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quantum Feynman-Kac formula, Markovian semigroups, elliptic operators, operator valued stochastic differential equations.
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\begin{document}

\title{  Feynman-Kac Representation and Markovianity of  Semigroups Generated by Noncommutative Elliptic Operators }

\author{Changsoo Bahn\footnote{ E-mail: changsoobahn@yahoo.co.kr} \\[0.3cm]
  Institute of Natural Science, Yonsei University,
 \\ Seoul 120-749, Korea \\[0.3cm]
and \\[0.3cm]
Yong Moon Park\footnote{ E-mail: ympark@yonsei.ac.kr} \\[0.3cm]
Department of Mathematics,  Yonsei University, \\ Seoul 120-749, Korea } %\\


%\address{Department of Mathematics,  Yonsei University, Seoul 120-749, Korea
%\\E-mail: changsoo@math.yonsei.ac.kr}


%\author{Yong Moon Park}

%\address{Department of Mathematics,  Yonsei University, Seoul 120-749, Korea
%\\E-mail: ympark@yonsei.ac.kr}
\date{}

\maketitle

\begin{abstract} We construct Markovian semigroups generated by noncommutative elliptic operators on a von Neumann algebra. We first introduce a general framework of quantum Feynman-Kac formula in terms of unitary evolutions and multipliers, and then apply the general result to our problem. We construct  multipliers which are determined by  operator valued stochastic differential equations and satisfy  cocycle relations with respect to a randomized flow on a von Neumann algebra. Our works are greatly influenced by that of Lindsay and Sinha [LS]. In order to ensure Markovianity of the semigroups, we modify the method employed by Lindsay and Sinha \cite{LS}.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

Recently Lindsay and Sinha \cite{LS} constructed semigroups generated by noncommutative elliptic operators on
a von Neumann algebra through a quantum version of Feynman-Kac formula.
However, in general the semigroups they constructed do not preserve the completely positive property.
In this paper, we introduce an another quantum version of Feynman-Kac formula such that
the completely positive preserving property of semigroups becomes manifest.
We then use the Feynman-Kac formula and a construction method similar to that employed in \cite{LS} to
construct Markovian semigroups generated by noncommutative elliptic operators.
The form of the quantum Feynman-Kac formula we introduce is closely related to that given in \cite{AFL}.

In order to describe the content of the paper and the problems involved, let us recall
the Feynman-Kac formula for diffusion type semigroups.
For given $b \in L^\infty (\R)$, let $H= -\frac 12 \frac {d^2}{dx^2} - b \frac {d}{dx}$.
Then the Feynman-Kac formula \cite{RS, PS} yields that for $f \in L^2(\R)$,
\begin{eqnarray} \label{1.1}
(e^{-tH} f) (x) &=& \int_\Omega m_t^b(x, \omega) f(x + B_t(\omega)) \, d\P(\omega), \\
m^b_t(x, \omega) &=& \exp \left \{ \int^t_0 b(x+ B_\tau ) \, dB_\tau - \frac 12 \int^t_0 b^2(x+ B_\tau) \, d\tau \right\},\nonumber
\end{eqnarray}
in which $(\Omega, \cB, \P)$ is the standard Wiener space,
and $B_\tau(\omega) = \omega(\tau)$, $ \omega \in \Omega$. Now, let us consider the semigroup $ P^b_t = e^{-tH}$
as an one-parameter semigroup  on the abelian von Neumann algebra $L^\infty (\R)$.
Then, the above representation can be written as
\begin{equation} \label{1.2}
P^b_t(f) = \E [\,j_t(f) m_t^b\,],
\end{equation}
where $j_t(f) (x) = f(x + B_t)$, and
\begin{equation} \label{1.3}
m^b_t = \exp \left\{ \int^t_0 j_\tau (b) \, dB_\tau - \frac 12 \int^t_0 j_\tau (b^2) \, d\tau\right\},
\end{equation}
and $\E$ is the Wiener expectation. Thus the translation group on $L^\infty (\R)$ is randomized and then the shifted
element is acted by a randomized multiplication operator, before being averaged to realize the semigroup $(P_t^b)$.
The expression (\ref{1.2}) can be written in more symmetric way as follow:
\begin{equation} \label{1.4}
P_t^b(f) = \E \left[\, \big( (m^b_t)^{1/2} \big)^- \, j_t(f) (m^b_t)^{1/2} \, \right],
\end{equation}
where the bar denotes the complex conjugation.

These structures together with various noncommutative extensions
have been studied by many authors (\cite{AFL}, \cite{AMV},
\cite{Fag}, \cite{HIP}, \cite{LS}, \cite{LW} and references there
in). In particular, Lindsay and Sinha \cite{LS} replaced the
function space on which the semigroup acts by an operator algebra.
More precisely, let $\cA$ be a von Neumann algebra acting on a
separable Hilbert space $\ch$ and let $(\alpha_t: t\in \R)$ be a
weak*- continuous group of *-automorphisms of $\cA$. For each $t
\ge 0$ and $x \in \cA$, define $j_t(x) : \Omega \to \cA$ by
$$
(j_t(x) ) (\omega) = \alpha_{\omega(t)} (x).
$$
Then each $j_t$ defines a weak*-continuous *-homomorphism of $\cA$ into $\cM = \cA \otimes L^\infty (\Omega)$.
The maps $(j_t)$ are extended to $\cM$ by
$$
J_t : x \otimes v \mapsto j_t(x) v, \quad x\in \cA, \, v \in L^\infty (\Omega).
$$
Notice that the multiplier $(m_t^b : t\ge 0)$ given (\ref{1.3}) satisfies the following stochastic differential equations
$$
d_t m_t^b (f) = j_t(b) m_t^b (f) dB_t, \quad m_0^b f = f,
$$
for $f\in L^2(\R)$ ( cf., \cite{PS}). For each $a =a^* \in \cA$, Lindsay and Sinha \cite{LS}
constructed the multiplier ($m_t^a : t \ge 0$) as the unique solution of the stochastic differential equations
\begin{equation} \label{1.5}
d_t m_t^a \varphi = J_t(x) m_t^a \varphi \, dB_t, \quad m_0^a \varphi = \varphi,
\end{equation}
for $\varphi \in \ch$, and defined
\begin{equation} \label{1.6}
P_t^a (x) \varphi := \E [ \, J_t(x) m_t^a \varphi \,], \quad \varphi \in \ch.
\end{equation}
It turned out that $\{ P_t^a\}$ is a weak*-continuous semigroup on
$\cA$ whose generator is an extension of $\frac12 \delta^2 + R_a
\delta$, where $\delta$ is the generator of $(\alpha_t)$ and $R_a$
is right multiplication operator : $R_a x = xa$, $ x\in \cA$. For
the details, see \cite{LS}.

