{\left( 1\right) }},\{\mathcal{H}^{\varphi
_{\left( 1\right) }},\mathcal{E}^{\varphi _{\left( 1\right) }},\mathcal{D}_
\mathcal{E}^{\varphi _{\left( 1\right) }}}\}\right) $ is a minimal
Stinespring construction associated with $\varphi _{\left( 1\right) }$, we
can suppose that for each $i\in \{1,...,n\}$, $\xi
_{i}=\tsum\limits_{k=1}^{n_{i}}\pi _{\varphi _{\left( 1\right) }}\left(
a_{ik}\right) V_{\varphi _{\left( 1\right) }}\eta _{ik},$ with $\eta
_{ik}\in \mathcal{H}_{\iota },$ and $a_{ik}\in \mathcal{A},$ $k=1,...,n_{i}$.

Let $\left[ c_{ij}\right] _{i,j=1}^{n}\in M_{n}\left( b(\mathcal{B)}\right) $
such that $\left[ c_{ij}\right] _{i,j=1}^{n}=_{\delta }0$. Clearly, $\left[
d_{ij}\otimes c_{ij}\right] _{i,j=1}^{n}$ $=_{\left( \lambda ,\delta \right)
}0$ for all $\left[ d_{ij}\right] _{i,j=1}^{n}\in M_{n}\left( \mathcal{A
\right) $, and then $\varphi ^{\left( n\right) }\left( \left[ d_{ij}\otimes
c_{ij}\right] _{i,j=1}^{n}\right) $ $=_{\iota }0$. From
\begin{eqnarray*}
&&\left\langle \left( \xi _{i}\right) _{i=1}^{n},E^{\left( n\right) }\left(
\left[ c_{ij}\right] _{i,j=1}^{n}\right) \left( \xi _{i}\right)
_{i=1}^{n}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\left\langle \xi _{i},E\left( c_{ij}\right) \xi
_{j}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{k=1}^{n_{i}}\tsu
\limits_{l=1}^{n_{j}}\left\langle \pi _{\varphi _{\left( 1\right) }}\left(
a_{ik}\right) V_{\varphi _{\left( 1\right) }}\eta _{ik},E\left(
c_{ij}\right) \pi _{\varphi _{\left( 1\right) }}\left( a_{jl}\right)
V_{\varphi _{\left( 1\right) }}\eta _{jl}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{k=1}^{n_{i}}\tsu
\limits_{l=1}^{n_{j}}\left\langle \eta _{ik},V_{\varphi _{\left( 1\right)
}}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a_{ik}^{\ast
}a_{jl}\right) E\left( c_{ij}\right) V_{\varphi _{\left( 1\right) }}\eta
_{jl}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{k=1}^{n_{i}}\tsu
\limits_{l=1}^{n_{j}}\left\langle \eta _{ik},\varphi \left( a_{ik}^{\ast
}a_{jl}\otimes c_{ij}\right) \eta _{jl}\right\rangle \\
&=&\tsum\limits_{k=1}^{n_{i}}\tsum\limits_{l=1}^{n_{j}}\left\langle \left(
\eta _{ik}\right) _{i=1}^{n},\varphi ^{\left( n\right) }\left( \left[
a_{ik}^{\ast }a_{jl}\otimes c_{ij}\right] _{i,j=1}^{n}\right) \left( \eta
_{jl}\right) _{j=1}^{n}\right\rangle =0,
\end{eqnarray*}
it follows that $E^{\left( n\right) }\left( \left[ c_{ij}\right]
_{i,j=1}^{n}\right) =_{\iota }0.$

Let $\left[ b_{ij}\right] _{i,j=1}^{n},\left[ c_{ij}\right] _{i,j=1}^{n}\in
M_{n}\left( b(\mathcal{B)}\right) $ such that $\left[ c_{ij}\right]
_{i,j=1}^{n}=_{\delta }0$. Then
\begin{eqnarray*}
&&\left\langle \left( \xi _{i}\right) _{i=1}^{n},E^{\left( n\right) }\left(
\left( \left[ b_{ij}\right] _{i,j=1}^{n}\right) ^{\ast }\left[ b_{ij}\right]
_{i,j=1}^{n}+\left[ c_{ij}\right] _{i,j=1}^{n}\right) \left( \xi _{i}\right)
_{i=1}^{n}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{r=1}^{n}\left\langle \xi
_{i},E\left( b_{ri}^{\ast }b_{rj}\right) \xi _{j}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{r=1}^{n}\left\langle
\tsum\limits_{k=1}^{n_{i}}\pi _{\varphi _{\left( 1\right) }}\left(
a_{ik}\right) V_{\varphi _{\left( 1\right) }}\eta _{ik},E\left( b_{ri}^{\ast
}b_{rj}\right) \tsum\limits_{l=1}^{n_{j}}\pi _{\varphi _{\left( 1\right)
}}\left( a_{jl}\right) V_{\varphi _{\left( 1\right) }}\eta _{jl}\right\rangle
\\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{r=1}^{n}\tsum\limits_{k=1}^{n_{i}
\tsum\limits_{l=1}^{n_{j}}\left\langle \eta _{ik},V_{\varphi _{\left(
1\right) }}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a_{ik}^{\ast
}a_{jl}\right) E\left( b_{ri}^{\ast }b_{rj}\right) V_{\varphi _{\left(
1\right) }}\eta _{jl}\right\rangle \\
&=&\tsum\limits_{i,j=1}^{n}\tsum\limits_{r=1}^{n}\tsum\limits_{k=1}^{n_{i}
\tsum\limits_{l=1}^{n_{j}}\left\langle \eta _{ik},\varphi \left(
a_{ik}^{\ast }a_{jl}\otimes b_{ri}^{\ast }b_{rj}\right) \eta
_{jl}\right\rangle \\
&=&\tsum\limits_{r=1}^{n}\tsum\limits_{i,j=1}^{n}\tsum\limits_{k=1}^{n_{i}
\tsum\limits_{l=1}^{n_{j}}\left\langle \eta _{ik},\varphi \left( \left(
a_{ik}\otimes b_{ri}\right) ^{\ast }\left( a_{jl}\otimes b_{rj}\right)
\right) \eta _{jl}\right\rangle \\
&=&\tsum\limits_{r=1}^{n}\left(
\tsum\limits_{i,j=1}^{n}\tsum\limits_{k=1}^{n_{i}}\tsum\limits_{l=1}^{n_{j}
\left\langle \eta _{ik},\varphi \left( \left( d_{ikr}\right) ^{\ast
}d_{jlr}\right) \eta _{jl}\right\rangle \right) \geq 0.
\end{eqnarray*
Therefore, $E$ is a local $\mathcal{CP}$-map, and then by \cite[Proposition
3.3]{MJ3}, it is continuous with respect to the families of $C^{\ast }
-seminorms $\{\left. p_{\delta }\right\vert _{b(\mathcal{B})}\}_{\delta \in
\Delta }\ $and $\{\left\Vert \cdot \right\Vert _{\iota }\}_{\iota \in
\Upsilon }.$

