text:
\section{$\protect\bigskip $Introduction}

Locally \ $C^{\ast }$-algebras are generalizations of $C^{\ast }$-algebras,
on which the topology instead of being given by a single $C^{\ast }$-norm 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. Sch
\"{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 a 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 $a=c^{\ast }c+d$ with $p_{\lambda }\left( d\right) =0$.
Thus, the notion of (local) completely positive maps appeared naturally
while studying linear maps between locally $C^{\ast }$-algebras. The
structure of strictly continuous completely positive maps between locally
C^{\ast }$-algebras is described in \cite{J}. The same is done for strongly
bounded completely positive maps of order zero \cite{MJ2}. Dosiev \cite{D1}
proved a Stinespring type theorem for unbounded local completely positive
and local completely contractive maps on unital locally $C^{\ast }
-algebras. A Radon-Nikodym type theorem for such maps was proved by Bhat,
Ghatak and Kumar \cite{BGK}. In \cite{MJ1}, we obtained a structure theorem
for unbounded local completely positive maps of local order zero.

Bhat and Osaka \cite{BO} proved some factorization properties for bounded
positive maps on $C^{\ast }$-algebras. In this paper, we extend the results
of \cite{BO} to unbounded local positive maps on 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 completely positive and local completely
contractive maps needed for understanding the main results of this paper. In
Section 3, we show that a linear map between locally $C^{\ast }$-algebras is
local $CP$ (completely positive) if and only if it is continuous and
completely positive (Proposition \ref{5}). Therefore, the local $CP$-map
\textit{\ }on unital locally $C^{\ast }$-algebras are exactly strictly
continuous completely positive maps on unital locally $C^{\ast }$-algebras
\cite[Remark 4.4]{J} and the structure theorem \cite[Theorem 4.6]{J} is
valid for local\textit{\ }$CP$-maps\textit{\ }on unital locally $C^{\ast }
-algebras. As in the case of bounded completely positive maps on $C^{\ast }
-algebras, we show that two unbounded local $CCP$ (local completely
contractive and local completely positive)-maps on unital locally $C^{\ast }
-algebras determine an unbounded local $CCP$-map on the minimal tensor
product and a minimal Stinespring dilation for the tensor product map can be
obtained in terms of the minimal Stinespring dilations for each map\
(Proposition \ref{3}). In section 4, we show that if $\mathcal{D}_{\mathcal{
}}$ is a Fr\'{e}chet quantized domain, then an unbounded local positive map
\phi $ on the minimal tensor product of the locally $C^{\ast }$-algebras
\mathcal{A}$ and $C^{\ast }(\mathcal{D}_{\mathcal{E}}),$ that is dominated
by $\varphi \otimes $id$_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}$ factorizes
as $\psi \otimes $id$_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}$, where $\psi $
is an unbounded local positive map dominated by $\varphi \ $(Theorem \ref{6
). As an application of this result, we show that given a local positive map
$\varphi :$ $\mathcal{A}\rightarrow $ $\mathcal{B}$, the local positive map
\varphi \otimes $id$_{M_{n}\left( \mathbb{C}\right) }$ is local decomposable
for $n\geq 2$ if and only if $\varphi $ is a local\ $CP$-map\ (Theorem \re
{7}). Also, we show that given unbounded\textbf{\ }local\textbf{\ }$CCP
-maps $\phi :\mathcal{A\otimes B}\rightarrow C^{\ast }(\mathcal{D}_{\mathcal
E\otimes F}}),$ $\varphi :\mathcal{A}\rightarrow C^{\ast }(\mathcal{D}_
\mathcal{E}})$ and $\psi :\mathcal{B}\rightarrow C^{\ast }(\mathcal{D}_
\mathcal{F}})$, if $\phi $ is dominated by $\varphi \otimes \psi $ and
\varphi $ is pure, then $\phi $ factorizes as $\varphi \otimes \widetilde
\psi }$, where $\widetilde{\psi }$ is an unbounded \ local $CCP$-map
dominated by $\psi $ (Theorem \ref{8}).

\section{Preliminaries}

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{bounded} if $\sup \{p_{\lambda
}\left( a\right) ;\lambda \in \Lambda \}<\infty $. The subset $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
$\mathcal{A}.$

Let us observe 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 $\pi _{\lambda _{2}\lambda _{1}}^{\mathcal{A}}:$ $\mathcal{A
_{\lambda _{2}}\rightarrow \mathcal{A}_{\lambda _{1}}$ defined by $\pi
_{\lambda _{2}\lambda _{1}}^{\mathcal{A}}\left( a+\mathcal{I}_{\lambda
_{2}}\right) =a+\mathcal{I}_{\lambda _{1}}$ for $a\in \mathcal{A}$. 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 $\mathcal{A}$ via the map $a\mapsto
\left( \pi _{\lambda }^{\mathcal{A}}\left( a\right) \right) _{\lambda \in
\Lambda }.$

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 \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 $. In the first case, we call
a $ as $\lambda $-self-adjoint, and in the second case, we call $a$ as
\lambda $-positive and write $a\geq _{\lambda }0$. 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 $\mathcal{
}_{\lambda }$ 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 $\mathcal{A}_{\lambda }.$

Let $\mathcal{A}$ and $\mathcal{B}$ be two locally $C^{\ast }$-algebras with
the topology defined by the family 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 size $n$ 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 a 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 $\left[ a_{ij}\right] _{i,j=1}^{n}\in M_{n}(\mathcal{A})$ .

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 }$CC$\textit{-map
)\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 }$CP$\textit{-map}
\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}

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

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

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

We say that a quantized domain $\mathcal{F}=\{\mathcal{K}_{\lambda };\lambda
\in \Lambda \}$ \ of $\mathcal{H}$ with its union space $\mathcal{D}_
\mathcal{F}}$ and $\mathcal{K}$ $=\overline{\tbigcup\limits_{\lambda \in
\Lambda }\mathcal{K}_{\lambda }}$ is a \textit{quantized subdomian} of
\mathcal{E}$, if $\mathcal{K}_{\lambda }\subseteq \mathcal{H}_{\lambda }$
for all $\lambda \in \Lambda $. In this case, we write $\mathcal{F}\subseteq
$ $\mathcal{E}$.

Let $\mathcal{E}^{i}=\{\mathcal{H}_{\lambda }^{i};\lambda \in \Lambda \}$ be
a quantized domain in a Hilbert space $\mathcal{H}^{i}$ for $i=1,2.$Given a
linear operator $V:\mathcal{D}_{\mathcal{E}^{1}}\rightarrow \mathcal{H}^{2}
, we write $V(\mathcal{E}^{1})\subseteq \mathcal{E}^{2}$ if $V(\mathcal{H
_{\lambda }^{1})\subseteq \mathcal{H}_{\lambda }^{2}$\ for all $\lambda \in
\Lambda $.

Let $\mathcal{H}$ and $\mathcal{K}$ be Hilbert spaces. A linear operator $T$
$:$ dom$(T)$ $\subseteq $ $\mathcal{H}$ $\rightarrow $ $\mathcal{K}$ 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 $ $\mathcal{K}\rightarrow \mathcal{H},$ wher
\begin{equation*}
\text{dom}(T^{\bigstar })=\{\xi \in \mathcal{K};\eta \rightarrow
\left\langle T\eta ,\xi \right\rangle _{\mathcal{K}}\ \text{is continuous
for every }\eta \in \text{dom}(T)\}
\end{equation*}
satisfying $\left\langle T\eta ,\xi \right\rangle _{\mathcal{K
}=\left\langle \eta ,T^{\bigstar }\xi \right\rangle _{\mathcal{H}}$ for all
\xi \in $dom$(T^{\bigstar })$ and $\eta \in $dom$(T).$

Let $\mathcal{E}=\{\mathcal{H}_{\lambda };\lambda \in \Lambda \}$ 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_{\lambda }=P_{\lambda }TP_{\lambda }\in B(\mathcal{H})\text{ for all
\lambda \in \Lambda \}
\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 $T(\mathcal{H}_{\lambda })\subseteq \mathcal{H}_{\mathcal{\lambda }}$
and $\left. T\right\vert _{\mathcal{H}_{\lambda }}\in B(\mathcal{H}_{\lambda
})$ for all $\lambda \in \Lambda $, 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_{\lambda }T\subseteq TP_{\lambda }\text{ for all }\lambda \in \Lambda
\}.
\end{equation*}
If $T\in C(\mathcal{D}_{\mathcal{E}})$, then $T\in C^{\ast }(\mathcal{D}_
\mathcal{E}})$ if and only if $T(\mathcal{H}_{\lambda }^{\bot }\cap \mathcal
D}_{\mathcal{E}})\subseteq \mathcal{H}_{\lambda }^{\bot }\cap \mathcal{D}_
\mathcal{E}}$ for all $\lambda \in \Lambda .\ $

If $T\in C^{\ast }(\mathcal{D}_{\mathcal{E}})$, then $\mathcal{D}_{\mathcal{
}}$ $\subseteq $ dom$(T^{\bigstar })$. Moreover, $T^{\bigstar }(\mathcal{H
_{\lambda })\subseteq \mathcal{H}_{\lambda }$ for all $\lambda \in \Lambda
. Now, let $T^{\ast }=\left. T^{\bigstar }\right\vert _{\mathcal{D}_
\mathcal{E}}}$. 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 $\lambda \in \Lambda $, the map $\left\Vert \cdot \right\Vert
_{\lambda }:C^{\ast }(\mathcal{D}_{\mathcal{E}})\rightarrow \lbrack 0,\infty
)$,
\begin{equation*}
\left\Vert T\right\Vert _{\lambda }=\left\Vert \left. T\right\vert _
\mathcal{H}_{\lambda }}\right\Vert =\sup \{\left\Vert T\left( \xi \right)
\right\Vert ;\xi \in \mathcal{H}_{\lambda },\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 $\{\left\Vert \cdot \right\Vert _{\lambda }\}_{\lambda \in
\Lambda }$ and $b(C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ is identified with
the $C^{\ast }$-algebra $\{T\in B\left( \mathcal{H}\right) ;P_{\lambda
}T=TP_{\lambda }\ $for all $\lambda \in \Lambda \}$\ via\ the map $T\mapsto
\widetilde{T}$, where $\widetilde{T}$ is the extension of $T$ to $\mathcal{H}
$ (see the proof of \cite[Lemma 3.1]{D})\textbf{.}

If $\mathcal{E}=\{\mathcal{H}_{\lambda };\lambda \in \Lambda \}$ is a
quantized domain in a Hilbert space $\mathcal{H}$, then $\mathcal{D}_
\mathcal{E}}$ can be regarded as a strict inductive limit of the direct
family of Hilbert spaces $\mathcal{E}=\{\mathcal{H}_{\lambda };\lambda \in
\Lambda \},$ $\mathcal{D}_{\mathcal{E}}=\lim\limits_{\rightarrow }\mathcal{H
_{\lambda }$, 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 is 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.