It is clear that in general the semigroup $(P_t^a)$ defined in (\ref{1.6}) does not preserve reality and positivity.
Thus we consider the Feynman-Kac formula given in (\ref{1.4}).
One may check that the multiplier $\big( (m_t^b)^{1/2}: t \ge 0\big)$ satisfies the stochastic differential equations
\begin{eqnarray} \label{1.7}
d_t (m_t^b)^{1/2} f &=& \frac 12 j_t(b) (m_t^b)^{1/2} f\, dB_t - \frac 18 j_t(b^2) (m_t^b)^{1/2}f \, dt, \\
(m_0^b)^{1/2} f &=& f, \nonumber
\end{eqnarray}
for $f \in L^2 (\R)$. For given $a=a^*\in \cA$, let $(\Tilde{m}_t^a : t\ge 0)$ be the multiplier corresponding
to the classical counter part $\big( (m_t^b)^{1/2} : t\ge 0)$. For each $t$,
one may extend $\Tilde{m}_t^a $
to $\cH = \ch \otimes L^2 (\Omega)$ by $\Tilde{M}_t^a (\varphi \otimes v) : = (\Tilde{m}_t^a \varphi) v$,
$ \varphi \in \ch$, $v \in L^2(\Omega)$.
It can be checked that for each $t$, $\Tilde{M}_t^a$ is unbounded operator on $\cH$, and so one has domain problems.

Thus we are looking for a bounded extension of $(\Tilde{m}_t^a)$.
Let $(\nE_t : t\ge 0)$ be the family of projection operators on
$\cH$ defined by
$$
\nE_t : \varphi \otimes v \mapsto \varphi\otimes \E[\, v | \cB_t\,], \quad \varphi \in \ch, \, v \in L^2 (\Omega),
$$
where $\cB_t$ is the $\sigma$-algebra generated by $B_s-B_0$, $0\le s \le t$.
Consider the multiplier ($M_t^a : t\ge 0$) satisfying the stochastic differential equations
\begin{eqnarray} \label{1.8}
d_t M_t^a \xi &=& J_t(a)(\nE_t M_t^a \xi)\, dB_t - \frac 12 J_t(a^2) \nE_t M_t^a \xi \, dt, \\
M_0^a \xi &=& \xi , \nonumber
\end{eqnarray}
for $\xi \in \cH$. One may compare (\ref{1.8}) and (\ref{1.7}) with $a=\frac b2$.
It turns out that for each $ a \in \cA$ and $ t\ge 0$, there
exists a unique solution $M_t^a \in B(\cH)$ satisfying (\ref{1.8}). For given $a=a^* \in \cA$, define
$$
P_t^a(x) \varphi := \nE_0 [\,(M_t^a)^* J_t(x) M_t^a \varphi \,], \quad x \in \cA, \varphi \in \ch.
$$
Then $(P_t^a : t\ge 0)$ becomes a weak*-continuous Markovian
semigroup on $\cA$ whose generator is an extension of the
following operator:
\begin{equation} \label{1.9}
G^a = \frac 12 \delta^2 + L_a \delta + R_a \delta + L_a R_a - \frac 12 L_a^2 - \frac 12 R_a^2,
\end{equation}
where $L_a$ and $R_a$ are the left and right multiplication operators respectively: $L_ax = ax$, $R_ax = xa$, $ x\in \cA$. For the details, see Theorem \ref{thm3.1} in Section 3.

Let us describe briefly the content of the paper. In Section 2, we introduce a general formalism of
quantum Feynman-Kac formula in the framework of a symmetric Fock space. Let $\Phi$ be the
symmetric Fock space over $L^2(\R_+: dx)$ and $\cH= \ch \otimes \Phi$. We first introduce the
notion of a unitary evolution \cite{HIP} and a multiplier (cocycle) of a given unitary evolution,
and then give conditions ( Assumption \ref{ass1} and \ref{ass2}) under which the Feynman-Kac formula given
in (\ref{(5)}) represents a weak*-continuous Markovian semigroup on $\cA$. We should mention that, if one
reformulates the Feynman-Kac formula we introduce in the algebraic language, it is essentially same as that given
in \cite{AFL}. However we formulate the Feynman-Kac formula so that it becomes very convenient in application.

In Section 3, we construct a multiplier $(M_t^a : t\ge 0)$ for each $a \in \cA$ and show that it
satisfies all the necessary conditions listed in Section 2. We use the construction method similar to
that employed in \cite{LS} with necessary modifications. In Section 4, we investigate some properties of
noncommutative elliptic operators of the type given in (\ref{1.9}). Let $\delta$ be a *-derivation on a
von Neumann algebra $\cA$, and $a=a^* \in \cA$. We show that $G^a$ is a complete dissipation(Theorem \ref{thm4.1})
\cite{Brat}.

Before closing the introduction, we remark that all the results in Section 2 and Section 3 can be
generalized to the case in which $\Phi$ is the symmetric Fock space over $L^2 (\R_+ : { \bf k}),$
 where ${\bf k}$ is a finite dimensional Hilbert space called the {\it noise dimensional space}.


\section{General Formalism}
Let $\ch$ be a separable Hilbert space, called the {\it initial space}. We denote by $\Phi$ the
boson Fock space over $L^2(\R_+)$. To each $t\in \R_+$ corresponds a tensor product
decomposition $\Phi = \Phi_{t]} \otimes \Phi_{[t}$. More generally, if $s < t$, we have a decomposition
$$
\Phi = \Phi_{s]}\otimes \Phi_{[s,t]} \otimes \Phi_{[t}.
$$
The space $\Phi_{s]}\,,\Phi_{[s,t]}$ and $\Phi_{[t}$ are the boson Fock spaces
over $L^2 ([0,s])$, $L^2([s,t])$ and $L^2([t,\infty))$ respectively. Denote by $\1_\Phi$ the vacuum
vector of $\Phi$. For each $0\le s \le t$, we identify $\Phi_{[s,t]}$ as a subspace of $\Phi$ by $u \in \Phi_{[s,t]}$
being identified with $\1_{s]}\otimes u \otimes \1_{[t}$ where $\1_{s]}$ and $\1_{[t}$ are the vacuum
vectors of $\Phi_{s]}$ and $\Phi_{[t}$ respectively. We use the following notations : for any $0\le s\le t$
$$
\cH :=\ch\otimes \Phi,\,\,\,\, \cH_{t]}:=\ch\otimes \Phi_{t]},\,\,\,\,
 \cH_{[s,t]} :=\ch\otimes \Phi_{[s,t]},\,\,\,\, \cH_{[t}:=\ch\otimes \Phi_{[t}.
$$
The space $\cH$ has also other tensor product decompositions. First we have $\cH = \cH_{t]} \otimes \Phi_{[t}$.
On the other hand, we also have $\cH=\Phi_{t]}\otimes \cH_{[t}$. We generally omit the tensor product sign in
these decompositions, writing simply $\varphi u\,(\varphi\in \ch, u \in \Phi)$ or
even $u_{t]} \varphi u_{[t} \, (u_{t]} \in \Phi_{t]}, \varphi \in \ch, u_{[t} \in \Phi_{[t}$) as if initial
vectors really did commute with vectors of Fock space. We do not distinguish $\varphi \1_{\Phi}$ from $\varphi$,
so that the initial space $\ch$ is identified with $\cH_{0]}$. For details, we refer the reader to \cite{Mey}
and \cite{Par}.

Denote by $\nE_t$ the projection on the closed subspace $\cH_{t]}$ of $\cH$. In particular, $\nE_0$
may be considered as the projection onto the initial space $\ch$.