Consequently,  $E$ extends to a unital local $\mathcal{CP}$-map from
\mathcal{B}$ to $C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi _{\left(
1\right) }}})$, denoted by $E$ too. By \cite[Corollary 4.1]{D1}, $E$ is also
a local $\mathcal{CC}$-map. Moreover, $E\left( b\right) \pi _{\varphi
_{\left( 1\right) }}\left( a\right) =\pi _{\varphi _{\left( 1\right)
}}\left( a\right) E\left( b\right) $ for all $b\in \mathcal{B}$ and $a\in
\mathcal{A}$, and
\begin{equation*}
\varphi \left( a\otimes b\right) \subseteq V_{\varphi _{\left( 1\right)
}}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a\right) E\left( b\right)
V_{\varphi _{\left( 1\right) }}
\end{equation*
for all $a\in \mathcal{A}$ and $b\in \mathcal{B}.$

To prove the uniqueness of $E$, suppose that $\widetilde{E}:\mathcal{B
\rightarrow C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi _{\left( 1\right)
}}})\ $is another local $\mathcal{CCP}$-map such that $\widetilde{E}\left(
b\right) \pi _{\varphi _{\left( 1\right) }}\left( a\right) =\pi _{\varphi
_{\left( 1\right) }}\left( a\right) \widetilde{E}\left( b\right) $ for all
b\in \mathcal{B}$ and $a\in \mathcal{A}$, and
\begin{equation*}
\varphi \left( a\otimes b\right) \subseteq V_{\varphi _{\left( 1\right)
}}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a\right) \widetilde{E
\left( b\right) V_{\varphi _{\left( 1\right) }}
\end{equation*
for all $a\in \mathcal{A}$ and $b\in \mathcal{B}$.$\ $Moreover, for each
b\in b(\mathcal{B)},\widetilde{E}\left( b\right) \in \pi _{\varphi _{\left(
1\right) }}\left( \mathcal{A}\right) ^{\prime }\cap C^{\ast }(\mathcal{D}_
\mathcal{E}^{\varphi _{\left( 1\right) }}})$. Then, for each $b\in b\left(
\mathcal{B}\right) ,$
\begin{eqnarray*}
\varphi _{\left( 1\right) \widetilde{E}\left( b\right) }\left( a\right) \xi
&=&V_{\varphi _{\left( 1\right) }}^{\ast }\pi _{\varphi _{\left( 1\right)
}}\left( a\right) \widetilde{E}\left( b\right) V_{\varphi _{\left( 1\right)
}}\xi =\varphi \left( a\otimes b\right) \xi \\
&=&V_{\varphi _{\left( 1\right) }}^{\ast }\pi _{\varphi _{\left( 1\right)
}}\left( a\right) E\left( b\right) V_{\varphi _{\left( 1\right) }}\xi
=\varphi _{\left( 1\right) E\left( b\right) }\left( a\right) \xi
\end{eqnarray*
for all $\xi \in \mathcal{D}_{\mathcal{E}}$ and for all $a\in \mathcal{A}$,
whence, by \cite[Theorem 4.5]{BGK}, $\widetilde{E}\left( b\right) =E\left(
b\right) $. Consequently, $\left. \widetilde{E}\right\vert _{b(\mathcal{B)
}=\left. E\right\vert _{b(\mathcal{B)}}$, and since $b(\mathcal{B)}$ is
dense in $\mathcal{B}$ and the maps $\widetilde{E}$ and $E$ are continuous,
\widetilde{E}=E.$
\end{proof}

The above theorem extends a result of Haapasalo, Heinosaari and Pellonpaa
\cite[Lemma 1]{HHP} in the context of unbounded local $\mathcal{CCP}$-maps.
As in \cite{HHP}, the quadruple $\left( \pi _{\varphi _{\left( 1\right)
}},E,V_{\varphi _{\left( 1\right) }},\{\mathcal{H}^{\varphi _{\left(
1\right) }},\mathcal{E}^{\varphi _{\left( 1\right) }},\mathcal{D}_{\mathcal{
}^{\varphi _{\left( 1\right) }}}\}\right) $ is called an $\mathcal{A}
-subminimal dilation of $\varphi $. In the same way, we define a $\mathcal{B}
$-subminimal dilation of $\varphi $.

\begin{theorem}
\label{C} Let $\varphi _{1}\in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))\ $and $\varphi _{2}\in \mathcal{CPCC
_{\text{loc,}P}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}})$. Suppose
that $\varphi _{1}$ and $\varphi _{2}$ are compatible.

\begin{enumerate}
\item If $\varphi _{1}$ is extremal in $\mathcal{CPCC}_{\text{loc,}P}
\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ or $\varphi _{2}$ is
extremal in

$\mathcal{CPCC}_{\text{loc,}P}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E
}))$, then they are a unique joint map.