\section{Local completely positive maps}

Let $\mathcal{A}$ and $\mathcal{B}$ be two locally $C^{\ast }$-algebras with
the topology defined by the family of $C^{\ast }$-seminorms $\left\{
p_{\lambda }\right\} _{\lambda \in \Lambda }$ and $\left\{ q_{\delta
}\right\} _{\delta \in \Delta }$, respectively.

\begin{proposition}
\label{4} Let $\varphi :\mathcal{A}\rightarrow $ $\mathcal{B}$ be a linear
map. If $\varphi $ is local positive, then $\varphi $ is continuous.
\end{proposition}

\begin{proof}
Since $\varphi $ is local positive, 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 $a=_{\lambda }0$. Define the map $\varphi
_{\delta }^{+}:\left( \mathcal{A}_{\lambda }\right) _{+}\rightarrow \mathcal
B}_{\delta }$ by $\varphi _{\delta }^{+}\left( \pi _{\lambda }^{\mathcal{A
}\left( a\right) \right) =\pi _{\delta }^{\mathcal{B}}\left( \varphi \left(
a\right) \right) $, and extend it to a linear map $\varphi _{\delta }
\mathcal{A}_{\lambda }\rightarrow \mathcal{B}_{\delta }$. This map is
positive, and so continuous. Therefore, there is $C_{\delta }>0\ $such tha
\begin{equation*}
q_{\delta }\left( \varphi \left( a\right) \right) =\left\Vert \pi _{\delta
}^{\mathcal{B}}\left( \varphi \left( a\right) \right) \right\Vert _{\mathcal
B}_{\delta }}=\left\Vert \varphi _{\delta }\left( \pi _{\lambda }^{\mathcal{
}}\left( a\right) \right) \right\Vert \leq C_{\delta }\left\Vert \pi
_{\lambda }^{\mathcal{A}}\left( a\right) \right\Vert _{\mathcal{A}_{\lambda
}}=C_{\delta }p_{\lambda }\left( a\right)
\end{equation*
for all $a\in \mathcal{A}$.
\end{proof}

Proposition \ref{4} it is a particular case of \cite[Lemma 4.4]{D1}.

\begin{remark}
\label{1}Let $\varphi :\mathcal{A}\rightarrow $ $\mathcal{B}$ be a local
positive linear map. Then, for each $\delta \in \Delta $, there exist
\lambda \in \Lambda $ and a positive map $\varphi _{\delta }:\mathcal{A
_{\lambda }\rightarrow \mathcal{B}_{\delta }$ such that $\varphi _{\delta
}\left( \pi _{\lambda }^{\mathcal{A}}\left( a\right) \right) =\pi _{\delta
}^{\mathcal{B}}\left( \varphi \left( a\right) \right) $ for all $a\in
\mathcal{A}$.
\end{remark}

\begin{proposition}
\label{5}Let $\mathcal{A}$ and $\mathcal{B}$\ be two locally $C^{\ast }
-algebras and $\varphi :\mathcal{A}\rightarrow $ $\mathcal{B}$ be a linear
map. Then $\varphi $ is local completely positive if and only if $\varphi $
is continuous and completely positive.
\end{proposition}

\begin{proof}
If $\varphi $ is local completely positive, then, by Proposition \ref{4},
for each $n\in \mathbb{N}$, the map $\varphi ^{\left( n\right) }$ is
continuous and by \cite[Proposition 2.1]{MJ1}, it is positive. Therefore,
\varphi $ is continuous and completely positive.

Conversely, suppose that $\varphi $ is continuous and completely positive.
Since $\varphi $ is continuous, for each $\delta \in \Delta $, there exist
\lambda \in \Lambda $ and a positive map $\varphi _{\delta }:\mathcal{A
_{\lambda }\rightarrow \mathcal{B}_{\delta }$ such that $\varphi _{\delta
}\left( \pi _{\lambda }^{\mathcal{A}}\left( a\right) \right) =\pi _{\delta
}^{\mathcal{B}}\left( \varphi \left( a\right) \right) $ for all $a\in
\mathcal{A}$. Let $\left[ a_{ij}\right] _{i,j=1}^{n}\in M_{n}\left( \mathcal
A}\right) $ such that $\left[ a_{ij}\right] _{i,j=1}^{n}\geq _{\lambda }0$.
This means that there exist $\left[ b_{ij}\right] _{i,j=1}^{n},$ $\left[
c_{ij}\right] _{i,j=1}^{n}\in M_{n}\left( \mathcal{A}\right) $ such that
\left[ a_{ij}\right] _{i,j=1}^{n}=\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}\ $and $p_{\lambda }^{n}\left( \left[ c_{ij}\right]
_{i,j=1}^{n}\right) =0$. The
\begin{eqnarray*}
\pi _{\delta }^{M_{n}(\mathcal{B)}}\left( \varphi ^{\left( n\right) }\left(
\left[ c_{ij}\right] _{i,j=1}^{n}\right) \right) &=&\left[ \pi _{\delta }^
\mathcal{B}}\left( \varphi \left( c_{ij}\right) \right) \right] _{i,j=1}^{n}
\left[ \varphi _{\delta }\left( \pi _{\lambda }^{\mathcal{A}}\left(
c_{ij}\right) \right) \right] _{i,j=1}^{n} \\
&=&\varphi _{\delta }^{\left( n\right) }\left( \pi _{\lambda }^{M_{n}\left(
\mathcal{A}\right) }\left( \left[ c_{ij}\right] _{i,j=1}^{n}\right) \right)
\left[ 0\right] _{i,j=1}^{n}
\end{eqnarray*
and, since $\varphi \ $is completely positive
\begin{equation*}
\pi _{\delta }^{M_{n}(\mathcal{B)}}\left( \varphi ^{\left( n\right) }\left(
\left[ a_{ij}\right] _{i,j}^{n}\right) \right) =\pi _{\delta }^{M_{n}
\mathcal{B)}}\left( \varphi ^{\left( n\right) }\left( \left( \left[ b_{ij
\right] _{i,j=1}^{n}\right) ^{\ast }\left[ b_{ij}\right] _{i,j=1}^{n}\right)
\right) \geq 0.
\end{equation*
If $\left[ a_{ij}\right] _{i,j}^{n}=_{\lambda }\left[ 0\right] _{i,j=1}^{n}$
$\ $then $p_{\lambda }^{n}\left( \left[ a_{ij}\right] _{i,j}^{n}\right) =0\
an
\begin{equation*}
\pi _{\delta }^{M_{n}(\mathcal{B)}}\left( \varphi ^{\left( n\right) }\left(
\left[ a_{ij}\right] _{i,j=1}^{n}\right) \right) =\varphi _{\delta }^{\left(
n\right) }\left( \pi _{\lambda }^{M_{n}\left( \mathcal{A}\right) }\left(
\left[ a_{ij}\right] _{i,j=1}^{n}\right) \right) =\left[ 0\right]
_{i,j=1}^{n}.
\end{equation*
Therefore,
\begin{equation*}
\varphi ^{\left( n\right) }\left( \left[ a_{ij}\right] _{i,j}^{n}\right)
\geq _{\delta }0
\end{equation*
for all $\left[ a_{ij}\right] _{i,j=1}^{n}\in M_{n}\left( \mathcal{A}\right)
$ such that $\left[ a_{ij}\right] _{i,j=1}^{n}\geq _{\lambda }0,\ $and
\varphi ^{\left( n\right) }\left( \left[ a_{ij}\right] _{i,j}^{n}\right)
=_{\delta }\left[ 0\right] _{i,j=1}^{n}$ for all $\left[ a_{ij}\right]
_{i,j=1}^{n}\in M_{n}\left( \mathcal{A}\right) $ such that $\left[ a_{ij
\right] _{i,j=1}^{n}=_{\lambda }\left[ 0\right] _{i,j=1}^{n}$ and for all
n, $ and so $\varphi $ is local completely positive.
\end{proof}

\begin{corollary}
\label{localpositive}Let $\varphi :\mathcal{A}\rightarrow $ $\mathcal{B}$ be
a linear map. Then $\varphi $ is local positive if and only if $\varphi $ is
continuous and positive.
\end{corollary}

\begin{remark}
\label{London} Let $\varphi :\mathcal{A}\rightarrow $ $\mathcal{B}$ be a
local completely positive map. If the locally $C^{\ast }$-algebra $\mathcal{
}$ is unital, then $\varphi $ is a strictly continuous completely positive\
map \cite[Remark 4.4]{J}. In particular, the structure theorem \cite[Theorem
4.6]{J} is valid for local completely positive maps on unital locally
C^{\ast }$-algebras.
\end{remark}

Let $\mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal
E}}))$ denote the set of all maps $\varphi :\mathcal{A}\rightarrow C^{\ast }
\mathcal{D}_{\mathcal{E}})$ which are local completely positive and local
completely contractive. 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 completely contractive and completely positive 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) \ $for all
a\in b(\mathcal{A}).$

The following result is a version of the Stinespring theorem for unbounded
local completely positive and local completely contractive maps.

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

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 associated to $\varphi $. Moreover, the minimal Stinespring
dilation associated to $\varphi $ is unique up to unitary equivalence in the
following sense, if $(\ \pi _{\varphi },V_{\varphi },$\ $\{\mathcal{H
^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\})$
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 associated to $\varphi $, then there is 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 $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi },\mathcal{E
^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ is a minimal
Stinespring dilation associated to $\varphi \in \mathcal{CPCC}_{\text{loc}}
\mathcal{A},C^{\ast }(\mathcal{D}_{\mathcal{E}}))$, 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 associated with $\left. \varphi
\right\vert _{b(\mathcal{A})}.$

Let $\mathcal{E}=\{\mathcal{H}_{\iota };\iota \in \Upsilon \}$ and $\mathcal
F}=\{\mathcal{K}_{\gamma };\gamma \in \Gamma \}\ $be quantized domains in
Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$, respectively. Then
\begin{equation*}
\mathcal{E}\otimes \mathcal{F}=\left\{ \mathcal{H}_{\iota }\otimes \mathcal{
}_{\gamma };\left( \iota ,\gamma \right) \in \Upsilon \times \Gamma \right\}
\end{equation*
is a quantized domain in the Hilbert space $\mathcal{H}\otimes \mathcal{K}$,
with the union space $\mathcal{D}_{\mathcal{E}\otimes \mathcal{F
}=\tbigcup\limits_{\left( \iota ,\gamma \right) \in \Upsilon \times \Gamma
\mathcal{H}_{\iota }\otimes \mathcal{K}_{\gamma }.$\ If $\mathcal{E}=\
\mathcal{H}\}$, then $\mathcal{H}\otimes \mathcal{F}=\left\{ \mathcal{H
\otimes \mathcal{K}_{\gamma };\gamma \in \Gamma \right\} $ is a quantized
domain in the Hilbert space $\mathcal{H}\otimes \mathcal{K}$ with the union
space $\mathcal{D}_{\mathcal{H}\otimes \mathcal{F}}=\tbigcup\limits_{\gamma
\in \Gamma }\mathcal{H}\otimes \mathcal{K}_{\gamma }$.