Let $\tau_t, \, t \in \R_+$ be the shift operators on $L^2 (\R_+)$ acting as follows:
$$
\tau_t f(s) = \left\{ \begin{array}{cc} f(s-t)& \text{if} \,\, t \le s \\
                               0 & \text{otherwise.}
                \end{array} \right.
 $$
Denote by $\theta_t$ the second quantization of $\tau_t$. The shift operators on $\Phi$ are isometric.
We extend the shift operators $\theta_t$ to $\cH$ by
$$
\Theta_t(\varphi \otimes u) = \varphi \otimes \theta_t u, \quad \varphi \in \ch ,\, u \in \Phi.
$$
Between the operators $\nE_t$ and $\Theta_t$, we have the relation \cite{Mey}
\begin{equation}\label{(1)}
\Theta_s \nE_t = \nE_{s+t} \Theta_s.
\end{equation}
In particular, $\nE_s \Theta_s = \Theta_s \nE_0 = \nE_0$.

We consider a family $\{ U_{s,t} : 0 \le s \le t\}$ of unitary operators acting on $\cH$ and
satisfying the following conditions:
\begin{eqnarray}
&(a)& U_{s,t} \,\,{\rm and}\,\, U^*_{s,t} \,\,{\rm are \,\, strongly\,\, continuous\,\, in} \,\,
 t \,\, {\rm for \,\, fixed } \,\, s,\nonumber \\
&(b)& U_{s,s} = \1_{{\rm B}(\cH)} \,\,\, {\rm for \,\, all} \,\, s, \label{(2)} \\
&(c)& U_{r,s} U_{s,t} = U_{r,t} \,\, {\rm for} \,\, r\le s \le t. \nonumber
\end{eqnarray}
Such family $\{U_{s,t} : s \le t\}$ is called a {\it unitary evolution} \cite{HIP}.

Let $\cA$ be a von Neumann algebra acting on the initial space $\ch$. Denote by $\cA'$ the commutant of $\cA$. Put
\begin{equation} \label{(3)}
J_{s,t}(x) := U_{s,t} (x\otimes \1_{{\rm B}(\Phi)} ) U_{s,t}^*, \quad x \in \cA.
\end{equation}
In the following, we use simplified notations $\Phi_t :=
\Phi_{t]}$, $\cH_t =\cH_{t]}$, $U_t := U_{0,t}$ and $J_t(x) =
J_{0,t}(x)$. Any operator defined on a tensorial factor of $\cH$
will be often identified with its canonical amplification to $\cH$
and denoted by the same symbol, i.e., $x = x\otimes \1_{{\rm
B}(\Phi)}$, $ x \in \cA$.

\begin{ass} \label{ass1}
Let $\{ U_{s,t} : s \le t\}$ be a unitary evolution and $\cA$ a von Neumann algebra acting on $\ch$.
We assume that the following properties hold: for any $0\le s \le t$,
\begin{enumerate}
\item[(a)] $U_{s,t}$ maps $\cH_{[s,t]}$ into itself, and acts as follow
on decomposed vectors $u_{s]}\phi_{[s,t]} v_{[t}$ ($u_{s]} \in \Phi_s, \, \phi_{[s,t]} \in \cH_{[s,t]}, \,
 v_{[t} \in \Phi_{[t},\, \otimes $ omitted)
$$
U_{s,t} (u_{s]} \phi_{[s,t]} v_{[t} ) = u_{s]} (U_{s,t} \phi_{[s,t]}) v_{[t},
$$
\item[(b)]  $\Theta_s U_{t-s} = U_{s,t} \Theta_s,$
\item[(c)] for any $x \in \cA$ and $y\in \cA'$,
$$
[J_t(x), y] =0,
$$
where $[A,B] := AB-BA, \, A,B \in {\rm B}(\cH)$.
\end{enumerate}
\end{ass}
One notice that Assumption \ref{ass1} (a) implies that $U_{s,t}$ is $t$-adapted (to the past)
and $s$-adapted (to the future) respectively \cite{Mey}. The semigroup property in (\ref{(2)}) and
Assumption \ref{ass1} (a)-(b) imply that $U_t$ satisfies the left cocycle \cite{Mey}.

A family $\{M_{s,t} : s \le t\}$ of bounded operators on $\cH$ is called a {\it multiplier} (or {\it cocycle} )
for the unitary evolution $\{U_{s,t} : s\le t\}$ if the following conditions are satisfied:
\begin{eqnarray}
&(a)& M_{s,t} \,\,{ \rm is \,\, strongly \,\, continuous \,\, in } \,\, t \,\, {\rm for \,\, fixed } \,\, s, \nonumber \\
&(b)& M_{s,s} = \1_{{\rm B}(\cH)} \,\, {\rm for \,\, all} \,\, s \in \R_+, \label{(4)}\\
&(c)& U_s M_{s,t} U_s^* M_s = M_t \,\, {\rm for}\,\, s \le t, \nonumber
\end{eqnarray}
where $M_t := M_{0,t}$.

\begin{ass} \label{ass2}
Let $\{M_{s,t} \}$ be a multiplier ( cocycle ) for the unitary evolution $\{U_{s,t} ; s\le t \}$.
We assume that the following properties hold: for $0\le s\le t$,
\begin{enumerate}
\item[(a)] $M_{s,t}$ maps $\cH_{[s}$ into $\cH_{[s}$ and acts as follow on decomposed vectors
$u_{s]} \phi_{[s} $ ($ u_{s]} \in \Phi_s$, $\phi_{[s} \in \cH_{[s}$, $\otimes$ omitted)
$$
M_{s,t} (u_{s]} \phi_{[s} ) = u_{s]} M_{s,t} \phi_{[s},
$$
\item[(b)] $\Theta_s M_{t-s} = M_{s,t} \Theta_s,$
\item[(c)] $\nE_t M_{s,t} = M_{s,t} \nE_t$,
\item[(d)] for any $y \in \cA'$, $[M_{s,t}, y]=0$,
\item[(e)] there is a continuous function $K: \R_+ \to \R_+ $ such that $\| M_t \| \le K(t)$.
\end{enumerate}
\end{ass}
We remark that $M_{s,t}$ is $s$-adapted, but not necessary $t$-adapted.
Instead of $t$-adaptedness of $M_{s,t}$, we have imposed the property (c).

 The following is the main result in this section.
\begin{thm} \label{thm1}
Let $\cA$ be a von Neumann algebra acting on $\ch$. Let $\{M_{s,t} : s \le t\}$ be a multiplier
(cocycle) for given unitary evolution $\{ U_{s,t} : s \le t\}$. For any $x \in \cA$  and $0 \le t$, define
\begin{equation}\label{(5)}
P_t(x) \varphi := \nE_0 [ M_t^* J_t(x) M_t \varphi  ], \quad \varphi \in \ch \subset \cH.
\end{equation}
Under Assumption \ref{ass1} and Assumption \ref{ass2}, the following properties hold:
\begin{enumerate}
\item[(a)] $\{P_t \}_{t\ge 0}$ is a completely positive preserving semigroup on $\cA$.
\item[(b)] reality preserving : $P_t(x^*) = P_t(x)^*$ for any $x \in \cA$ and $t \ge 0$.
\item[(c)] The semigroup $\{P_t \}_{t \ge 0}$ is weak*-continuous.
\end{enumerate}
\end{thm}
\begin{rema}\label{rema1}
The expression (\ref{(5)}) can be considered as a Feynman-Kac representation of the semigroup $\{P_t\}_{t\ge 0}$.
See also Theorem 2.3 of \cite{AFL}.
\end{rema}