\item If $\varphi _{1}$ is extremal in $\mathcal{CPCC}_{\text{loc,}P}
\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ and $\varphi _{2}$ is
extremal in

$\mathcal{CPCC}_{\text{loc,}P}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E
}))$, then their unique joint map is extremal

in $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max }\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{E}})).$

\item If $\varphi _{1}$ or $\varphi _{2}$ is a local contractive $\ast
-morphism, then

$\varphi _{1}\left( a\right) \varphi _{2}\left( b\right) =\varphi _{2}\left(
b\right) \varphi _{1}\left( a\right) $ for all $a\in \mathcal{A}$ and $b\in
\mathcal{B}$ and the

unique joint map $\psi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes
_{\max }\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ is of the form

$\psi \left( a\otimes b\right) =\varphi _{1}\left( a\right) \varphi
_{2}\left( b\right) $ for all $a\in \mathcal{A}$ and $b\in \mathcal{B}$.
\end{enumerate}
\end{theorem}

\begin{proof}
$\left( 1\right) $\ Since $\varphi _{1}$ and $\varphi _{2}$ are compatible,
there is $\varphi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes
_{\max }\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such that
\varphi _{\left( 1\right) }=\varphi _{1}$ and $\varphi _{\left( 2\right)
}=\varphi _{2}$. Suppose that $\varphi $ is not unique and $\varphi _{1}$ is
extremal in $\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D
_{\mathcal{E}}))$. Let $\psi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal
A\otimes }_{\max }\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ such
that $\psi _{\left( 1\right) }=\varphi _{1},$ $\psi _{\left( 2\right)
}=\varphi _{2}$ and $\varphi \neq \psi $ and $\left( \pi _{\varphi
_{1}},V_{\varphi _{1}},\{\mathcal{H}^{\varphi _{1}},\mathcal{E}^{\varphi
_{1}},\mathcal{D}_{\mathcal{E}^{\varphi _{1}}}\}\right) $ be a minimal
Stinespring dilation of $\varphi _{1}$. By Theorem \ref{Help}, there exist
E,\widetilde{E}\in $ $\mathcal{CPCC}_{\text{loc,}1}(\mathcal{B},C^{\ast }
\mathcal{D}_{\mathcal{E}^{\varphi _{1}}}))$ such that
\begin{equation*}
\varphi \left( a\otimes b\right) \subseteq V_{\varphi _{1}}^{\ast }\pi
_{\varphi _{1}}\left( a\right) E\left( b\right) V_{\varphi _{1}}\text{ and
\psi \left( a\otimes b\right) \subseteq V_{\varphi _{1}}^{\ast }\pi
_{\varphi _{1}}\left( a\right) \widetilde{E}\left( b\right) V_{\varphi _{1}}.
\end{equation*
Then
\begin{equation*}
V_{\varphi _{1}}^{\ast }\left( E\left( b\right) -\widetilde{E}\left(
b\right) \right) V_{\varphi _{1}}\xi =\varphi \left( 1_{\mathcal{A}}\otimes
b\right) \xi -\psi \left( 1_{\mathcal{A}}\otimes b\right) \xi =\varphi
_{\left( 2\right) }\left( b\right) \xi -\psi _{\left( 2\right) }\left(
b\right) \xi =0
\end{equation*
for all $\xi \in \mathcal{D}_{\mathcal{E}}$ and for all $b\in b(\mathcal{B)}
.$\ $From this relation and taking into account that $E\left( b\right)
\widetilde{E}\left( b\right) \in $ $\pi _{\varphi _{1}}\left( \mathcal{A
\right) ^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi _{1}}})$
for all $b\in b(\mathcal{B)}$, and $\varphi _{1}$ is extremal in $\mathcal
CPCC}_{\text{loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$, we
conclude that $E\left( b\right) =\widetilde{E}\left( b\right) $ for all
b\in b(\mathcal{B)}$. Since $E$ and $\widetilde{E}\ $are continuous and $b
\mathcal{B)}$ is dense in $\mathcal{B}$, from $\left. E\right\vert _{b
\mathcal{B})}=\left. \widetilde{E}\right\vert _{b(\mathcal{B})}$, we deduce
that $E=$ $\widetilde{E}$. Therefore, $\varphi =\psi $, a contradiction.