The map $\Phi :C^{\ast }(\mathcal{D}_{\mathcal{E}})\otimes _{\text{alg
}C^{\ast }(\mathcal{D}_{\mathcal{F}})\rightarrow C^{\ast }(\mathcal{D}_
\mathcal{E}\otimes \mathcal{F}})$ given by
\begin{equation*}
\Phi \left( T\otimes S\right) \left( \xi \otimes \eta \right) =T\xi \otimes
S\eta ,T\in C^{\ast }(\mathcal{D}_{\mathcal{E}}),S\in C^{\ast }(\mathcal{D}_
\mathcal{F}}),\xi \in \mathcal{D}_{\mathcal{E}},\eta \in \mathcal{D}_
\mathcal{F}}
\end{equation*
identifies $C^{\ast }(\mathcal{D}_{\mathcal{E}})\otimes _{\text{alg}}C^{\ast
}(\mathcal{D}_{\mathcal{F}})\ $with a $\ast $-subalgebra of $C^{\ast }
\mathcal{D}_{\mathcal{E}\otimes \mathcal{F}})$. The minimal tensor product
of the locally $C^{\ast }$-algebras $C^{\ast }(\mathcal{D}_{\mathcal{E}})$
and $C^{\ast }(\mathcal{D}_{\mathcal{F}})$ is the locally $C^{\ast }
-algebra $C^{\ast }(\mathcal{D}_{\mathcal{E}})\otimes C^{\ast }(\mathcal{D}_
\mathcal{F}})$ obtained by the completion of the $\ast $-subalgebra $C^{\ast
}(\mathcal{D}_{\mathcal{E}})\otimes _{\text{alg}}C^{\ast }(\mathcal{D}_
\mathcal{F}})$ in $C^{\ast }(\mathcal{D}_{\mathcal{E}\otimes \mathcal{F}})$
(see for example \cite{G}).

Let $\mathcal{A}$ and $\mathcal{B}$ be two locally $C^{\ast }$-algebras.
Recall that $\mathcal{A}$ and $\mathcal{B}$ can be identified with a locally
$C^{\ast }$-subalgebra in $C^{\ast }(\mathcal{D}_{\mathcal{E}})$ and
C^{\ast }(\mathcal{D}_{\mathcal{F}})$, respectively for some quantized
domains $\mathcal{D}_{\mathcal{E}}$ and $\mathcal{D}_{\mathcal{F}}$. The
minimal or spatial tensor product of the locally $C^{\ast }$-algebras
\mathcal{A}$ and $\mathcal{B}$ is the locally $C^{\ast }$-algebra $\mathcal{
}\otimes \mathcal{B}$ obtained by the completion of the $\ast $-subalgebra
\mathcal{A}\otimes _{\text{alg}}\mathcal{B}$\ in $C^{\ast }(\mathcal{D}_
\mathcal{E}\otimes \mathcal{F}})$. In fact, $\mathcal{A}\otimes \mathcal{B}$
can be identified with the projective limit of the projective system of
C^{\ast }$-algebras $\{\mathcal{A}_{\lambda }\otimes \mathcal{B}_{\delta
};\pi _{\lambda _{1}\lambda _{2}}^{\mathcal{A}}\otimes \pi _{\delta
_{1}\delta _{2}}^{\mathcal{B}},\lambda _{1}\geq \lambda _{2},\delta _{1}\geq
\delta _{2}\}_{\left( \lambda ,\delta \right) \in \Lambda \times \Delta }$
(see \cite{Ph,F}).

If $\mathcal{A}$ is a $C^{\ast }$-algebra acting nondegenerately on a
Hilbert space $\mathcal{H}$ and $\mathcal{B}$ is a locally $C^{\ast }
-algebra that is identified with a locally $C^{\ast }$-subalgebra in
C^{\ast }(\mathcal{D}_{\mathcal{F}})$, then the minimal tensor product
\mathcal{A}\otimes \mathcal{B}$ of $\mathcal{A}$ and $\mathcal{B}$ is the
completion of the $\ast $-subalgebra $\mathcal{A}\otimes _{\text{alg}
\mathcal{B}$\ in $C^{\ast }(\mathcal{D}_{\mathcal{H}\otimes \mathcal{F}})$.

Let $\mathcal{A}$ and $\mathcal{B}$ be two locally $C^{\ast }$-algebras,
\pi _{1}:$ $\mathcal{A\rightarrow }C^{\ast }(\mathcal{D}_{\mathcal{E}})$ and
$\pi _{2}:$ $\mathcal{B\rightarrow }C^{\ast }(\mathcal{D}_{\mathcal{F}})$ be
two local contractive $\ast $-morphisms. By the functorial property of the
minimal tensor product of $C^{\ast }$-algebras and taking into account the
above discussion we conclude that there is a unique local $\ast $-morphism
\pi _{1}\otimes \pi _{2}:\mathcal{A\otimes B\rightarrow }C^{\ast }(\mathcal{
}_{\mathcal{E}\otimes \mathcal{F}})$ such that
\begin{equation*}
\left( \pi _{1}\otimes \pi _{2}\right) \left( a\otimes b\right) =\pi
_{1}\left( a\right) \otimes \pi _{2}\left( b\right)
\end{equation*
where $\left( \pi _{1}\left( a\right) \otimes \pi _{2}\left( b\right)
\right) \left( \xi \otimes \eta \right) =\pi _{1}\left( a\right) \xi \otimes
\pi _{2}\left( b\right) \eta $ for all $\xi \in \mathcal{D}_{\mathcal{E
},\eta \in \mathcal{D}_{\mathcal{F}}$.

\begin{proposition}
\label{3} Let $\mathcal{A}$ and $\mathcal{B}$ be two unital locally $C^{\ast
}$-algebras and let $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A
,C^{\ast }(\mathcal{D}_{\mathcal{E}}))$ and $\psi \in \mathcal{CPCC}_{\text
loc}}(\mathcal{B},C^{\ast }(\mathcal{D}_{\mathcal{F}}))$. Then there is a
unique map $\varphi \otimes \psi \in $ $\mathcal{CPCC}_{\text{loc}}(\mathcal
A\otimes B},C^{\ast }(\mathcal{D}_{\mathcal{E}\otimes \mathcal{F}}))$ such
that
\begin{equation*}
\left( \varphi \otimes \psi \right) \left( a\otimes b\right) =\varphi \left(
a\right) \otimes \psi \left( b\right) \ \text{for all }a\in \mathcal{A\
\text{and }b\in \mathcal{B}.
\end{equation*
Moreover, if $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi }
\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ is a
minimal Stinespring dilation associated to $\varphi $ and $\left( \pi _{\psi
},V_{\psi },\{\mathcal{H}^{\psi },\mathcal{E}^{\psi },\mathcal{D}_{\mathcal{
}^{\psi }}\}\right) $ is a minimal Stinespring dilation associated to $\psi
, then $\left( \pi _{\varphi }\otimes \pi _{\psi },V_{\varphi }\otimes
V_{\psi },\{\mathcal{H}^{\varphi }\otimes \mathcal{H}^{\psi },\mathcal{E
^{\varphi }\otimes \mathcal{E}^{\psi },\mathcal{D}_{\mathcal{E}^{\varphi
}\otimes \mathcal{E}^{\psi }}\}\right) \ $is a minimal Stinespring dilation
associated to $\varphi \otimes \psi .$
\end{proposition}

\begin{proof}
Since $\pi _{\varphi }:\mathcal{A\rightarrow }C^{\ast }(\mathcal{D}_
\mathcal{E}^{\varphi }})$ and $\pi _{\psi }:\mathcal{B\rightarrow }C^{\ast }
\mathcal{D}_{\mathcal{E}^{\psi }})$ are local contractive $\ast $-morphisms,
there is a local contractive $\ast $-morphism $\pi _{\varphi }\otimes \pi
_{\psi }:\mathcal{A\otimes B\rightarrow }$ $C^{\ast }(\mathcal{D}_{\mathcal{
}^{\varphi }\otimes \mathcal{E}^{\psi }})$ such that
\begin{equation*}
\left( \pi _{\varphi }\otimes \pi _{\psi }\right) \left( a\otimes b\right)
=\pi _{\varphi }\left( a\right) \otimes \pi _{\psi }\left( b\right) ,
\end{equation*
where $\left( \pi _{\varphi }\left( a\right) \otimes \pi _{\psi }\left(
b\right) \right) \left( \xi \otimes \eta \right) =\pi _{\varphi }\left(
a\right) \xi \otimes \pi _{\psi }\left( b\right) \eta $ for all $\xi \in
\mathcal{D}_{\mathcal{E}^{\varphi }},\eta \in \mathcal{D}_{\mathcal{E}^{\psi
}}$. Since $V_{\varphi }:\mathcal{H}\rightarrow \mathcal{H}^{\varphi }$ and
V_{\psi }:\mathcal{K}\rightarrow \mathcal{H}^{\psi }$ are contractions, we
get a contraction $V_{\varphi }\otimes V_{\psi }:\mathcal{H}\otimes \mathcal
K}\rightarrow \mathcal{H}^{\varphi }\otimes \mathcal{H}^{\psi }$.$\ $Since
V_{\varphi }\left( \mathcal{E}\right) \subseteq \mathcal{E}^{\varphi }$ and
V_{\psi }\left( \mathcal{F}\right) \subseteq \mathcal{E}^{\psi },$ $\left(
V_{\varphi }\otimes V_{\psi }\right) \left( \mathcal{H}_{\iota }\otimes
\mathcal{K}_{\mathcal{\gamma }}\right) \subseteq \mathcal{H}_{\iota
}^{\varphi }\otimes \mathcal{H}_{\mathcal{\gamma }}^{\varphi }\ $for all
\left( \iota ,\mathcal{\gamma }\right) \in \Upsilon \times \Gamma $, thus
\begin{equation*}
\left( V_{\varphi }\otimes V_{\psi }\right) \left( \mathcal{E}\otimes
\mathcal{F}\right) \subseteq \mathcal{E}^{\varphi }\otimes \mathcal{E}^{\psi
}.
\end{equation*
Therefore, we may consider the map $\varphi \otimes \psi :$ $\mathcal
A\otimes B\rightarrow }C^{\ast }(\mathcal{D}_{\mathcal{E}\otimes \mathcal{F
})$ given b
\begin{equation*}
\left( \varphi \otimes \psi \right) \left( c\right) =\left. \left(
V_{\varphi }\otimes V_{\psi }\right) ^{\ast }\left( \pi _{\varphi }\otimes
\pi _{\psi }\right) \left( c\right) \left( V_{\varphi }\otimes V_{\psi
}\right) \right\vert _{\mathcal{D}_{\mathcal{E}^{\varphi }\otimes \mathcal{E
^{\psi }}}\text{.}
\end{equation*
Since $\pi _{\varphi }\otimes \pi _{\psi }$ is a local contractive $\ast
-morphism and $V_{\varphi }\otimes V_{\psi }$ is a contraction, $\varphi
\otimes \psi $ is a local completely positive and local completely
contractive map from $\mathcal{A\otimes B}$ to $C^{\ast }(\mathcal{D}_
\mathcal{E}\otimes \mathcal{F}})$. Clearly, $\left( \varphi \otimes \psi
\right) \left( a\otimes b\right) =\varphi \left( a\right) \otimes \psi
\left( b\right) \ $for all $a\in \mathcal{A\ }$and $b\in \mathcal{B}$.