\noindent {\bf Proof of Theorem \ref{thm1}}. (a) By Assumption \ref{ass2} (e), $\| P_t(x) \| \le K(t)^2 \| x\|$,
and so $P_t$ is bounded for each $t \in \R_+$. By (\ref{(2)}) (b) and (\ref{(4)}) (b), $P_0(x) =x$ for any $x \in \cA$.
Let us prove the semigroup property.  Notice that by Assumption \ref{ass2} (a)
and (b), $M_{t-s}^* = \Theta_s^* M_{s,t}^* \Theta_s$, and so $\Theta_s M_{t-s}^* = M_{s,t}^* \Theta_s$.
It follows from (\ref{(3)}), Assumption \ref{ass1} (a) -(b) and Assumption \ref{ass2} (a)-(b) that for
any $x \in \cA$, $w_{s]}\1_{[s}  \in \Phi_s \subset \Phi$, $ \varphi \in \ch$ and $s \le t$,
\begin{eqnarray*}
&& M_{s,t}^* J_{s,t}(x) M_{s,t} w_{s]}\1_{[s} \varphi   \\
&& \quad = \left[  M_{s,t}^* J_{s,t}(x) M_{s,t} \Theta_s \varphi \1_{\Phi}\right] w_{s]} \\
&& \quad = \left[ \Theta_s  M_{t-s}^* J_{t-s}(x) M_{t-s}  \varphi  \1_{\Phi}\right] w_{s]},
\end{eqnarray*}
and so by (\ref{(1)})
\begin{eqnarray*}
&& \nE_s  \left[  M_{s,t}^* J_{s,t}(x) M_{s,t} w_{s]}\1_{[s}  \varphi \right] \\
&& \quad =\Theta_s\left[ \nE_0 M_{t-s}^* J_{t-s}(x) M_{t-s}  \varphi \1_{\Phi} \right] w_{s]} \\
&& \quad = (P_{t-s} (x) \varphi ) w_{s]}\1_{[s}.
\end{eqnarray*}
Let us use the abbreviated notation, $\varphi := \varphi \1_{[s}$, etc., for $\varphi \in \ch$.
 Since $\{ w_{s]} \varphi : w_{s]} \in \Phi_s,  \varphi\in \ch \}$ is total in $\cH_s$,  it follows from
 the above relation that for $\varphi \in \ch$
\begin{eqnarray*}
&&  \nE_s  \left[  M_{s,t}^* J_{s,t}(x) M_{s,t} U_s^* M_s   \varphi \right] \\
&& \quad = P_{t-s} (x) U_s^* M_s \varphi.
\end{eqnarray*}
Here we have used the fact that  $U_s^*$ and $M_s$ map $\cH_s$ into $\cH_s$ by Assumption \ref{ass1} (a)
and Assumption \ref{ass2} (c).
We use the cocycle relation (c) in (\ref{(4)}), Assumption \ref{ass1} (a), Assumption \ref{ass2} (c)
and the fact that $U_s^* J_t(x) U_s = J_{s,t}(x) $ to obtain that
\begin{eqnarray*}
P_t(x) \varphi & =& \nE_0 \left[ M_s^* U_s M_{s,t}^* U_s^* J_t(x) U_s M_{s,t} U_s^* M_s \varphi \right] \\
&=& \nE_0 \left\{ M_s^* U_s \nE_s \left[ M_{s,t}^* J_{s,t}(x) M_{s,t} U_s^* M_s \varphi \right] \right\} \\
&=& \nE_0 \left\{ M_s^* U_s P_{t-s} (x) U_s^* M_s \varphi \right \}\\
&=& \nE_0 \left[ M_s^* J_s \left(P_{t-s} (x)\right) M_s \varphi \right] \\
&=& P_s \left( P_{t-s} (x) \right) \varphi.
\end{eqnarray*}
Thus the semigroup property holds. Since $J_t(x)$, $x \in \cA$, and $M_t$ commute with any $y \in \cA'$
by assumption \ref{ass1} (c) and Assumption \ref{ass2} (d), $P_t (x) \in \cA$ for any $x \in \cA$ by (\ref{(5)}).

The completely positive preserving property follows from the definition $P_t$ in (\ref{(5)}). For any $x_1, x_2,
\cdots, x_n \in \cA$, $ y_1, y_2, \cdots , y_n \in \cA$, $ n \in \N$, and $ \varphi \in \ch$,
\begin{eqnarray*}
&&\langle \varphi, \sum_{i,j} y_i^* P_t (x_i^* x_j) y_j \varphi \rangle_\ch \\
&& \quad = \langle \varphi \otimes \1_\Phi, (\sum^n_{i=1} y_i^* M_t^* U_t x_i^*) (\sum_{j=1}^n x_j U_t^* M_t y_j )
(\varphi \otimes \1_\Phi ) \rangle_\cH\\
&& \quad \ge 0.
\end{eqnarray*}

(b) Since $\nE_0 \varphi =\varphi$ for $\varphi \in \ch$, it follows from (\ref{(5)}) that for any $ x \in \cA$
and $t \ge 0$
$$
P_t(x) = \nE_0 M_t^* J_t(x) M_t \nE_0.
$$
Thus (b) follows from the above.

(c) We next consider weak*-continuity of $P_t$. We will use the argument similar to that used in \cite{LS}.
We exploit the coincidence of $\sigma$-weak and weak operator topologies on the bounded subsets of $\cA$ \cite{BR}.
First fix $t\ge 0$ and let $(x_n)$ be  be a sequence in the closed unit ball of $ \cA$, converging in the weak operator
topology to $x$. Then, for $\varphi, \xi \in \ch$,
$$
\langle \varphi, P_t ( x_n) \xi \rangle_\ch =  \langle U_t^* M_t\varphi , (x_n \otimes \1_\Phi ) U_t^* M_t \xi \rangle_\cH.
$$
Since $(x_n \otimes \1_\Phi )$ converge in the weak operator
topology to $x\otimes \1_\Phi$, $P_t(x_n)$ converges in the weak
operator topology to $P_t(x)$. Since $ \| P_t (x_n) \| \le \|
M_t\|^2$, $P_t(x_n)$ converges in the $\sigma$-weak topology. Thus
$\phi \circ P_t$ is weak operator continuous on the closed unit
ball of $\cA$,  and therefore $\sigma$-weakly continuous for each
$\phi \in \cA_*$. Hence each $P_t$ is $\sigma$-weakly continuous.