$\left( 2\right) $ Since $\varphi _{1}$ and $\varphi _{2}$ are extremal, the
joint map of $\varphi _{1}\ $and $\varphi _{2}$ is unique. Let $\psi \in
\mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max }\mathcal{B},C^{\ast
}(\mathcal{D}_{\mathcal{E}}))$ be the joint map of $\varphi _{1}\ $and
\varphi _{2}.$ Suppose that $\psi $ is not extremal. There exist $\psi
_{1},\psi _{2}\in \mathcal{CPCC}_{\text{loc,}P}(\mathcal{A\otimes }_{\max
\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}})),\psi _{1}\neq \psi _{2}$
such that $\psi =\frac{1}{2}\psi _{1}+\frac{1}{2}\psi _{2}$, and so $\varphi
_{i}=\psi _{\left( i\right) }=\frac{1}{2}\left( \psi _{1}\right) _{\left(
i\right) }+\frac{1}{2}\left( \psi _{2}\right) _{\left( i\right) },i=1,2.$

Let $\left( \pi _{\varphi _{1}},V_{\varphi _{1}},\{\mathcal{H}^{\varphi
_{1}},\mathcal{E}^{\varphi _{1}},\mathcal{D}_{\mathcal{E}^{\varphi
_{1}}}\}\right) $ be a minimal Stinespring dilation of $\varphi _{1}$. From
\ \varphi _{1}\geq \frac{1}{2}\left( \psi _{i}\right) _{\left( 1\right)
},i=1,2$,$\ $and Radon Nikodym theorem \cite[Theorem 4.5]{BGK}, we deduce
that there exists $T_{i}\in \pi _{\varphi _{1}}\left( \mathcal{A}\right)
^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\varphi _{1}}}),i=1,2$,
such that
\begin{equation*}
\left( \psi _{i}\right) _{\left( 1\right) }\subseteq V_{\varphi _{\left(
1\right) }}^{\ast }\pi _{\varphi _{\left( 1\right) }}\left( a\right)
T_{i}V_{\varphi _{\left( 1\right) }}=V_{\varphi _{\left( 1\right) }}^{\ast
}\left( T_{i}\right) ^{\frac{1}{2}}\pi _{\varphi _{\left( 1\right) }}\left(
a\right) \left( T_{i}\right) ^{\frac{1}{2}}V_{\varphi _{\left( 1\right)
}},i=1,2.
\end{equation*
For each $\iota \in \Upsilon $, put $\left[ \left( T_{i}\right) ^{\frac{1}{2
}\mathcal{H}_{\iota }^{\varphi _{\left( 1\right) }}\right] =\mathcal{H
_{\iota }^{i},i=1,2$. Clearly, $\mathcal{H}_{\iota _{1}}^{i}\subseteq
\mathcal{H}_{\iota _{2}}^{i}\ $for all $\iota _{1},\iota _{2}\in \Upsilon \
with $\iota _{1}\leq \iota _{2}$, $i=1,2$. Let $\mathcal{H}^{i}=\left[
\left( T_{i}\right) ^{\frac{1}{2}}\mathcal{H}^{\varphi _{\left( 1\right) }
\right] ,$ $i=1,2.$ Then, $\{\mathcal{H}^{i};\mathcal{E}^{i};\mathcal{D}_
\mathcal{E}^{i}}\}$, where $\mathcal{E}^{i}\mathcal{=\{H}_{\iota }^{i
\mathcal{\}}_{\iota \in \Upsilon }$, is a quantized domain in the Hilbert
space $\mathcal{H}^{i},\mathcal{\ }i=1,2.\ $Since, $\left( T_{i}\right) ^
\frac{1}{2}}V_{\varphi }\left( \mathcal{E}\right) \subseteq \mathcal{E}^{i};
\ $\left( \psi _{i}\right) _{\left( 1\right) }\left( a\right) \subseteq
V_{\varphi _{1}}^{\ast }\left( T_{i}\right) ^{\frac{1}{2}}\pi _{\varphi
_{1}}\left( a\right) \left( T_{i}\right) ^{\frac{1}{2}}V_{\varphi _{1}},$\ \
\ for all $a\in \mathcal{A}$, and$\ \ \left[ \pi _{\varphi _{1}}\left(
\mathcal{A}\right) \left( T_{i}\right) ^{\frac{1}{2}}V_{\varphi _{1}
\mathcal{H}_{_{\tau }}\right] =\left[ \left( T_{i}\right) ^{\frac{1}{2}}\pi
_{\varphi _{1}}\left( \mathcal{A}\right) V_{\varphi _{1}}\mathcal{H}_{\iota
\right] =$ $\left[ \left( T_{i}\right) ^{\frac{1}{2}}\mathcal{H}^{\varphi
_{\left( 1\right) }}\right] =\mathcal{H}_{\iota }^{i}$ for all $\iota \in
\Upsilon $, it follows that $\left( \pi _{\varphi _{1}},\left( T_{i}\right)
^{\frac{1}{2}}V_{\varphi _{1}},\{\mathcal{H}^{i},\mathcal{E}^{i},\mathcal{D
_{\mathcal{E}^{i}}\}\right) ,i=1,2$, is a minimal Stinespring dilation of
\left( \psi _{i}\right) _{\left( 1\right) },i=1,2$. Then, by Theorem \re
{Help}, there exists $E_{i}\in $ $\mathcal{CPCC}_{\text{loc,}1}(\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{E}^{i}}))$ such tha
\begin{equation*}
\psi _{i}\left( a\otimes b\right) \subseteq V_{\varphi _{1}}^{\ast }\left(
T_{i}\right) ^{\frac{1}{2}}\pi _{\varphi _{1}}\left( a\right) E_{i}\left(
b\right) \left( T_{i}\right) ^{\frac{1}{2}}V_{\varphi _{1}}=V_{\varphi
_{1}}^{\ast }\pi _{\varphi _{1}}\left( a\right) \left( T_{i}\right) ^{\frac{
}{2}}E_{i}\left( b\right) \left( T_{i}\right) ^{\frac{1}{2}}V_{\varphi _{1}}
\end{equation*}
for all $a\in \mathcal{A}$ and for all $b\in \mathcal{B},i=1,2$, and
E_{i}\left( b\right) \in \pi _{\varphi _{1}}\left( \mathcal{A}\right)
^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{i}})$for all $b\in b
\mathcal{B}),i=1,2$. Since $\varphi _{2}\ $is extremal, from $\varphi
_{2}=\psi _{\left( 2\right) }=\frac{1}{2}\left( \psi _{1}\right) _{\left(
2\right) }+\frac{1}{2}\left( \psi _{2}\right) _{\left( 2\right) }$, we
deduce that $\left( \psi _{1}\right) _{\left( 2\right) }=\left( \psi
_{2}\right) _{\left( 2\right) }$. Therefore, for each $b\in b(\mathcal{B)},$
\begin{eqnarray*}
0 &=&\left( \psi _{1}\right) _{\left( 2\right) }\left( b\right) -\left( \psi
_{2}\right) _{\left( 2\right) }\left( b\right)  \\
&=&\left. V_{\varphi _{1}}^{\ast }\pi _{\varphi _{1}}\left( a\right) \left(
\left( T_{1}\right) ^{\frac{1}{2}}E_{1}\left( b\right) \left( T_{1}\right) ^
\frac{1}{2}}-\left( T_{2}\right) ^{\frac{1}{2}}E_{2}\left( b\right) \left(
T_{2}\right) ^{\frac{1}{2}}\right) V_{\varphi _{1}}\right\vert _{\mathcal{D
_{\mathcal{E}}}
\end{eqnarray*
and $\left( T_{1}\right) ^{\frac{1}{2}}E_{1}\left( b\right) \left(
T_{1}\right) ^{\frac{1}{2}}-\left( T_{2}\right) ^{\frac{1}{2}}E_{2}\left(
b\right) \left( T_{2}\right) ^{\frac{1}{2}}\in \pi _{\varphi _{1}}\left(
\mathcal{A}\right) ^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{E}})$.
From the above relation, and taking into account that $\varphi _{1}\ $is
extremal and $E_{1}$ and $E_{2}$ are continuous, we deduce that $\left(
T_{1}\right) ^{\frac{1}{2}}E_{1}\left( b\right) \left( T_{1}\right) ^{\frac{
}{2}}=\left( T_{2}\right) ^{\frac{1}{2}}E_{2}\left( b\right) \left(
T_{2}\right) ^{\frac{1}{2}}$ for all $b\in \mathcal{B}$. Consequently, $\psi
_{1}=\psi _{2}$, a contradiction. Therefore, $\psi $ is extremal.