To show that $\left( \pi _{\varphi }\otimes \pi _{\psi },V_{\varphi }\otimes
V_{\psi },\{\mathcal{H}^{\varphi }\otimes \mathcal{H}^{\psi },\mathcal{E
^{\varphi }\otimes \mathcal{E}^{\psi },\mathcal{D}_{\mathcal{E}^{\varphi
}\otimes \mathcal{E}^{\psi }}\}\right) \ $is a minimal Stinespring dilation
associated with $\varphi \otimes \psi $ it remains to show that $\mathcal{H
_{\iota }^{\varphi }\otimes \mathcal{H}_{\gamma }^{\psi }=\left[ \left( \pi
_{\varphi }\otimes \pi _{\psi }\right) \left( \mathcal{A\otimes B}\right)
\left( V_{\varphi }\otimes V_{\psi }\right) \left( \mathcal{H}_{\iota
}\otimes \mathcal{K}_{\gamma }\right) \right] $ for all $\left( \iota
\mathcal{\gamma }\right) \in \Upsilon \times \Gamma $.

Let $\left( \iota ,\mathcal{\gamma }\right) \in \Upsilon \times \Gamma $.
Then
\begin{eqnarray*}
\left[ \left( \pi _{\varphi }\otimes \pi _{\psi }\right) \left( \mathcal
A\otimes B}\right) \left( V_{\varphi }\otimes V_{\psi }\right) \left(
\mathcal{H}_{\iota }\otimes \mathcal{K}_{\gamma }\right) \right] &=&\left[
\pi _{\varphi }\left( \mathcal{A}\right) V_{\varphi }\left( \mathcal{H
_{\iota }\right) \otimes \pi _{\psi }\left( \mathcal{B}\right) V_{\psi
}\left( \mathcal{K}_{\gamma }\right) \right] \\
&=&\mathcal{H}_{\iota }^{\varphi }\otimes \mathcal{H}_{\gamma }^{\psi },
\end{eqnarray*
as required.

To show the uniqueness of the map $\varphi \otimes \psi $, let $\phi \in
\mathcal{CPCC}_{\text{loc}}(\mathcal{A\otimes B},C^{\ast }(\mathcal{D}_
\mathcal{E}\otimes \mathcal{F}}))$ such that $\phi \left( a\otimes b\right)
=\varphi \left( a\right) \otimes \psi \left( b\right) \ $for all $a\in
\mathcal{A\ }$and $b\in \mathcal{B}$, $\phi \neq \varphi \otimes \psi $.
Since $\varphi \otimes \psi $ and $\phi $ are continuous and $\mathcal
A\otimes }_{\text{alg}}\mathcal{B}$ is dense, from $\phi \left( a\otimes
b\right) =\varphi \left( a\right) \otimes \psi \left( b\right) =\left(
\varphi \otimes \psi \right) \left( a\otimes b\right) $ for all $a\in
\mathcal{A\ }$and $b\in \mathcal{B}$ it follows that $\phi =\varphi \otimes
\psi $, a contradiction, and the uniqueness of the map $\varphi \otimes \psi
$ is proved.
\end{proof}

\section{Main results}

\ Suppose that $\{\mathcal{H};\mathcal{E};\mathcal{D}_{\mathcal{E}}\}$ is a
Fr\'{e}chet quantized domain in a Hilbert space $\mathcal{H}\ $(that is,
\mathcal{E=\{H}_{n}\mathcal{\}}_{n\in \mathbb{N}}$). For each $n\in \mathbb{
}^{\ast },$ let $\mathcal{H}_{n}^{c}$ be the orthogonal complement of the
closed subspace\ $\mathcal{H}_{n-1}$ in $\mathcal{H}_{n}$, and put $\mathcal
H}_{0}^{c}=\mathcal{H}_{0}$. Then $\mathcal{H}_{n}=\tbigoplus\limits_{k\leq
n}\mathcal{H}_{k}^{c}$. For each $n\in \mathbb{N}$ and for each $\xi \in
\mathcal{H}_{k}^{c},k\leq n$, the rank one operator $\theta _{\xi ,\xi }
\mathcal{D}_{\mathcal{E}}\rightarrow \mathcal{D}_{\mathcal{E}},$ $\theta
_{\xi ,\xi }(\eta )=\xi \left\langle \xi ,\eta \right\rangle \ $is an
element in $C^{\ast }(\mathcal{D}_{\mathcal{E}})$. The closed two sided
\ast $-ideal of $C^{\ast }(\mathcal{D}_{\mathcal{E}})$ generated by these
rank one operators is denoted by $K(\mathcal{D}_{\mathcal{E}})$ and it is
called the locally $C^{\ast }$-algebra of all compact operators on $\mathcal
D}_{\mathcal{E}}$. Clearly, for each $n\in \mathbb{N},K(\mathcal{D}_
\mathcal{E}})_{n}$ $\subseteq K\left( \mathcal{H}_{n}\right) $, which it is
a closed two sided $\ast $-ideal of $C^{\ast }(\mathcal{D}_{\mathcal{E
})_{n}.$

Let $\mathcal{A}$ be a unital locally $C^{\ast }$-algebra, $\{\mathcal{H}
\mathcal{E};\mathcal{D}_{\mathcal{E}}\}$ be a Fr\'{e}chet quantized domain
in a Hilbert space $\mathcal{H}$ with $\mathcal{E=\{H}_{n}\mathcal{\}}_{n\in
\mathbb{N}}$, $a\in \mathcal{A}$ and $c\in \mathcal{A\otimes }C^{\ast }
\mathcal{D}_{\mathcal{E}})$. If $\xi \in \mathcal{H}_{n}^{c}$ and $a\otimes
\theta _{\xi ,\xi }\geq _{\left( \lambda ,n\right) }c$ $\geq _{\left(
\lambda ,n\right) }0$, then there is $b\in \mathcal{A}$ such that
c=_{\left( \lambda ,n\right) }b\otimes \theta _{\xi ,\xi }\ $. Indeed, from
a\otimes \theta _{\xi ,\xi }\geq _{\left( \lambda ,n\right) }c$ $\geq
_{\left( \lambda ,n\right) }0$ we deduce that
\begin{equation*}
\pi _{\lambda }^{\mathcal{A}}\left( a\right) \otimes \left. \theta _{\xi
,\xi }\right\vert _{\mathcal{H}_{n}}=\pi _{\left( \lambda ,n\right) }^
\mathcal{A\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E}})}\left( a\otimes
\theta _{\xi ,\xi }\right) \geq \pi _{\left( \lambda ,n\right) }^{\mathcal
A\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E}})}\left( c\right) \geq 0
\end{equation*
and by \cite[Lemma 2.1]{BO}, there is $b_{\lambda }\in \mathcal{A}_{\lambda
} $ such that $\pi _{\left( \lambda ,n\right) }^{\mathcal{A\otimes }C^{\ast
}(\mathcal{D}_{\mathcal{E}})}\left( c\right) =b_{\lambda }\otimes \left.
\theta _{\xi ,\xi }\right\vert _{\mathcal{H}_{n}}.$ Therefore, there is
b\in \mathcal{A}$ such that
\begin{equation*}
\pi _{\left( \lambda ,n\right) }^{\mathcal{A\otimes }C^{\ast }(\mathcal{D}_
\mathcal{E}})}\left( c\right) =b_{\lambda }\otimes \left. \theta _{\xi ,\xi
}\right\vert _{\mathcal{H}_{n}}=\pi _{\lambda }^{\mathcal{A}}\left( b\right)
\otimes \left. \theta _{\xi ,\xi }\right\vert _{\mathcal{H}_{n}}=\pi
_{\left( \lambda ,n\right) }^{\mathcal{A\otimes }C^{\ast }(\mathcal{D}_
\mathcal{E}})}\left( b\otimes \theta _{\xi ,\xi }\right) .
\end{equation*}

\begin{theorem}
\label{6} Let $\mathcal{A}$ and $\mathcal{B}$\ be two unital locally
C^{\ast }$-algebras, $\{\mathcal{H};\mathcal{E};\mathcal{D}_{\mathcal{E}}\}$
be a Fr\'{e}chet quantized domain in a Hilbert space $\mathcal{H}$ with
\mathcal{E=\{H}_{n}\mathcal{\}}_{n\in \mathbb{N}},$ $\varphi :$ $\mathcal{A}$
$\rightarrow $ $\mathcal{B\ }$and $\phi :\mathcal{A\otimes }C^{\ast }
\mathcal{D}_{\mathcal{E}})\rightarrow \mathcal{B\otimes }C^{\ast }(\mathcal{
}_{\mathcal{E}})$ be two linear maps. Then the following conditions are
equivalent:

\begin{enumerate}
\item $\phi $ and $\varphi \otimes $id$_{C^{\ast }(\mathcal{D}_{\mathcal{E
})}-\phi $ are local positive.