Next fix $x \in \cA$ and $(t_n)$ be a real positive sequence
converging to $0$. Then for $\varphi, \xi \in \ch$,  we  write
\begin{eqnarray*}
& & \langle \varphi, \{ P_{t_n} (x) - x\} \xi \rangle_\ch \\
& & \quad = \langle M_{t_n} \varphi, J_{t_n} (x) M_{t_n} \xi \rangle_\cH - \langle \varphi, x \xi \rangle_\cH \\
& & \quad = \langle M_{t_n} \varphi- \varphi, J_{t_n} (x) M_{t_n} \xi \rangle_\cH + \langle \varphi, J_{t_n}(x) M_{t_n}\xi \rangle_\cH - \langle \varphi, x \xi \rangle_\cH \\
& & \quad = \langle M_{t_n} \varphi- \varphi, J_{t_n} (x)( M_{t_n} \xi -\xi)\rangle_\cH + \langle M_{t_n}\varphi- \varphi, J_{t_n}(x) \xi \rangle_\cH \\
& & \qquad + \langle \varphi, J_{t_n}(x)(M_{t_n} \xi -\xi)\rangle_\cH +  \langle \varphi, (J_{t_n}(x)-x)\xi\rangle_\cH
\end{eqnarray*}
Notice that $\| J_{t_n} (x) \| \le \|x\| $. Thus the first  three
terms in the above converge  to 0 as $t_n \to 0$ by the continuity
of $M_t$ in (\ref{(4)}). The last term converges to $0$ as $t_n
\to  0$ by the continuity of $U_t$ and $U^*_t$ in (\ref{(2)}).
Hence $t \mapsto P_t(x)$ is weak operator continuous. Since the
family $(P_t(x))$ is locally bounded by Assumption \ref{ass2} (e),
$t \mapsto P_t(x)$ is $\sigma$-weakly continuous. Thus the
semigroup $\{P_t \}_{t\ge0}$ is weak* continuous. $\quad \square$


\section{Semigroups Generated by Noncommutative Elliptic Operators}

Let $(\Omega, \cB, \P)$ be the standard Wiener space. Thus
$\Omega$ is the space of continuous  functions on $[0,\infty)$
which vanish at $0$, $\cB$ is a $\sigma$-algebra and $\P$ is
Wiener measure. The coordinate process $\omega\in \Omega \mapsto
\omega(t) $ is a standard Brownian motion which will be denoted
$B_t$. It generates the Wiener filtration $(\cB_{[s,t]}) _{t\ge s
>0}, $ where $B_{[s,t]}$ is the $\sigma$-algebra generated by
$B_\tau- B_s$, $ s\le \tau \le t$.

The Fock space $\Phi$ over $L^2 (\R_+)$ can be identified with
$L^2 (\Omega, \cB, \P)$ \cite{Mey}.  The space $\Phi_{[s,t]}$, for
instance, is $L^2(\cB_{[s,t]})$. The shift map on $L^2(\Omega)$
satisfies the relation
\begin{equation}\label{3.1}
\theta_t B_s = B_{s+t} - B_t.
\end{equation}
With the above identification, we will use the notation in Section 2. If necessary, we identify
 elements of $\cH = \ch \otimes L^2 (\Omega, \cB, \P)$ as functions on $\Omega$: $\cH = L^2(\Omega, \ch)$.

Let $(\alpha_t: t \in \R)$ be a weak*-continuous group of
*-automorphisms of von Neumann  algebra $\cA$ acting on the
initial Hilbert space $\ch$. We assume that the automorphism is
implemented, $\alpha_t(a) = \Tilde{U_t} a \Tilde{U_t}^*$ by a
strongly continuous one parameter group of unitary operators on
$\ch$. As in \cite{LS}, we randomize $\Tilde{U_t}$ and extend to
$\cH$ by continuous linear extension of the prescription
\begin{equation} \label{3.2}
U_t : v \otimes \varphi \mapsto v \Tilde{U}_{B_t} \varphi, \quad v \in \Phi,\, \varphi \in \ch.
\end{equation}
Let $\{ U_{s,t} : s \le t\}$ be the family of unitary operators on
$\cH$, where for decomposed  vectors $u_s \varphi v_{[s} $ $( u_s
\in \Phi_s, \varphi \in \ch, v_{[s} \in \Phi_{[s})$, $ U_{s,t}$
acts as
\begin{equation} \label{3.3}
U_{s,t} u_s \varphi v_{[s} = u_s (\Tilde{U}_{B_t- B_s} \varphi ) v_{[s}.
\end{equation}
It follows from (\ref{(3)}) that for any $x \in \cA$
\begin{equation} \label{3.4}
J_{s,t}(x) = \alpha_{B_t- B_s} (x).
\end{equation}
We recall that for decomposed vectors $\varphi \otimes v$ $(\varphi \in \ch, v \in \Phi)$
\begin{eqnarray*}
\Theta_t &:& \varphi \otimes v \mapsto \varphi \otimes \theta_t v , \\
\nE_t &:&  \varphi \otimes  v \mapsto \varphi \otimes \E[ v | \cB_{[0,t ]}],
\end{eqnarray*}
where $\E$ is the Wiener expectation. One notice that $\theta_t v
= v \circ \gamma_t$, where $\gamma_t$ is the shift on paths
defined by $(\gamma_t \omega )(s) = \omega(t+s) - \omega(t) $. See
(\ref{3.1}).

\begin{lemm} \label{lemm3.1}
The family  $\{ U_{s,t} : s\le t\}$ of unitary operators is a
unitary evolution satisfying the properties in (\ref{(2)}).
Moreover the properties (a) -(c) in Assumption \ref{ass1} also
hold.
\end{lemm}

\begin{pf}
The strong continuity of $U_{s,t}$ and $U_{s,t}^*$ can be proved by using the spectral theorem
for the infinitesimal generator of $\Tilde{U}_t$ and an explicit computation on Wiener integral.
We leave the details to the reader. The properties (b) and (c) in (\ref{(2)}) follows from (\ref{3.3}).

Next, we consider the properties in Assumption \ref{ass1}. Notice that for
any $\varphi \in \ch$, $\Tilde{U}_{B_t-B_s} \varphi \in \cH_{[s,t]}$. Thus the properties (a) and (b) in
Assumption \ref{ass1} follow from (\ref{3.3}) and (\ref{3.1}). Since $\alpha_t (\cA) \subset \cA$,
it follows from (\ref{3.4}) that for any $x \in \cA$ and $y \in \cA'$, $ J_{s,t}(x) $ and $y$ commute.
This completes the proof.
\end{pf}

Next we are looking for the multipliers (cocycles) for the unitary
evolution $ \{ U_{s,t} : s\le t \} $ corresponding to
noncommutative elliptic operators on $\cA$. In the following, we
identify $\cH$ and $\cH_s $ with $L^2 (\Omega : \ch)$ and $L^2
(\cB_{[0,s]} : \ch)$ respectively : $\cH= L^2(\Omega : \ch)$ and
$\cH_s =L^2 (\cB_{[0,s]} : \ch)$, $ s \in \R_+$.  We also denote
$L^{\infty} (\cB_{[0,s]} : \cA)$ by $\cM_s $. An $\ch$-valued
stochastic process $(f_t )$  said to be {\it adapted} if $f_t \in
L^2 (\cB_{[0,t]} : \ch)$ for each $t \in \R_+$ and $t \mapsto \|
f_t \| _\cH$ is measurable.