$\left( 3\right) \ $Suppose that $\varphi _{1}$ is a local contractive $\ast
$-morphism. Then $P\ $is a projection in $\varphi _{1}\left( \mathcal{A
\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}})$. Therefore,
we can suppose that for each $\iota \in \Upsilon ,\left[ P\mathcal{H}_{\iota
}\right] =\mathcal{H}_{\iota }$, and then $\left[ \varphi _{1}\left(
\mathcal{H}_{\iota }\right) \right] =\mathcal{H}_{\iota }$. Then $\left(
\varphi _{1},\text{id}_{\mathcal{D}_{\mathcal{E}}},\{\mathcal{H},\mathcal{E}
\mathcal{D}_{\mathcal{E}}\}\right) $ is a minimal Stinespring dilation of
\varphi _{1}$. Since $\varphi _{1}\ $is extremal\ in $\mathcal{CPCC}_{\text
loc,}P}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$, by $(1)$, there
exists a unique joint map $\psi \in \mathcal{CPCC}_{\text{loc,}P}(\mathcal
A\otimes }_{\max }\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$, and by
Theorem \ref{Help}, there exists a unique $E\mathcal{\in CPCC}_{\text{loc,
1}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$\ such that $E\left(
b\right) \varphi _{1}\left( a\right) =\varphi _{1}\left( a\right) E\left(
b\right) $ for all $b\in \mathcal{B}$ and for all $a\in \mathcal{A}$, and
\begin{equation*}
\psi \left( a\otimes b\right) =\varphi _{1}\left( a\right) E\left( b\right)
=E\left( b\right) \varphi _{1}\left( a\right)
\end{equation*
for all $a\in \mathcal{A}$ and for all $b\in \mathcal{B}$. From the above
relation, we deduce that $\varphi _{2}=E,$ and so $\varphi _{1}$ and
\varphi _{2}$ commute.
\end{proof}

\begin{corollary}
Let $\varphi _{1}:\mathcal{A}\rightarrow C^{\ast }(\mathcal{D}_{\mathcal{E
}))\ $be a unital local contractive $\ast $-morphism and $\varphi _{2}
\mathcal{B}\rightarrow C^{\ast }(\mathcal{D}_{\mathcal{E}})\ $be a unital
\mathcal{CCP}$-map. Then, $\varphi _{1}$ and $\varphi _{2}$ are compatible
if and only if the ranges of $\varphi _{1}$ and $\varphi _{2}\ $commute.
\end{corollary}


\section{Introduction}

Much of the research in dialogue systems from the last few years has focused on replacing all (or some) of its components with Deep Neural Network (DNN) architectures~\cite{pipelinedialogue,speechrec,li2017adversarial,serban2016building,serban2016hierarchical,neural_conv}. These DNN models are trained end-to-end with large corpora of human-to-human dialogues, and essentially learn to mimic human conversations.