\item There exists a local positive map $\psi :\mathcal{A\rightarrow B\ }
such that $\varphi -\psi $ is local positive and $\phi =\psi \otimes $ id
_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}$.
\end{enumerate}
\end{theorem}

\begin{proof}
$(1)\Rightarrow (2)$ Since the maps $\phi $ and $\varphi \otimes $id
_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}-\phi $ are local positive, the map
\varphi \otimes $id$_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}$ is local
positive. Then, $\varphi $ is local positive, and by Corollary \re
{localpositive}
\begin{equation*}
0\leq \phi \left( a\otimes T\right) \leq \varphi \left( a\right) \otimes T
\end{equation*
for all $a\in \mathcal{A}\ $with $a\geq 0$ and for all $T\in C^{\ast }
\mathcal{D}_{\mathcal{E}})$ with $T\geq 0.$ Therefore, for each $\left(
\delta ,n\right) \in \Delta \times \mathbb{N},$ there are $\lambda _{0}\in
\Lambda $ and $C_{0}>0$ such tha
\begin{equation*}
\left\Vert \phi \left( a\otimes T\right) \right\Vert _{\left( \delta
,n\right) }\leq \left\Vert \varphi \left( a\right) \otimes T\right\Vert
_{\left( \delta ,n\right) }=\left\Vert \varphi \left( a\right) \right\Vert
_{\delta }\left\Vert T\right\Vert _{n}\leq C_{0}p_{\lambda _{0}}\left(
a\right) \left\Vert T\right\Vert _{n}
\end{equation*
for all $a\in \mathcal{A}\ $with $a\geq 0$ and for all $T\in C^{\ast }
\mathcal{D}_{\mathcal{E}})$ with $T\geq 0$. Therefore, there exist maps
\phi _{\left( \delta ,n\right) }:$ $\mathcal{A}_{\lambda _{0}}\mathcal
\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E}})_{n}\rightarrow \mathcal{B
_{\delta }\mathcal{\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E}})_{n}$ and
\varphi _{\delta }:\mathcal{A}_{\lambda _{0}}\rightarrow \mathcal{B}_{\delta
}$ such that
\begin{equation*}
\pi _{\left( \delta ,n\right) }^{\mathcal{B\otimes }C^{\ast }(\mathcal{D}_
\mathcal{E}})}\circ \phi =\phi _{\left( \delta ,n\right) }\circ \pi _{\left(
\lambda _{0},n\right) }^{\mathcal{A\otimes }C^{\ast }(\mathcal{D}_{\mathcal{
}})}\text{ and }\pi _{\delta }^{\mathcal{B}}\circ \varphi =\varphi _{\delta
}\circ \pi _{\lambda _{0}}^{\mathcal{A}}.
\end{equation*
Moreover, $\phi _{\left( \delta ,n\right) }$ and $\varphi _{\delta }\otimes
id$_{C^{\ast }(\mathcal{D}_{\mathcal{E}})_{n}}-\phi _{\left( \delta
,n\right) }$ are positive. Thus, by the proof of \cite[Theorem 2]{BO}, there
is a positive map $\widetilde{\psi }_{\delta }:\mathcal{A}_{\lambda
_{0}}\rightarrow \mathcal{B}_{\delta }$ such that
\begin{equation*}
\phi _{\left( \delta ,n\right) }\left( \pi _{\lambda _{0}}^{\mathcal{A
}\left( a\right) \otimes \left. \theta _{\xi ,\xi }\right\vert _{\mathcal{H
_{n}}\right) =\widetilde{\psi }_{\delta }\left( \pi _{\lambda _{0}}^
\mathcal{A}}\left( a\right) \right) \otimes \left. \theta _{\xi ,\xi
}\right\vert _{\mathcal{H}_{n}}
\end{equation*
for all $a\in \mathcal{A}$ such that $a\geq _{\lambda _{0}}0$ and $\xi \in
\mathcal{H}_{k}^{c},k\leq n$, and $\varphi _{\delta }-\widetilde{\psi
_{\delta }$ is positive. Since $K(\mathcal{D}_{\mathcal{E}})_{n}\ $is
generated by rank one operators $\theta _{\xi ,\xi },\xi \in \mathcal{H
_{k}^{c},$ $k\leq n,$
\begin{equation*}
\phi _{\left( \delta ,n\right) }\left( \pi _{\lambda _{0}}^{\mathcal{A
}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H}_{n}}\right)
\widetilde{\psi }_{\delta }\left( \pi _{\lambda _{0}}^{\mathcal{A}}\left(
a\right) \right) \otimes \left. T\right\vert _{\mathcal{H}_{n}}
\end{equation*
for all $a\in \mathcal{A}$ and for all $T\in K(\mathcal{D}_{\mathcal{E}}).$

Suppose that $\mathcal{B}_{\delta }$ acts nondegenerately on a Hilbert space
$\mathcal{K}$. Let $\{u_{i}\}_{i\in I}$ be an approximate unit for $K
\mathcal{D}_{\mathcal{E}})_{n}$, $\eta \in \mathcal{K}$ and $\xi \in
\mathcal{H}_{k}^{c},k\leq n,\left\Vert \xi \right\Vert \neq 0.$ Since
\mathcal{H}_{n}=\tbigoplus\limits_{k\leq n}\mathcal{H}_{k}^{c}$ and $K
\mathcal{D}_{\mathcal{E}})_{n}$ is a closed two sided $\ast $-ideal of
C^{\ast }(\mathcal{D}_{\mathcal{E}})_{n}$, we have
\begin{eqnarray*}
&&\left\langle \phi _{\left( \delta ,n\right) }\left( \pi _{\lambda _{0}}^
\mathcal{A}}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H
_{n}}\right) \left( \eta \otimes \xi \right) ,\left( \eta \otimes \xi
\right) \right\rangle \\
&=&\lim\limits_{i}\left\langle \phi _{\left( \delta ,n\right) }\left( \pi
_{\lambda _{0}}^{\mathcal{A}}\left( a\right) \otimes u_{i}\left.
T\right\vert _{\mathcal{H}_{n}}u_{i}\right) \left( \eta \otimes \xi \right)
,\eta \otimes \xi \right\rangle \\
&=&\lim\limits_{i}\left\langle \left( \widetilde{\psi }_{\delta }\left( \pi
_{\lambda _{0}}^{\mathcal{A}}\left( a\right) \right) \otimes u_{i}\left.
T\right\vert _{\mathcal{H}_{n}}u_{i}\right) \left( \eta \otimes \xi \right)
,\eta \otimes \xi \right\rangle \\
&=&\lim\limits_{i}\left( \left\langle \widetilde{\psi }_{\delta }\left( \pi
_{\lambda _{0}}^{\mathcal{A}}\left( a\right) \right) \eta ,\eta
\right\rangle \otimes \left\langle u_{i}\left. T\right\vert _{\mathcal{H
_{n}}u_{i}\left. \theta _{\frac{1}{\left\Vert \xi \right\Vert }\xi ,\frac{1}
\left\Vert \xi \right\Vert }\xi }\right\vert _{\mathcal{H}_{n}}\left( \xi
\right) ,\xi \right\rangle \right) \\
&=&\left\langle \widetilde{\psi }_{\delta }\left( \pi _{\lambda _{0}}^
\mathcal{A}}\left( a\right) \right) \eta ,\eta \right\rangle \otimes
\left\langle \left. T\right\vert _{\mathcal{H}_{n}}\left. \theta _{\frac{1}
\left\Vert \xi \right\Vert }\xi ,\frac{1}{\left\Vert \xi \right\Vert }\xi
}\right\vert _{\mathcal{H}_{n}}\left( \xi \right) ,\xi \right\rangle \\
&=&\left\langle \widetilde{\psi }_{\delta }\left( \pi _{\lambda _{0}}^
\mathcal{A}}\left( a\right) \right) \eta ,\eta \right\rangle \otimes
\left\langle \left. T\right\vert _{\mathcal{H}_{n}}\left( \xi \right) ,\xi
\right\rangle \\
&=&\left\langle \left( \widetilde{\psi }_{\delta }\left( \pi _{\lambda
_{0}}^{\mathcal{A}}\left( a\right) \right) \otimes \left. T\right\vert _
\mathcal{H}_{n}}\right) \left( \eta \otimes \xi \right) ,\eta \otimes \xi
\right\rangle
\end{eqnarray*
for all $a\in \mathcal{A}$ and for all $T\in C^{\ast }(\mathcal{D}_{\mathcal
E}})$. Therefore,
\begin{equation*}
\phi _{\left( \delta ,n\right) }\left( \pi _{\lambda _{0}}^{\mathcal{A
}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H}_{n}}\right)
\widetilde{\psi }_{\delta }\left( \pi _{\lambda _{0}}^{\mathcal{A}}\left(
a\right) \right) \otimes \left. T\right\vert _{\mathcal{H}_{n}}
\end{equation*
for all $a\in \mathcal{A}$ and for all $T\in C^{\ast }(\mathcal{D}_{\mathcal
E}}).$

Let $\psi _{\delta }:\mathcal{A}\rightarrow \mathcal{B}_{\delta },\psi
_{\delta }=\widetilde{\psi }_{\delta }\circ \pi _{\lambda _{0}}^{\mathcal{A
} $. Clearly, $\psi _{\delta }$ is a local positive map, and
\begin{equation*}
\pi _{\left( \delta ,n\right) }^{\mathcal{B\otimes }C^{\ast }(\mathcal{D}_
\mathcal{E}})}\left( \phi \left( a\otimes T\right) \right) =\psi _{\delta
}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H}_{n}}
\end{equation*
for all $a\in \mathcal{A}$ and $T\in $ $C^{\ast }(\mathcal{D}_{\mathcal{E}})
.

Let $\delta _{1},\delta _{2}\in \Delta $ with $\delta _{1}\geq \delta _{2},$
$n\in \mathbb{N}$ and $a\in \mathcal{A}$. Since
\begin{eqnarray*}
\pi _{\delta _{1}\delta _{2}}^{\mathcal{B}}\left( \psi _{\delta _{1}}\left(
a\right) \right) \otimes \left. T\right\vert _{\mathcal{H}_{n}} &=&\pi
_{\left( \delta _{1},n\right) \left( \delta _{2},n\right) }^{\mathcal
B\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E}})}\left( \psi _{\delta
_{1}}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H}_{n}}\right) \\
&=&\pi _{\left( \delta _{1},n\right) \left( \delta _{2},n\right) }^{\mathcal
B\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E}})}\left( \pi _{\left( \delta
_{1},n\right) }^{\mathcal{B\otimes }C^{\ast }(\mathcal{D}_{\mathcal{E
})}\left( \phi \left( a\otimes T\right) \right) \right) \\
&=&\pi _{\left( \delta _{2},n\right) }^{\mathcal{B\otimes }C^{\ast }
\mathcal{D}_{\mathcal{E}})}\left( \phi \left( a\otimes T\right) \right)
=\psi _{\delta _{2}}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H
_{n}}
\end{eqnarray*
for all $T\in $ $C^{\ast }(\mathcal{D}_{\mathcal{E}})$, it follows that $\pi
_{\delta _{1}\delta _{2}}^{\mathcal{B}}\left( \psi _{\delta _{1}}\left(
a\right) \right) =\psi _{\delta _{2}}\left( a\right) $. Therefore, there is
a linear map $\psi :\mathcal{A\rightarrow B}$ such that
\begin{equation*}
\psi \left( a\right) =\left( \psi _{\delta }\left( a\right) \right) _{\delta
\in \Delta }\text{.}
\end{equation*
Moreover, since for each $\delta \in \Delta $, $\psi _{\delta }$ is local
positive, $\psi $ is local positive and
\begin{eqnarray*}
\pi _{\left( \delta ,n\right) }^{\mathcal{B\otimes }C^{\ast }(\mathcal{D}_
\mathcal{E}})}\left( \left( \psi \otimes \text{id}_{C^{\ast }(\mathcal{D}_
\mathcal{E}})}\right) \left( a\otimes T\right) \right) &=&\psi _{\delta
}\left( a\right) \otimes \left. T\right\vert _{\mathcal{H}_{n}} \\
&=&\pi _{\left( \delta ,n\right) }^{\mathcal{B\otimes }C^{\ast }(\mathcal{D
_{\mathcal{E}})}\left( \phi \left( a\otimes T\right) \right)
\end{eqnarray*
for all $a\in \mathcal{A}$ and $T\in $ $C^{\ast }(\mathcal{D}_{\mathcal{E}})
, and for all $\left( \delta ,n\right) \in \Delta \times \mathbb{N}$.
Therefore,
\begin{equation*}
\phi =\psi \otimes \text{id}_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}\text{.}
\end{equation*}

To show that $\varphi -\psi $ is local positive, let $\delta \in \Delta $.
We seen that there exist $\lambda _{0}\in \Lambda $ such that $\varphi
_{\delta }-\widetilde{\psi }_{\delta }$ is positive, where $\pi _{\delta }^
\mathcal{B}}\circ \varphi =\varphi _{\delta }\circ \pi _{\lambda _{0}}^
\mathcal{A}}$ and $\pi _{\delta }^{\mathcal{B}}\circ \psi =\psi _{\delta }
\widetilde{\psi }_{\delta }\circ \pi _{\lambda _{0}}^{\mathcal{A}}$.
Clearly, $\left( \varphi -\psi \right) \left( a\right) \geq _{\delta }0$
whenever $a\geq _{\lambda _{0}}0$ and $\left( \varphi -\psi \right) \left(
a\right) =_{\delta }0$ whenever $a=_{\lambda _{0}}0$. Therefore, $\varphi
-\psi $ is local positive.