\begin{prop} \label{prop3.1}
For $ 0\le s \le t$, $ a \in \cM_s $, $ \xi \in \cH$, the stochastic integral equation
\begin{equation}\label{3.5}
f(a, \xi ; s, t) = \xi + \int^t_s J_{s, \tau} (a) \left[\nE_\tau f(a, \xi; s, \tau)\right]\, dB_\tau -\frac12 \int^t _s J_{s,\tau}
(a^2)\left[\nE_\tau f(a, \xi; s, \tau)\right] \,d\tau
\end{equation}
has a unique solution in $\cH$, where the first integral is an Ito
integral for a function taking values in $\ch$, and the second
integral is a Bochner integral defined pointwise for each path.
\end{prop}

\begin{pf}
We use the method similar to that employed in the proof of
Proposition 2.1 of \cite{LS}.  For given $\xi \in \cH$ and $ a \in
\cM_s$, put
\begin{equation} \label{3.6}
f^{(0)}_{s,t} : = \xi.
\end{equation}
As an inductive hypothesis, assume that for $n \ge 1$ the recursive procedure
\begin{equation} \label{3.7}
 f^{(n)}_{s,t} : = \int^t_s J_{s,\tau} (a) \left[ \nE_\tau f^{(n-1)} _{s,\tau}\right] dB_\tau - \frac 12 \int^t_s J_{s,\tau}(a^2) \left[ \nE_\tau f^{(n-1)}_{s,\tau}\right] d\tau,
\end{equation}
defines $t$-adapted, $\ch$-valued processes $(f_{s,t}^{(n)} )_{t\ge s}$ satisfying
\begin{equation} \label{3.8}
\| f^{(n)}_{s,t} \|^2_\cH   \le \frac 1 {n!}
\big[\,4  m(\|a \|^2)\, m (t-s) \,\big]^n \|\xi \|^2,
\end{equation}
where
\begin{eqnarray*}
\| a\| &:=& \sup _{\omega \in \Omega} \| a(\omega) \|_\cA, \\
m(t) &:=& max\{ t, t^2 \}, \quad t \in \R_+.
\end{eqnarray*}
We note that for fixed $s \in \R_+ $, $ J_{s, \tau}(a) \in \cM_\tau$ for any $a \in \cM_s$ and $\nE_\tau f_{s,\tau} ^{(n-1)} \in \cH_\tau$ , $ n \ge 1$. The inductive hypothesis (\ref{3.8}) is clearly satisfied when $n=0$. It is also easy to show that $(f^{(1)} _{s,t} )_{t\ge s}$ is $t$-adapted.
See the proof below.

Assume that the bound (\ref{3.8}) holds for $n=k$. It follows from the Ito isometry and the H\"{o}lder inequality that
\begin{eqnarray}
&& \| f^{(k+1)}_{s,t} \|^2_\cH = \E \big[\, \| f^{(k+1)}_{s,t} \|^2_\ch \,\big] \nonumber \\
\quad &&\le  \E \left[\, 2 \| \int^t_s J_{s,\tau} (a) (\nE_\tau f^{(k)}_{s,\tau})\, dB_\tau \|^2_\ch + 2 \| \frac12 \int^t_s J_{s,\tau} (a^2) \nE_\tau f^{(k)}_{s,\tau} \,d\tau \|^2_\ch \,\right]\nonumber \\
\quad && \le  2 \int^t_s \E \big[\, \|  J_{s,\tau} (a) \nE_\tau f^{(k)}_{s,\tau} \|^2_\ch\,\big] \, d\tau + (t-s) \int^t_s \E \big[\, \| J_{s,\tau}(a^2) \nE_\tau f_{s,\tau} ^{(k)} \|^2_\ch \,\big] \,d\tau. \qquad \qquad \label{3.9}
\end{eqnarray}
Consider the case for $t-s <1$. By (\ref{3.8}), the above is bounded by
\begin{eqnarray*}
 &&2 \|a\|^2 \|\xi\|^2 \big(4m(\|a \|^2)\big)^k \int^t_s  \frac {(\tau-s )^k } {k!}\, d\tau
 + \|a\|^4 \|\xi \|^2\big(4m(\|a\|^2)\big)^k  \int ^t_s \frac {(\tau-s)^k } {k!} d\tau \\
&& \quad \le \frac {\big(4m(\|a\|^2)\big)^{k+1} (t-s)^{k+1}} {(k+1)!} \| \xi \|^2
\end{eqnarray*}
If $t-s \ge 1$, by (\ref{3.8}) the expression (\ref{3.9}) is bounded by
\begin{eqnarray*}
 &&2 \|a\|^2 \|\xi\|^2 \big(4m(\|a \|^2)\big)^k \int_{s}^t  \frac {(\tau-s )^{2k} } {k!}\, d\tau \\ &&\qquad \quad
+ \|a\|^4 \|\xi \|^2 \big(4m(\|a \|^2)\big)^k (t-s) \int_{s}^t  \frac {(\tau-s )^{2k} } {k!} \,d\tau  \\
&&\le \frac {\big(4m(\|a \|^2)(t-s)^2\big)^{k+1}} {(k+1)!} \|\xi \|^2.
\end{eqnarray*}
Therefore $(f_{s,t}^{(k+1)} : t \ge s) $ is well defined,
$t$-adapted  process satisfying (\ref{3.8}). Thus the recursive
procedure is justified, and
\begin{equation*}
\sum_{n \ge 0} \| f^{(n)}_{s,t} \| \le \| \xi \| \sum_{n\ge 0}
\frac 1 {\sqrt{n!}} \big[4 m(\|a\|^2) m (t-s)  \big]^{n/2} < \infty.
\end{equation*}
Hence $\sum_{n\ge 0} f^{(n)}_{s,t}$ defined an $\ch$-valued process $(f_{s,t}: t \ge s)$.  Since for each $N \ge 1$,
$$
\sum_{n=0}^N f^{(n)}_{s,t} = \xi + \int^t_s J_{s,\tau} (a) \nE_\tau \Big(\sum_{n=0}^{N-1}
f^{(n)} _{s,\tau}\Big ) \,dB_\tau - \frac 12 \int^t_s J_{s,\tau}(a^2) \nE_\tau \Big(
\sum_{n=0}^{N-1} f^{(n)}_{s,\tau}\Big)\, d\tau,
$$
$(f_{s,t} )$ satisfies the equation (\ref{3.5}).