Although these models can represent the input context, the need for a dedicated external memory to remember  information in context was pointed out and mechanisms were introduced in models like Memory Networks \cite{sukhbaatar2015end,bordes2016learning,gulcehre2018dynamic}, and the Neural Turing Machine \cite{graves2014neural}. Although these models, in theory, are better at maintaining the \emph{state} using their memory component, they require longer training time and excessive search for hyperparameters.

In this paper we explore the possibility of incorporating external information in dialogue systems as a mechanism to supplement the standard context encoding and facilitate the generation to be more specific with faster learning time. Furthermore, especially in the case of chit-chat systems, knowledge can be leveraged from different topics ( education, sports, news, travel, etc.).
Current memory-based architectures cannot efficiently handle access to large unstructured external knowledge sources

In this work, we build on the popular Encoder-Decoder model, and incorporate external knowledge as an additional continuous vector. The proposed Extended Encoder-Decoder (Ext-ED) architecture learns to predict the embedding of the relevant external context during training and, during testing, uses an augmented \emph{state} of external context and encoder final state to generate the next utterance. We evaluate our model with experiments on Reddit News dataset, and consider using either the Wikipedia summary~\cite{scheepers2017compositionality} or the NELL KB~\cite{carlson2010toward} as a source of external context.

\section{Related Work}
\label{related}
Incorporating supplemental knowledge in neural conversational agents has been addressed in a few recent works on dialogue systems. Previous research however was mostly in the context of goal-driven dialogue systems, where the knowledge-base is highly structured, and queried to obtain very specific information (e.g. bus schedules, restaurant information, more broad tourist information).

Few goal-oriented dialogues research use external information directly from the web or relevant data sources. An exception is \cite{longenhanced}, which searches the web for relevant information that pertains to the input context and provides them as suggestions (like advising on places to visit while the user intends to visit a city). Similarly, instead of dynamically querying the web, \cite{ghazvininejad2017knowledge} pre-trains the model from facts in Foursquare \cite{foursquare} to select relevant facts as suggestions.

Moving closer to unstructured domains, but still within task-driven conversations, \cite{lowe2015incorporating} proposes a way of retrieving relevant responses based on external knowledge sources. The model selects relevant knowledge from Ubuntu man pages and uses it to retrieve a relevant context-response pair that is inline with the knowledge extracted. Similarly, \cite{young2017augmenting} extracts relations within the context and parses over it to score the message response pairs. The relational knowledge provides a way of incorporating useful knowledge or \emph{common sense} as termed by the authors.

Though \cite{guu2017generating} did not directly make use of an external knowledge source, they used an \emph{edit vector} that aids in editing a sampled prototype sentence. This is relevant to our proposed model as the generated response is conditioned on a supplementary vector similar to the external context vector discussed later in this paper.

\section{Technical Background}
\label{back}
\subsection{Recurrent Neural Networks}
The Recurrent Neural Network (RNN) is a variant of neural network used for learning representations of inputs, $\bm{x_{1:T}}$, that have an inherent sequential structure (speech, video, dialogue etc.). In natural language processing, RNNs are used to learn language models that generalize over n-gram models \cite{katzNgram}. The RNN maintains a hidden state, $h_t$, that is an abstraction of inputs observed until time-step $t$ of the input sequence, and uses $x_t$ to operate on them. RNN uses two projection functions, $U$ and $W$, for computing operations on input and hidden states respectively. A third function, $V$, to map $h_t$ to the output, $y_t$, the output of the RNN at every time step $t$. $y_t$, is a distribution over the next token given the previous tokens, and is computed as a function of $h_t$. The functions of the RNN can be explained as shown in Equations \ref{state} and \ref{output},

\begin{equation}
h_t = \emph{g}\left(U\cdot x_t + W\cdot h_{t-1} + b\right),
\label{state}
\end{equation}
\begin{equation}
y_t = V\cdot h_t + d,
\label{output}
\end{equation}
where $y_t$ is the output, $x_t$ is the vector representation of input token, $h_t$ is the internal state of the RNN at time $t$ and $g$ is a non-linear function (like \emph{tanh} or \emph{sigmoid}). RNNs are trained with Back Propagation Through Time (BPTT) \cite{rumelhart} to compute weight updates using the derivative of a loss function with respect to the parameters over all previous time-steps.

\subsection{Seq2Seq Dialogue Architecture}

Generative dialogue models \cite{seq2seq,serban2015survey} extends the language model learned by RNNs to generate natural language that are conditioned not only on the previous words generated in the response but also on a representation of the input context. The ability of such a learning module to \emph{understand} an input sequence of words (that we call context ($\bm{c_{1:T}^i}$)) and \emph{generate} a response $\bm{r_{1:T}^i}$ tantamount to solving the \textit{dialogue task}.

\cite{neural_conv} first proposed a vanilla LSTM \cite{lstm} dialogue model that encodes a given context with an LSTM (\emph{Encoder}) and feeds it to another LSTM (\emph{Decoder}) that generates a response token-by-token. Here the choice of encoder and decoder modules can be any recurrent architectures like GRU \cite{cho2014gru}, RNN, Bi-directional LSTM \cite{bidirectional}, etc. The model is then trained to learn a conditional distribution over the vocabulary for generating the next token in response to the $i^{th}$ context as shown in Equation \ref{generative}:
\begin{equation}
P\left(\bm{r_{1:T}^i}\mid \bm{c_{1:T}^i}\right) = \Pi_{k=1}^TP\left(r^i_k\mid \bm{r_{1:k-1}^i},\bm{c_{1:T}^i}\right).
\label{generative}
\end{equation}

With neural language models, this form can aid in maintaining long term dependencies and the next word distribution can be made to conditionally depend on an arbitrarily long context. Sophisticated models have made significant architectural improvements to aid better modelling of the contexts \cite{serban2016hierarchical}.