$\left( 2\right) \Rightarrow \left( 1\right) $ By Proposition \ref{3}, $\phi
$ is local positive, and since
\begin{equation*}
\varphi \otimes \text{id}_{C^{\ast }(\mathcal{D}_{\mathcal{E}})}-\phi
=\left( \varphi -\psi \right) \otimes \text{id}_{C^{\ast }(\mathcal{D}_
\mathcal{E}})}
\end{equation*
and $\varphi -\psi $ are local positive, $\varphi \otimes $id$_{C^{\ast }
\mathcal{D}_{\mathcal{E}})}-\phi $ is local positive.
\end{proof}

\begin{corollary}
\label{positive} Let $\mathcal{A}$ and $\mathcal{B}$\ be two unital locally
C^{\ast }$-algebras, $\mathcal{H}$ be a Hilbert space $\mathcal{H},$
\varphi :$ $\mathcal{A}$ $\rightarrow $ $\mathcal{B\ }$and $\phi :\mathcal
A\otimes }B(\mathcal{H})\rightarrow \mathcal{B\otimes }B(\mathcal{H})$ be
two linear maps. Then $\phi $ and $\varphi \otimes $id$_{B(\mathcal{H
))}-\phi $ are local positive if and only if there is a local positive map
\psi :\mathcal{A\rightarrow B\ }$such that $\varphi -\psi $ is local
positive and $\phi =\psi \otimes $ id$_{B(\mathcal{H})}$.
\end{corollary}

As an application of Theorem \ref{6}, we show that given a local positive
map $\varphi :$ $\mathcal{A}\rightarrow $ $\mathcal{B}$, the local positive
map $\varphi \otimes $id$_{M_{n}\left( \mathbb{C}\right) }$ is local
decomposable for some $n\geq 2$ if and only if $\varphi $ is a local\ $CP
-map.

\begin{definition}
\cite{MJ3} A linear map $\varphi :\mathcal{A}\rightarrow \mathcal{B}$ is
called \textit{local }$\mathit{n}$\textit{-copositive if }the map $\varphi
\otimes t:\mathcal{A\otimes }M_{n}\left( \mathbb{C}\right) \rightarrow
\mathcal{B\otimes }M_{n}\left( \mathbb{C}\right) $ defined by
\begin{equation*}
\left( \varphi \otimes t\right) \left( \left[ a_{ij}\right]
_{i,j=1}^{n}\right) =\left[ \varphi \left( a_{ji}\right) \right] _{i,j=1}^{n}
\end{equation*
is local positive, where $t$ denotes the transpose map on $M_{n}\left(
\mathbb{C}\right) $.

We say that $\varphi $ is local completely copositive if for each $\delta
\in \Delta $, there exists $\lambda \in \Lambda $ such that $\left( \varphi
\otimes t\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 $\left( \varphi \otimes t\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{definition}

\begin{remark}
\cite{MJ3}\label{copositive}Let $\mathcal{A}$ and $\mathcal{B}$\ be two
locally $C^{\ast }$-algebras and $\varphi :\mathcal{A}\rightarrow \mathcal{B}
$ be a \textit{local }$\mathit{n}$\textit{-copositive} map. Then:

\begin{enumerate}
\item $\varphi $ is \textit{\ }local positive and so it is continuous and
positive.

\item \textit{for each }$\delta \in \Delta $, there exist $\lambda \in
\Lambda $ and an $n$-copositive map $\varphi _{\delta }:\mathcal{A}_{\lambda
}\rightarrow \mathcal{B}_{\delta }$ such that $\pi _{\delta }^{\mathcal{B
}\circ \varphi =\varphi _{\delta }\circ \pi _{\lambda }^{\mathcal{A}}.$
\end{enumerate}
\end{remark}

\begin{definition}
\cite{MJ3} A linear map $\varphi :\mathcal{A}\rightarrow \mathcal{B}$ is
local decomposable if it is sum of a local completely positive map and a
\textit{local completely copositive }map.
\end{definition}

If $\varphi :\mathcal{A}\rightarrow \mathcal{B}$ is local decomposable, then
for each\textit{\ }$\delta \in \Delta $, there exist $\lambda \in \Lambda $
and a decomposable positive map $\varphi _{\delta }:\mathcal{A}_{\lambda
}\rightarrow \mathcal{B}_{\delta }$ such that $\pi _{\delta }^{\mathcal{B
}\circ \varphi =\varphi _{\delta }\circ \pi _{\lambda }^{\mathcal{A}}.$

\begin{theorem}
\label{7}Let $\mathcal{A}$ and $\mathcal{B}$\ be two unital locally $C^{\ast
}$-algebras and $\varphi :\mathcal{A\rightarrow B}$ be a linear map. If for
some $n\geq 2$, $\varphi \otimes $id$_{M_{n}\left( \mathbb{C}\right) }
\mathcal{A\otimes }M_{n}\left( \mathbb{C}\right) \rightarrow \mathcal
B\otimes }M_{n}\left( \mathbb{C}\right) $ is local decomposable, then
\varphi $ is local completely positive.
\end{theorem}

\begin{proof}
We adapt the proof of \cite[Theorem 3.1 ]{BO}. If $\varphi \otimes $id
_{M_{n}\left( \mathbb{C}\right) }$ is decomposable, there are a local
completely positive map $\phi $ and a local completely copositive map $\psi $
such that $\varphi \otimes $id$_{M_{n}\left( \mathbb{C}\right) }=\phi +\psi
.$\ $By Corollary \ref{positive}, there exist two local positive maps $\phi
_{1}:$ $\mathcal{A\rightarrow B}$ and $\psi _{1}:$ $\mathcal{A\rightarrow B\
}$such that $\phi =\phi _{1}\otimes $id$_{M_{n}\left( \mathbb{C}\right) }$
and $\psi =\psi _{1}\otimes $id$_{M_{n}\left( \mathbb{C}\right) }$.

Since $\phi =\phi _{1}\otimes $id$_{M_{n}\left( \mathbb{C}\right) }$ and
\phi $ is local completely positive, $\phi _{1}$ is local completely
positive.

Since $\psi $ and $\psi _{1}$ are continuous (as $\psi $ is local completely
copositive and $\psi _{1}$ is local positive) and $\psi =\psi _{1}\otimes $i
$_{M_{n}\left( \mathbb{C}\right) }$,\ for each $\delta \in \Delta ,$ there
exist $\lambda \in \Lambda $, and completely positive maps $\psi _{\delta }
\mathcal{A}_{\lambda }\rightarrow \mathcal{B}_{\delta }$ and $\psi _{1\delta
}:$ $\mathcal{A}_{\lambda }\rightarrow \mathcal{B}_{\delta }$\ such that
\pi _{\delta }^{\mathcal{B}}\circ \psi =\psi _{\delta }\circ \pi _{\lambda
}^{\mathcal{A}},$ $\pi _{\delta }^{\mathcal{B}}\circ \psi _{1}=\psi
_{1\delta }\circ \pi _{\lambda }^{\mathcal{A}}\ $and $\psi _{\delta }=\psi
_{1\delta }\otimes $id$_{M_{n}\left( \mathbb{C}\right) }$. Then, since
n\geq 2$, by \cite[Lemma 3.2]{BO}, for each $\delta \in \Delta ,$ $\psi
_{\delta }=0$. Consequently, $\psi =0$, and so $\varphi \otimes $id
_{M_{n}\left( \mathbb{C}\right) }=\phi =\phi _{1}\otimes $id$_{M_{n}\left(
\mathbb{C}\right) }$, whence $\varphi =\phi _{1}.$
\end{proof}

Let $\{\mathcal{H};\mathcal{E};\mathcal{D}_{\mathcal{E}}\}$ be a quantized
domain in the Hilbert space $\mathcal{H\ }$with $\mathcal{E=\{H}_{\iota
\mathcal{\}}_{\iota \in \Upsilon }$.

For a local contractive $\ast $-morphism $\pi :\mathcal{A\rightarrow
C^{\ast }(\mathcal{D}_{\mathcal{E}})$
\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*}

\begin{remark}
\begin{enumerate}
\item $\pi \left( \mathcal{A}\right) ^{^{\prime }}$is a\textbf{\ }von
Neumann algebra. Indeed
\begin{eqnarray*}
\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}\} \\
&=&\{T\in B\left( \mathcal{H}\right) ;T\pi \left( a\right) \subseteq \pi
\left( a\right) T\text{ for all }a\in b(\mathcal{A)}\} \\
&=&\{T\in B\left( \mathcal{H}\right) ;T\left. \pi \right\vert _{b(\mathcal{A
}}\left( a\right) =\left. \pi \right\vert _{b(\mathcal{A)}}\left( a\right)
\text{ for all }a\in b(\mathcal{A)}\} \\
&=&\left. \pi \right\vert _{b(\mathcal{A)}}\left( b(\mathcal{A})\right)
^{\prime }
\end{eqnarray*
where $\left. \pi \right\vert _{b(\mathcal{A)}}:b(\mathcal{A)\rightarrow
B\left( \mathcal{H}\right) $ is the $\ast $-representation of the $C^{\ast }
-algebra $b(\mathcal{A)}$ of all bounded elements of $\mathcal{A},$ $\left.
\left. \pi \right\vert _{b(\mathcal{A)}}\left( a\right) \right\vert _
\mathcal{D}_{\mathcal{E}}}=\pi \left( a\right) .$

\item $\pi \left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D
_{\mathcal{E}})$ \ is identified with a\textbf{\ }von Neumann algebra on
\mathcal{H}$. Indeed,
\begin{eqnarray*}
\pi \left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }(\mathcal{D}_
\mathcal{E}}) &=&\{T\in B\left( \mathcal{H}\right) \cap C^{\ast }(\mathcal{D
_{\mathcal{E}});T\pi \left( a\right) \subseteq \pi \left( a\right) T\text{
for all }a\in \mathcal{A}\} \\
&=&\{S\in b(C^{\ast }(\mathcal{D}_{\mathcal{E}}));S\pi \left( a\right) =\pi
\left( a\right) S\ \text{ for all }a\in \mathcal{A}\} \\
&=&b(\pi \left( \mathcal{A}\right) ^{c})
\end{eqnarray*
where $\pi \left( \mathcal{A}\right) ^{c}=\{S\in C^{\ast }(\mathcal{D}_
\mathcal{E}});S\pi \left( a\right) =\pi \left( a\right) S$ for all $a\in
\mathcal{A}\}$.