In order to show the uniqueness,
let $(g_t)$ be any solution of (\ref{3.5}). Then
\begin{eqnarray*}
(g_t- \xi) &=&  \int^t_s J_{s,\tau} (a) \nE_\tau g _{\tau}\, dB_\tau - \frac 12 \int^t_s J_{s,\tau}(a^2) \nE_\tau g_{\tau} \,d\tau \\
&=& \int^t_s J_{s,\tau} (a) \nE_\tau \xi \,dB_\tau +\int^t_s J_{s,\tau} (a) \nE_\tau ( g _{\tau}-\xi) \,dB_\tau  \\
&&- \frac 12 \int^t_s J_{s,\tau}(a^2) \nE_\tau \xi \, d\tau - \frac 12 \int^t_s J_{s,\tau}(a^2) \nE_\tau (g_{\tau} -\xi)\, d\tau.
\end{eqnarray*}
We use the triangle inequality, the Ito isometry and the H\"{o}lder inequality to obtain
\begin{eqnarray*}
&& \E [\, \| g_t - \xi \|^2_\ch\, ] \\&&\le  4 \|a \|^2 (t-s)  \| \xi \|^2  + 4\| a\|^2 \int^t_s \E [\,\| g_\tau -\xi \|^2_\ch \,]\,d\tau \\
&& \qquad + 2(t-s)^2 \|a\|^4  \|\xi \|^2 + 2(t-s) \|a\|^4 \int_s^t \E[\, \|g_\tau -\xi \|^2_\ch\,]\, d\tau \\
&&\le  8 m(\|a\|^2)m(t-s) \|\xi \|^2  + (4+2(t-s)) m(\|a\|^2) \int^t_s \E[\, \|g_\tau -\xi \|^2_\ch\,]\, d\tau.
\end{eqnarray*}
Applying the Gronwall Lemma  leads to the bound
\begin{eqnarray} \nonumber
\| g_t -\xi \|^2_\cH& =&\E[\|g_t -\xi \|^2_\ch]  \\
&\le& 8m(\|a\|^2)m(t-s) \| \xi \|^2\, exp \big\{ 8 m(\|a\|^2)m(t-s) \big\},\label{3.10}
\end{eqnarray}
Now let $h_t =f_{s,t} - g_t$. By the  Ito isometry and the H\"{o}lder inequality,
\begin{eqnarray*}
\E[\,\|h_t \|^2_\ch\,] & \le & 2 \int_s^t \E [\,\| J_{s,\tau} (a) \nE_\tau h_\tau  \|_\ch^2\,]\, d\tau
 + (t-s) \int_{s}^t \E [\,\| J_{s,\tau} (a^2) \nE_\tau h_\tau \|^2_\ch\,]\, d\tau\\
&\le& (2+(t-s)) m(\|a\|^2) \int^t_s \E[\,\| h_\tau \|^2_\ch\,]\, d\tau.
\end{eqnarray*}
Iterating this relation and applying the bound (\ref{3.10}) to
the estimate $\|h_{t_n} \|^2_\ch \le 2 \{ \| f_{s,t_n} -\xi
\|^2_\ch + \|g_{t_n} -\xi\|^2_\ch \}$ gives
\begin{eqnarray*}
\E[\, \|h_t \|^2_\ch\,] &\le & [(2+(t-s))m(\|a\|^2) ]^n \int^t_s \cdots \int^{t_{n-1}}_s
  32m(\|a\|^2)m(t_{n}-s)\\
&& \quad \quad \cdot \|\xi \|^2 exp\big\{ 8m(\|a\|^2) m(t_{n}-s)) \big\}\, dt_{n} \cdots dt_1 \\[0.3cm]
&\le &  [(2+(t-s))m(\|a\|^2) ]^n 32 m(\|a\|^2) m(t-s) \|\xi \|^2 \\[0.2cm]
&& \quad \quad \cdot exp\big\{ 8m(\|a\|^2) m(t-s)) \big\}  \frac {(t-s)^n}{n!}
\end{eqnarray*}
for any $n \in \N$. Hence $h_t =0$ and so $g_t=f_{s,t}$ on $\cH$.
\end{pf}

We now cast Proposition \ref{prop3.1} in terms of operators.
Notice that by (\ref{3.5}) -(\ref{3.7}), $\xi \mapsto f(a,\xi;
s,t)$ is linear. Thus for each $a \in \cM_s$ and $0\le s\le t$,
let us define
\begin{eqnarray} \label{3.11}
M^a_{s,t} &:& \cH \to \cH \\
&& \xi \mapsto f(a, \xi ; s,t). \nonumber
\end{eqnarray}
By Proposition \ref{prop3.1} and the bound (\ref{3.10}), $M^a_{s,t}$ is a bounded operator on $\cH$.

\begin{prop} \label{prop3.2}
(a) For $ a \in \cA$, the family of bounded operators $M_{s,t}^a$, $s \le t$, defined as in (\ref{3.11}) satisfies the stochastic differential equation
\begin{eqnarray}\label{3.12*}
d_t M_{s,t}^a  &=& J_{s,t} (a) \nE_t M_{s,t}^a dB_t - \frac 12 J_{s,t}(a^2) \nE_t M_{s,t} ^a dt, \\
M^a_{s,s} &=& \1_{{\rm B}(\cH)} \nonumber
\end{eqnarray}
on $\cH$ in the strong operator sense.

(b) For $a \in \cA$, the operators $M_{s,t}^a$, $s\le t$, are
strongly continuous in $t$ for fixed $s$,  and satisfy the cocycle
relation
$$
M_t^a = U_s M_{s,t}^a U_s^* M_s^a
$$
for $0\le s \le t$, where $M_t^a = M_{0,t}^a$, $t\ge 0$.
\end{prop}
\begin{rema} \label{rema3.1*}
The expression (\ref{3.12*}) is somewhat misreading. The correct
expression would be  the following : for $\xi \in \cH$
\begin{eqnarray*}
d_t M_{s,t}^a \xi  &=& J_{s,t} (a) [\,\nE_t M_{s,t}^a\xi \,]\, dB_t - \frac 12 J_{s,t}(a^2) [\,\nE_t M_{s,t} ^a \xi ]\, dt, \\
M^a_{s,s}\xi  &=& \xi.
\end{eqnarray*}
For a notational brevity, we will use the expression (\ref{3.12*})
in the sequel.
\end{rema}

\noindent{\it Proof of Proposition \ref{prop3.2} } (a) This is a consequence of Proposition \ref{prop3.1} and
the definition of $M_{s,t}^a $ in (\ref{3.11}).

(b) Define $\{g_{r, t}: t \ge r\} $ by
$$
g_{r, t} = \left\{ \begin{array}{lc}
             f(a,\xi;r ,t) \quad & {\rm for}\,\, r \le t \le s \\
             M^{J_{r, s}(a)}_{s,t} f(a, \xi ; r,s) \quad &{\rm for}\,\, t \ge s.
              \end{array}
       \right.
$$
Since $J_{s,\tau}(J_{r,s}(a))= J_{r, \tau}(a) $ for $ r \le s \le
\tau$, $(g_{r, t}) $ satisfies  the same stochastic integral
equation as $(M_{r,t}^a \xi )$ and so
\begin{equation} \label{3.13}
M_{r,t}^a = M_{s,t}^{J_{r,s}(a)} M_{r,s}^a.
\end{equation}
Thus it follows from (\ref{3.13}) and (\ref{3.10}) that for $r \le s \le t$
\begin{eqnarray*}
\|M_{r,t}^a - M_{r,s}^a \| &\le & \|M_{s,t}^{J_{r,s}(a)} - \1 \|\, \|M_{r,s}^a \| \\
&\le & C_1 m(t-s)^{1/2} exp \{ C_2 m(t-s)  + C_3 m(s-r)\}
\end{eqnarray*}
for some constants $C_1, C_2 $ and $C_3$ depending only on
$\|a\|$. Hence $M_{s,t}^a$, $s\le t$,  are norm-continuous and so
strongly continuous.

For $a\in \cA$ and $\xi \in \cH$, define $(g_t: t \ge 0) $ by
$$
g_t = \left\{ \begin{array}{lc}
             f(a,\xi; 0,t) \quad & { \rm for}\,\,0 \le t \le s \\
             U_s M^a_{s,t} U_s^* f(a, \xi ; 0,s) \quad & { \rm for}\,\, t \ge s
              \end{array}
       \right.
$$
Since $U_s \nE_\tau = \nE_\tau U_s$ for $\tau \ge s$, we have
\begin{eqnarray*}
g_t &=& U_s \Big\{ U_s^* f(a, \xi ; 0,s) + \int^t_s J_{s,\tau} (a) \left(\nE_\tau M^a_{s,\tau} U^*_s f(a, \xi ; 0,s )\right)\, dB_\tau \\
&&-\frac12 \int^t_s J_{s,\tau}(a^2) \nE_\tau M^a_{s,\tau} U^*_s f(a,\xi ;0,s) \,d\tau \Big\} \\
&=&  f(a, \xi ; 0,s) + \int^t_s J_{\tau} (a)\left(\nE_\tau U_s  M^a_{s,\tau} U^*_s f(a, \xi ; 0,s ) \right)\,dB_\tau \\
&&-\frac12 \int^t_s J_{\tau}(a^2) \nE_\tau U_s M^a_{s,\tau} U^*_s f(a,\xi;0,s) \,d\tau  \\
&=& \xi + \int^s_0 J_\tau (a) \nE_\tau f(a, \xi ;0,\tau)\, dB_\tau -\frac12 \int^s_0 J_\tau(a^2) \nE_\tau f(a,\xi; 0,\tau) \,d\tau  \\
 && + \int^t_s J_{\tau} (a) (\nE_\tau g_\tau) \,dB_\tau
 -\frac12 \int^t_s J_{\tau}(a^2) \nE_\tau g_\tau\, d\tau  \\
&=&  \xi + \int^t_0 J_\tau(a) (\nE_\tau g_\tau )\,dB_\tau -\frac12 \int^t_0 J_\tau (a^2) \nE_\tau g_\tau \,d\tau.
\end{eqnarray*}
Hence, by the uniqueness for this stochastic integral equation
(Proposition \ref{prop3.1}),   $g_t =f(a, \xi; 0,t) = M^a_t \xi$.
This proved the cocycle relation. $\quad \square$