\section{Extended Encoder Decoder Architecture}
\label{model}
The primary objective of the proposed architecture is to supplement response generation with external knowledge relevant to the context. Most of the knowledge sources that are available are free-form and lack suitable structure for easy querying of relevant information. In this work, we attempt to incorporate such unstructured knowledge corpora for dialogue generation in Seq2Seq models.

\begin{figure}[H]
    \centering
    \includegraphics[width=0.55\textwidth]{images/EED.png}
    \caption{Architecture Diagram of Extended Encoder Decoder Model.}
    \label{eed}
\end{figure}

\subsection{The Model}
\label{modeldefinition}

The Extended Encoder Decoder (Ext-ED) model, shown in Figure \ref{eed}, uses an encoder LSTM (parameterized as $\bm{\Theta}_{Enc}$) to encode the $i^{th}$ context, $\bm{c_{1:T}^i}$, and a fully connected layer ($f$) to predict an external context vector ($\bm{ec^i}$) conditioned on the encoded context. The predicted external context vector, $\bm{ec^i}$, is provided to the decoder LSTM ($\bm{\Theta}_{Dec}$) at every step, augmented with the encoder final state and previous predicted token, to generate the next token in the response:
\begin{equation}
    P\left(\bm{ec^i} \mid \bm{c_{1:T}^i}\right) = f\left(\bm{\Theta_{Enc}\left(c_{1:T}^i\right)}\right)
    \label{external}
\end{equation}
\begin{equation}
    P\left(\bm{r_{1:T}^i}\mid \bm{c_{1:T}^i}\right) = \Pi_{k=1}^TP\left(r_k^i\mid \bm{r_{1:k-1}^i},\bm{c_{1:T}^i}, \bm{ec^i}\right).
    \label{decodereqn}
\end{equation}

The decoder is provided with an encoding of the context along with the external knowledge encoding, as $\bm{ec^i}$ acts as information supplement to the knowledge available in the context as shown in Equations \ref{external} and \ref{decodereqn}.

During training, the gradients for Ext-ED parameters ($f,\Theta_{Enc}, \Theta_{Dec}$) are computed by backpropagating the gradients for parameters with respect to losses in Equations \ref{loss1}, \ref{loss2} and \ref{loss3}:

\begin{equation}
    \mathcal{L}_1 = \sum_{k=1}^T Q\left(r^i \mid \hat{ec}^i, c^i\right) \log P\left(r^i \mid c^i\right),
    \label{loss1}
\end{equation}
\begin{equation}
    \mathcal{L}_2 = \Vert \hat{\bm{ec}}^i - \bm{ec^i} \Vert_2,
    \label{loss2}
\end{equation}
\begin{equation}
    \mathcal{L}_3 = - \sum_{k=1}^T Q\left(r^i \mid c^i, \bm{0}\right) \log P\left(r^i \mid c^i\right).
    \label{loss3}
\end{equation}

 Here $\mathcal{L}_1$ is the log-likelihood that is used to make the model distribution mimic the data distribution. $\mathcal{L}_2$ trains $f$ to correctly predict $ec^i$, and $\mathcal{L}_3$ trains $\Theta_{Dec}$ to make use of the external context by forcing it the model distribution to diverge when not provided with the external context ($ec^i$ is set to ${\bm 0}$ vector). In the loss equations, \emph{P} and \emph{Q} represent the data and the model learned distributions respectively, and $\bm{ec^i}$ and $\bm{\hat{ec}^i}$ represent true and model($f$) predicted external knowledge encoding.

\subsection{External Context Vector}
\label{modeldetails}
We use Wikipedia summary \cite{scheepers2017compositionality} and NELL knowledge base (KB)  \cite{carlson2010toward} to compute the external knowledge encoding for every context in the context-response pairs. Algorithm \ref{extcontalg} oultines the pseudocode for computing the external context vector ($\bm{ec^i}$). For $i^{th}$ input context, the methods \emph{Return\_All\_Values\_for\_Entity} or  \emph{Wiki\_Summary\_Query} is used to extract the external knowledge vector, $ec^i$, from NELL KB or Wikipedia summary sources.

\SetAlFnt{\small\sf}
   \begin{algorithm}[h]
    \label{extcontalg}
    \caption{Get\_External\_Context \_Vector ($\bm{c^i_{1:T}}$)}

    \SetKw{return}{return}
    \SetKw{from}{from}
    \SetKw{downto}{downto}
    \SetKw{step}{step}

    $\bm{ec^i}$ $\gets$ zero\_vector \\
    $\#\_Ext\_Tokens$ $\gets$ 0 \\
   \For{t in range (1,T)}{
    \If{$\bm{c_t^i}$ is not a \textbf{stop word}}{
    External\_Tokens$_{List}$ $\gets$ Wiki\_Summary\_Query($\bm{c_t^i}$) \\ \% Return\_All\_Values\_for\_Entity($\bm{c_t^i}$)\\
    \For{token in External\_Tokens$_{List}$}{
    $\bm{ec^i}$ $\gets$ $\bm{ec^i}$ + GloVe\_Embedding(token)\\
    $\#\_Ext\_Tokens$ $\gets$ $\#\_Ext\_Tokens$ + 1
    }
    }
}
        return $\frac{\bm{ec^i}}{\#\_Ext\_Tokens}$
        \end{algorithm}

The external context encoding, $\bm{ec^i}$, is a fixed length continuous embedding of the knowledge from external sources, as having all the words sampled (represented as a Bag of Words ) proved to be a severe computational overhead because of sparsity.