On the other hand, $\pi \left( \mathcal{A}\right) ^{c}$\ is a locally von
Neumann algebra \cite[p.4198]{D}, and then, $b(\pi \left( \mathcal{A}\right)
^{c})$ is identified with a von Neumann algebra on $\mathcal{H}$ \cite
Proposition 3.2]{D}.

\item By \cite[Proposition 3.2]{D}, $b(C^{\ast }(\mathcal{D}_{\mathcal{E}}))$
is identified with a von Neumann algebra on $\mathcal{H}$, which is
spatially isomorphic to the von Neumann algebra $\{T\in B\left( \mathcal{H
\right) ;P_{\iota }T=TP_{\iota },\forall \iota \in \Upsilon \}.$

\item Von Neumann algebras $\pi \left( \mathcal{A}\right) ^{^{\prime }}\cap
C^{\ast }(\mathcal{D}_{\mathcal{E}})$ and $\left. \pi \right\vert _{b
\mathcal{A)}}\left( b(\mathcal{A})\right) ^{\prime }\cap b(C^{\ast }
\mathcal{D}_{\mathcal{E}}))$ are isomorphic.
\end{enumerate}
\end{remark}

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

\ \ \ Let $\varphi ,\psi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast
}(\mathcal{D}_{\mathcal{E}}))$ such that $\psi $ is dominated by $\varphi $.
If $(\pi _{\varphi },V_{\varphi },$\ \ \ $\{\mathcal{H}^{\varphi },\mathcal{
}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\})$ is a minimal
Stinespring dilation associated to $\varphi $, then, by Radon Nikodym type
theorem \cite[Theorem 4.5]{BGK}, 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 $a\in \mathcal{A}.$

\begin{definition}
Let $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{
}_{\mathcal{E}}))$. We say that $\varphi $ is 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 $\alpha $ such
that $\psi =\alpha \varphi .$
\end{definition}

Let $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{
}_{\mathcal{E}}))$ and $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H
^{\varphi },\mathcal{E}^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi
}}\}\right) $ be a minimal Stinespring dilation associated to $\varphi $.
Then $\varphi $ 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
\}.$

\begin{proposition}
Let $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},C^{\ast }(\mathcal{
}_{\mathcal{E}}))$. If $\varphi $ is pure, then it is a bounded operator
valued completely positive map.
\end{proposition}

\begin{proof}
Let $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi },\mathcal{E
^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ be a minimal
Stinespring dilation associated to $\varphi $. Since $\varphi $ is pure,
\pi _{\varphi }\left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }
\mathcal{D}_{\mathcal{E}^{\varphi }})=\{\alpha $id$_{\mathcal{D}_{\mathcal{E
}};\alpha \in \mathbb{C}\}$. On the other hand, for each $\iota \in \Upsilon
,$ $\left. P_{\iota }\right\vert _{\mathcal{D}_{\mathcal{E}^{\varphi }}}\in
\pi _{\varphi }\left( \mathcal{A}\right) ^{^{\prime }}\cap C^{\ast }
\mathcal{D}_{\mathcal{E}^{\varphi }})$, and so $P_{\iota }=$id$_{\mathcal{H
^{\varphi }}$. Therefore, for each $\iota \in \Upsilon ,$ $\mathcal{H
_{\iota }^{\varphi }=\mathcal{H}^{\varphi }$, $\mathcal{D}_{\mathcal{E
^{\varphi }}=\mathcal{H}^{\varphi }$ and $C^{\ast }(\mathcal{D}_{\mathcal{E
^{\varphi }})=B\left( \mathcal{H}^{\varphi }\right) $. Consequently, $\pi
_{\varphi }\left( a\right) \in B\left( \mathcal{H}^{\varphi }\right) $ for
all $a\in \mathcal{A}.\ $Therefore, $V_{\varphi }^{\ast }\pi _{\varphi
}\left( a\right) V_{\varphi }\in B\left( \mathcal{H}\right) $ for all $a\in
\mathcal{A}$, and
\begin{equation*}
\varphi \left( a\right) =\left. V_{\varphi }^{\ast }\pi _{\varphi }\left(
a\right) V_{\varphi }\right\vert _{\mathcal{D}_{\mathcal{E}}}\in b\left(
C^{\ast }(\mathcal{D}_{\mathcal{E}})\right)
\end{equation*
for all $a\in \mathcal{A}$.
\end{proof}

\begin{theorem}
\label{8} Let $\mathcal{A}$ and $\mathcal{B}$ be two unital locally $C^{\ast
}$-algebras, $\varphi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{A},\mathcal{D
_{\mathcal{E}}),$ $\psi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{B},C^{\ast
}(\mathcal{D}_{\mathcal{F}}))$ and $\phi \in \mathcal{CPCC}_{\text{loc}}
\mathcal{A\otimes B},C^{\ast }(\mathcal{D}_{\mathcal{E\otimes F}}))$. If
\phi $ is dominated by $\varphi \otimes \psi $ and $\varphi $ is pure, then
there exists $\widetilde{\psi }\in \mathcal{CPCC}_{\text{loc}}(\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{F}}))$ dominated by $\psi $ such that $\phi
=\varphi \otimes \widetilde{\psi }.$
\end{theorem}

\begin{proof}
Let $\left( \pi _{\varphi },V_{\varphi },\{\mathcal{H}^{\varphi },\mathcal{E
^{\varphi },\mathcal{D}_{\mathcal{E}^{\varphi }}\}\right) $ and $\left( \pi
_{\psi },V_{\psi },\{\mathcal{H}^{\psi },\mathcal{E}^{\psi },\mathcal{D}_
\mathcal{E}^{\psi }}\}\right) $ be the minimal Stinespring dilations
associated to $\varphi $ and $\psi $. Since $\varphi $ is pure, $\mathcal{D
_{\mathcal{E}^{\varphi }}=\mathcal{H}^{\varphi }$ and $C^{\ast }(\mathcal{D
_{\mathcal{E}^{\varphi }})=B\left( \mathcal{H}^{\varphi }\right) $. By
Proposition \ref{3}, $\ (\pi _{\varphi }\otimes \pi _{\psi },V_{\varphi
}\otimes V_{\psi },\{\mathcal{H}^{\varphi }\otimes \mathcal{H}^{\psi }
\mathcal{H}^{\varphi }\otimes \mathcal{E}^{\psi },$ $\mathcal{D}_{\mathcal{H
^{\varphi }\otimes \mathcal{E}^{\psi }}\})$ is a minimal Stinespring
dilation associated to $\varphi \otimes \psi $. We have:
\begin{eqnarray*}
\left( \left( \pi _{\varphi }\otimes \pi _{\psi }\right) \left( \mathcal
A\otimes B}\right) \right) ^{\prime } &=&\left( \left( \left. \pi _{\varphi
}\right\vert _{b\mathcal{(A)}}\otimes \left. \pi _{\psi }\right\vert _{
\mathcal{(B)}}\right) \left( b(\mathcal{A)\otimes }_{\text{alg}}b\mathcal{(B
}\right) \right) ^{\prime } \\
&=&\left( \left. \pi _{\varphi }\right\vert _{b\mathcal{(A)}}\left( b
\mathcal{A)}\right) \right) ^{\prime }\overline{\otimes }\left( \left. \pi
_{\psi }\right\vert _{b\mathcal{(B)}}\left( b\mathcal{(B)}\right) \right)
^{\prime }
\end{eqnarray*
where "$\overline{\otimes }$ " denotes the tensor product of von Neumann
algebras, and
\begin{eqnarray*}
b(C^{\ast }(\mathcal{D}_{\mathcal{H}^{\varphi }\otimes \mathcal{E}^{\psi
}})) &=&\{R\in B(\mathcal{H}^{\varphi }\otimes \mathcal{H}^{\psi });R(\text
id}_{\mathcal{H}^{\varphi }}\otimes P_{\iota })=(\text{id}_{\mathcal{H
^{\varphi }}\otimes P_{\iota })R,\forall \iota \in \Upsilon \} \\
&=&B(\mathcal{H}^{\varphi })\overline{\otimes }\{S\in B(\mathcal{H}^{\psi
});SP_{\iota }=P_{\iota }S,\forall \iota \in \Upsilon \} \\
&=&B(\mathcal{H}^{\varphi })\overline{\otimes }b(C^{\ast }(\mathcal{D}_
\mathcal{E}^{\psi }})).
\end{eqnarray*
Thus,
\begin{eqnarray*}
&&\left( \left( \pi _{\varphi }\otimes \pi _{\psi }\right) \left( \mathcal
A\otimes B}\right) \right) ^{\prime }\cap b(C^{\ast }(\mathcal{D}_{\mathcal{
}^{\varphi }\otimes \mathcal{E}^{\psi }})) \\
&=&\left( \left( \left. \pi _{\varphi }\right\vert _{b\mathcal{(A)}}\left( b
\mathcal{A)}\right) \right) ^{\prime }\overline{\otimes }\left( \left. \pi
_{\psi }\right\vert _{b\mathcal{(B)}}\left( b\mathcal{(B)}\right) \right)
^{\prime }\right) \cap \left( B(\mathcal{H}^{\varphi })\overline{\otimes
b(C^{\ast }(\mathcal{D}_{\mathcal{E}^{\psi }}))\right) \\
&=&\left( \pi _{\varphi }\left( \mathcal{A}\right) ^{\prime }\cap B(\mathcal
H}^{\varphi })\right) \overline{\otimes }\left( \left( \left. \pi _{\psi
}\right\vert _{b\mathcal{(B)}}\left( b\mathcal{(B)}\right) \right) ^{\prime
}\cap b(C^{\ast }(\mathcal{D}_{\mathcal{E}^{\psi }}))\right) \\
&&\text{(since }\varphi \text{ is pure) } \\
&=&\{\alpha \text{id}_{\mathcal{H}^{\varphi }};\alpha \in \mathbb{C
\}\otimes \left( \pi _{\psi }\left( \mathcal{B}\right) ^{\prime }\cap
C^{\ast }(\mathcal{D}_{\mathcal{E}^{\psi }})\right) .
\end{eqnarray*
Therefore,
\begin{equation*}
\left( \left( \pi _{\varphi }\otimes \pi _{\psi }\right) \left( \mathcal
A\otimes B}\right) \right) ^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{H
^{\varphi }\otimes \mathcal{E}^{\psi }})=\{\alpha \text{id}_{\mathcal{H
^{\varphi }};\alpha \in \mathbb{C}\}\otimes \left( \pi _{\psi }\left(
\mathcal{B}\right) ^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\psi
}})\right) .
\end{equation*
Since $\phi \ $and $\varphi \otimes \psi $ are local completely contractive
and local completely positive and $\phi $ is dominated by $\varphi \otimes
\psi $, by Radon Nikodym theorem \cite[Theorem 4.5]{BGK}, there is a unique
positive contractive linear operator $R\in \left( \left( \pi _{\varphi
}\otimes \pi _{\psi }\right) \left( \mathcal{A\otimes B}\right) \right)
^{\prime }\cap C^{\ast }(\mathcal{D}_{\mathcal{H}^{\varphi }\otimes \mathcal
E}^{\psi }})$ such that $\phi =\left( \varphi \otimes \psi \right) _{R}$.
Therefore, there is $T\in \pi _{\psi }\left( \mathcal{B}\right) ^{\prime
}\cap C^{\ast }(\mathcal{D}_{\mathcal{E}^{\psi }})$ such that $R=$id$_
\mathcal{H}^{\varphi }}\otimes T$ and
\begin{eqnarray*}
\phi \left( a\otimes b\right) &=&\left( \varphi \otimes \psi \right) _{\text
id}_{\mathcal{H}^{\varphi }}\otimes T}\left( a\otimes b\right) \\
&=&\left( V_{\varphi }^{\ast }\otimes V_{\psi }^{\ast }\right) \left( \text
id}_{\mathcal{H}^{\varphi }}\otimes T\right) \left( \pi _{\varphi }\otimes
\pi _{\psi }\right) \left( a\otimes b\right) \left( V_{\varphi }\otimes
V_{\psi }\right) \\
&=&V_{\varphi }^{\ast }\pi _{\varphi }\left( a\right) V_{\varphi }\otimes
V_{\psi }^{\ast }T\pi _{\psi }\left( b\right) V_{\psi }=\varphi \left(
a\right) \otimes \psi _{T}\left( b\right)
\end{eqnarray*
for all $a\in \mathcal{A}$ and $b\in \mathcal{B}$.$\ $Hence, there is
\widetilde{\psi }=\psi _{T}\in \mathcal{CPCC}_{\text{loc}}(\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{F}}))$ such that $\phi =\varphi \otimes
\widetilde{\psi }$.\ Moreover, $\widetilde{\psi }$ is dominated by $\psi $.
\end{proof}

\begin{corollary}
Let $\mathcal{A}$ be a unital $C^{\ast }$-algebra, $\mathcal{B}$ be a unital
locally $C^{\ast }$-algebra, $\varphi \in \mathcal{CP}(\mathcal{A},B
\mathcal{H})),$ $\psi \in \mathcal{CPCC}_{\text{loc}}(\mathcal{B},C^{\ast }
\mathcal{D}_{\mathcal{F}}))$ and $\phi \in \mathcal{CPCC}_{\text{loc}}
\mathcal{A\otimes B},C^{\ast }(\mathcal{D}_{\mathcal{H\otimes F}}))$. If
\phi $ is dominated by $\varphi \otimes \psi $ and $\varphi $ is pure, then
there is $\widetilde{\psi }\in \mathcal{CPCC}_{\text{loc}}(\mathcal{B
,C^{\ast }(\mathcal{D}_{\mathcal{F}}))$ which is dominated by $\psi $ and
such that $\phi =\varphi \otimes \widetilde{\psi }$.
\end{corollary}

\begin{acknowledgement}
I wish to thank A. Dosiev and P. Santhosh Kumar for useful discussions on
the details of the present work. Also, I would like to thank the referee for
his/her careful reading and useful comments.
\end{acknowledgement}

redpajama_set_name: RedPajamaArXiv


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text:
Innovative Container Hardware Stores Launch In Cape Town
The Hardware Co.
5 months ago 3 min read
Western Cape based family-owned hardware chain, Brights Hardware, has embarked on an innovative mentorship and entrepreneur nurturing project that sees micro hardware stores being opened in shipping containers in low-income areas of Cape Town.
The Container Store initiative that Brights started in collaboration with The Hardware Co. kicked off in 2019, with the first container hardware up and running in Khayelitsha by October 2020. The second container hardware is currently being erected nearby – with completion scheduled for September 2021.
"The first container hardware store is an impressive double storey space created out of two shipping containers stacked on top of each other," explains Orlando Luis, CEO of Brights Hardware store. "The bottom container is 12 metres long and the top is 6 metres, and they are joined by a steel staircase. The lower-level sells building materials and the upstairs container offers smaller electrical and hardware products."
The Hardware Co. container stores belong to Yanga Njozela, a young entrepreneur who approached Brights Hardware to assist him with setting up his hardware business. Luis says that he met Njozela in 2016 when he opened his construction company. "He asked if I would mentor and support him in this venture and after a few years of building a business relationship and learning the operations and systems, Yanga requested Brights' assistance to open his own container hardware.
"Yanga purchased the two shipping containers and rented a piece of ground. We assisted him to fully convert and rig out the containers with shelving, computer systems, electrical connections etc. We also provide him all his products at wholesale prices," explains Luis. "Yanga now also has his own delivery truck to collect stock from the closest Brights Store and is able to quote and supply his customers with the same retail pricing that Brights offers."
Brights has also assisted Yanga with infrastructure and construction support, IT and computer support, as well as financial terms support as part of their role as entrepreneurial support and mentorship. According to Orlando, Njozela manages all the rest of his business logistics, income and expenditure, profits and cash flow, staffing etc. and is a completely independent business entity.
"We are very grateful for the assistance that Orlando and Brights has afforded us," says Njozela. "The container hardware has always been a dream of ours with the aim to change the landscape of "informal" businesses in the townships and to empower young people from the communities we operate in."
"Our vision is to build lives, build communities and build Africa. And the Container Hardware concept has allowed us the opportunity to do just that. We now have a great team of young people working with us and the community have received the hardware store incredibly well, affirming that The Hardware Co. has been able to make quality hardware supplies accessible to the residents of Khayelitsha. We are currently in the process of opening up our second store in Harare, Khayelitsha and are excited about this new venture and our future in the hardware store space."
For more information contact Yanga Njozela on 084 703 0745 or visit www.thehardwareco.co.za. You can also follow The Hardware Co on Facebook: @thehardwareco as well as Instagram: @buildwith_thehardwareco.
Tags: Community Container Hardware Stores entrepreneur Hardware stores
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redpajama_set_name: RedPajamaCommonCrawl


text:
HomeGENERALArtDocklands Marina Park | Cork Ireland | OKRA
Docklands Marina Park | Cork Ireland | OKRA
August 23, 2012,AEST Damian Holmes Art, Asia, Landscape Architecture, Urban Planning
Ireland's first large scale urban wetlands and park surrounding a modern sports stadium, which it is hoped will act as a catalyst project for the redevelopment of Cork Docklands, has been awarded by Cork City Council to OKRA.
The winning concept design, led by the Dutch landscape architects OKRA in cooperation with the Irish landscape architects REDscape, took the City Council's brief for the redevelopment of Marina Park to a new level and offers an exciting vision of dynamic landscapes and ecosystems tailored to a modern and developing city. The design concept includes a sequence of urban water gardens, watercourses and wetland areas that will recycle storm water from the adjacent docklands and create a sustainable environment for the new city park on the River Lee.
Contemporary urban spaces will be generated around Pairc Ui Chaoimh so it can, along with facilitating the GAA supporters on match days, also host Cork's cultural events, festivities and gatherings thereby turning the project into a catalyst for the regeneration of the south docklands area. Integration of the GAA's proposals for the stadium into the park will form an important element of the design work.
The transformation of the 35-hectare site will be implemented over a number of years on a phased basis as funding allows and will also ensure the conservation of historic structures such as Barrington's folly. The landscape architects OKRA and REDscape, in association with O'Connor Sutton Cronin (engineers), Venhoeven CS (architects), Howley Hayes (conservation architects), Fehily Timoney (environment and ecology), Modus Operandi (art consultants), and Davis Langdon (quantity surveyors) will work closely with Cork City Council on this project.
Pat Ledwidge, Director of Services, Strategic Planning and Economic Development in City Council said that: "This proposed regional park will achieve a number of objectives for the city including the provision of a world class modern urban park facility for the citizens of Cork while facilitating the upgrading of Pairc Ui Chaoimh into a high quality modern sport centre which will enhance the match day experience for all involved".
IMAGES & TEXT: OKRA
urban design planning
Designs of the Year 2013 Nominations
January 14, 2013,AEDT Damian Holmes
Orientation by Light, Maasstad Hospital | Rotterdam Netherlands Stijlgroep landscape and urban design
July 5, 2011,AEST Damian Holmes
Is Great Park a soaring vision or just hot air? – latimes.com
October 2, 2009,AEST Damian Holmes


– Burmese cats are occasionally referred to as copper cats, which we think is a fairly cool nickname. They are generally well-known for their playful and affectionate temperament, and as all of us know many cat breeds are simply not very comfortable around people, that's not the case with the Burmese.
Burma (formerly Myanmar) is a huge country in south eastern Asia which borders on Thailand and China, and has close ties to India via the sea. The country has a long and turbulent history, and has been colonized by the British not long past. It is via British colonization and ties into the western world that Burmese cats found their way to homes around the world. In fact the colonization of much of the world from the 19th century did much to distribute the majority of the cat breeds, without ties to the western world we would not like these cats now.
If you have one of these cats, then maybe you will want to build a small makeshift altar for your little buddy's great ancestor, the illustrious Wong Mau. It's known that early Burmese people worshipped these cats and also worshipped their ancestors, who knows this might make your cat feel right at home.
There are three distinct breeds of Burmese-the Contemporary, the Conventional, and also the European; they're a little bit different. The European variety is thinner, for instance, and the conventional look most like the original cat attracted over from Burma (the famous and prolific Wong Mau). Each one these cats are rather muscular and also have a brief coat, making perfect sense understanding that Burma is a tropical country where a long haired cat would not be advantageous.
Exotic cats are somewhat similar in looks to the Siamese cat, which should be of no great surprise when you realize that Thailand (the home of the Siamese cat) lies just to the east of Burma. In closing, these Burmese are extremely human like, and consequently very popular cats all over the world.
Find out the most recent images of 9 Awesome Things You Can Learn From Abyssinian Kittens For Sale Ontario | Abyssinian Kittens For Sale Ontario here, and also you can get the image here simply image posted uploaded by Teman Saya that saved in our collection.