The continuous embedding of external context provides an additional conditioning with relevant external knowledge that is used to generate the next utterance. Although this is the intended hypothesis, there are certain expected characteristics that are desirable of external knowledge sources for them to be useful:
\begin{itemize}
    \item The knowledge vectors being away from the mean of their distribution.
    \item The knowledge vectors having high variance.
\end{itemize}

We analyzed the knowledge vectors constructed using the two knowledge sources and the distribution of distances of $\bm{ec^i}$ from their mean. The variance of this distribution in NELL KB was \emph{1.44} and from that of Wikipedia summary was \emph{0.73}. Mean distance was very low (\emph{0.77}) in the case of Wikipedia compared to that of NELL (\emph{2.16}). We observed that the vectors not being spread out made them less useful than the encoded context itself in the initial experiments.

\section{Experiments and Discussion}
\label{exper}
We evaluated Ext-ED by training with Reddit News dataset, and incorporated Wikipedia Summary and NELL KB as sources of external knowledge. The objective of the experiments are three-fold: (1) to evaluate the ability of the model to make use of the external context to condition the response; (2) to analyze the training time with the additional knowledge provided; and (3) to observe any tangible differences in training with the two knowledge sources.

\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{images/train_plot.png}
    \caption{Convergence of Sequence loss (cross-entropy loss in sequential outputs) over different models during training.(-L3 in legend denote exclusion of $\mathcal{L}_3$ loss from gradient computation.)}
   \label{train}
\end{figure}

For the first analysis, we trained Ext-ED with external context ($\mathbb{R}^{100}$) and validated it without providing it (see Ext-ED - $\mathcal{L}_3$ Ablation in Table \ref{scores}). Without the inclusion of $\mathcal{L}_3$, Ext-ED did not find the external context useful and the performance was not very different from a Vanilla Seq2Seq dialogue model (Figure \ref{train}). The encoder context had enough variance to be a viable information source and hence the external context was ignored. This can be observed from similar learning curves of \emph{Vanilla Seq2Seq, Ext-ED - L3 (Wiki)} models in Figure \ref{train}.

With propagating back gradients with respect to $\mathcal{L}_3$, we observed that the model learns to use the external context, but, as discussed in Section \ref{modeldetails}, the variance in the external context vectors constructed using the two knowledge sources was too low. To fix this, we scaled the external context vectors with $\mathcal{N}(4,1)$. This improved the variance in the knowledge that subsequently improved the usefulness of these vectors which was also observed in Figure \ref{train}.

\begin{table}[h]
    \centering
    \begin{tabular}{|p{3.5cm}|p{1cm}|p{1.4cm}|}
        \hline
         \textbf{Model} & \textbf{PPL} & \textbf{BLEU-4}  \\
        \hline
         Vanilla Seq2Seq & 38.09 & 0.437 \\
         Ext-ED - $\mathcal{L}_3$ (Wiki) & 38.37 & 0.435 \\
         Ext-ED - $\mathcal{L}_3$ Ablation & 37.06 & 0.425 \\
         Ext-ED (Wiki) & {\bf 30.26} & {\bf 0.53} \\
         Ext-ED (NELL KB) & {\bf 29.07} & {\bf 0.525}\\
         Ext-ED Ablation & 601.8 & 0.274 \\
         \hline
    \end{tabular}
    \caption{Comparison of BLEU (sentence\_bleu) and Perplexity scores on validation set across different models.}
    \label{scores}
\end{table}

The provision of external knowledge improved the Perplexity and BLEU-4 scores as shown in Table \ref{scores}. Though the improvements are reasonable, the metrics used are not strong indicators for evaluating the influence of external contexts in dialogue. But, they do indicate that the prediction accuracy is improved with the inclusion of external knowledge sources. The poor perplexity for Ext-ED Ablation is because the model is conditioned to predict the next utterance  using $\bm{ec^i}$ and when not provided the context alone is not sufficiently informative. Another way to interpret this would be to see that the external context and and the dialogue context provide complementary information for better predicting the next utterance. Further, the experiments illustrated that the model, when provided with an \emph{informative} source of knowledge (the one that has higher variance), will let the model converge faster. One possible hypothesis is that $\bm{ec^i}$, which has high variance and is provided as input in every step of decoding, is learned before the RNN parameters converge. The information in $\bm{ec^i}$ is relevant to the context and subsequently helps in training the decoder faster.

\section{Conclusion}
\label{conclusion}

The proposed model, Extended Encoder Decoder, offers a framework to incorporate unstructured external knowledge to generate dialogue utterances. The experiments helped in understanding the need for external knowledge sources for improving learning time, and helped characterize the value of external knowledge sources. The experiments showed that external knowledge improved the learning time of the models. In future work, we aim to add more experiments with dialogue tasks that require understanding a supplementary source of knowledge to solve the task. Also, we plan to look at specialized tasks that naturally evaluate the influence of external knowledge, to help the model to generate diverse responses.
\section*{Acknowledgements}
The authors gratefully acknowledge
financial support for portions of this work by
the Samsung Advanced Institute of Technology
(SAIT) and the Natural Sciences and Engineering
Research Council of Canada (NSERC).
\bibliographystyle{acl_natbib_nourl}