\vspace{0.2cm} Proposition \ref{prop3.2} implies that for any
$a\in \cA$, $\{M_{s,t}^a : s \le t\}$ is  a multiplier ( cocycle )
for the unitary evolution $\{U_{s,t} : s\le t\}$ defined in
(\ref{3.3}).

\begin{prop} \label{prop3.3}
For any $a \in \cA$, the multiplier ( cocycle ) $\{M_{s,t}^a : s
\le t\}$ for the  unitary evolution $\{U_{s,t} : s\le t\}$
satisfies the properties (a)-(e) in Assumption \ref{ass2}.
\end{prop}
\begin{pf}
For $a \in \cA$ and $0 \le s\le t$, let $\{M_{s,t}^{(n)} \}_{n
\ge0}$ be the family of  bounded operators on $\cH$ defined by
$\xi \mapsto f^{(n)}_{s,t}$, where for $\xi \in \cH$, $\{
f^{(n)}_{s,t}\}$ is the family of vectors in $\cH$ defined
recursively in (\ref{3.7}). The results in the proof of
Proposition \ref{prop3.2} imply that the family $\{M_{s,t}^{(n)}
\}$ satisfies the recursive relation (strong operator sense)
\begin{eqnarray}
M^{(0)}_{s,t} &=& \1_{\cH},  \nonumber \\
M^{(n)}_{s,t}  &=& \int^t_s J_{s,t}(a) \nE_\tau M_{s,\tau}^{(n-1)} \, dB_\tau - \frac12 \int^t_s J_{s,t}(a^2) \nE_\tau M_{s,\tau}^{(n-1)} \, dt, \label{3.14}
\end{eqnarray}
and
\begin{equation} \label{3.15}
M_{s,t}^a = \sum^\infty_{n=0} M_{s,t} ^{(n)}.
\end{equation}
It is clear that $\{M_{s,t}^{(0)} : s\le t\}$ satisfies (a) - (d)
in Assumption \ref{ass2}.  As an inductive hypothesis, assume that
for given $n \ge 1$, $\{M_{s,t}^{(n-1)} : s\le t\}$ satisfies the
properties (a) -(d) in Assumption \ref{ass2}.

For $s \le \tau \le t$, $\nE_\tau M_{s,t}^{(n-1)}$ maps $\cH_{[s}$
into $\cH_{[s}$ and acts  as follow on decomposed vectors $u_s
\varphi_{[s}$ ($u_s \in \Phi_s,\, \varphi_{[s} \in \cH_{[s}$)
$$
\nE_\tau M_{s,t}^{(n-1)} (u_s \varphi_{[s}) = u_s \nE_\tau M_{s,t}^{(n-1)} \varphi_{[s},
$$
and so it follows from (\ref{3.14}) and (\ref{3.15}) that
$M_{s,t}^a$ satisfies the property (a) in  Assumption \ref{ass2}.
By the inductive hypothesis and (\ref{(1)}), we have
\begin{eqnarray*}
J_{s,t}(a) \nE_\tau M_{s,\tau}^{(n-1)} \Theta_s &=& J_{s,\tau} (a) \Theta_s \nE_{\tau-s} M_{\tau-s}^{(n-1)} \\
&=& \Theta_s J_{\tau-s}(a) \nE_{\tau-s} M_{\tau-s}^{(n-1)}.
\end{eqnarray*}
Since $(B_\tau - B_s) \Theta_s  =  \Theta_s B_{\tau-s}$ by taking
the adjoint of the relation of (\ref{3.1}),  it follows from
(\ref{3.14}) and the above relation that
\begin{eqnarray*}
M^{(n)}_{s,t} \Theta_s &=& \Theta_s \left\{ \int^{t-s}_0 J_\tau (a) \nE_\tau M_\tau^{(n-1)} \,dB_\tau - \frac12 \int^{t-s}_s J_\tau (a^2) \nE_\tau M_\tau^{(n-1)} \, d\tau \right\} \\
&=& \Theta_s M_{t-s}^{(n)}.
\end{eqnarray*}
Thus by (\ref{3.15}), $\{M_{s,t}^a\}$ satisfies the property (b) in Assumption \ref{ass2}.

Since $\{ M_{s,t}^{(n-1)} ; s\le t\}$ satisfies the properties (c)
and (d) by the induction  hypothesis, $\{M_{s,t} ^a : s\le t \}$
also satisfies the properties by (\ref{3.14}) and (\ref{3.15}).
The property (e) in Assumption \ref{ass2} holds by the bound
(\ref{3.10}). This completes the proof of the proposition.
\end{pf}

The following is the main result in this section:
\begin{thm} \label{thm3.1}
(a) Let $a\in \cA$. If the weak*-continuous automorphism group
$\{\alpha_t\}$ is unitarily  implemented, then $(P^a_t)$ defined
by
\begin{equation}\label{3.16}
(P^a_t(x) ) \varphi = \nE_0 \left[ (M_t^a)^* J_t(x) M_t^a \varphi \right], \quad \varphi \in \ch,
\end{equation}
is a weak*-continuous, completely positive preserving semigroup on $\cA$.

(b) Let $a=a^* \in \cA$. The semigroup $(P_t^a)$ satisfies the following properties : for any $t \ge 0$
\begin{enumerate}
\item[(i)] identity preserving : $P_t^a (\1) = \1$.
\item[(ii)] reality preserving : $P^a_t (x^*) = (P_t^a (x))^*$, $x \in \cA$,
\item[(iii)] Markovianity : $0\le x \le \1 \implies 0 \le P^a_t(x) \le \1 $, $ x\in \cA$.
\item[(iv)] contractivity : $\| P_t \| \le 1$, $ t \ge 0$.
\end{enumerate}
Thus $(P_t^a)$ is a Markovian semigroup in the sense of \cite{DL} and \cite{AMV}.

(c) The weak*-generator of $(P_t^a)$ is an extension of the operator $G^a$ defined by
\begin{eqnarray}
D(G^a) & =& D(\delta^2), \nonumber \\
G^a & = &\frac 12 \delta^2 + L_{a^*} \delta+ R_a \delta + L_{a^*} R_a -\frac12 (L_{a^*})^2
-\frac 12(R_a)^2 \label{3.17}
\end{eqnarray}
where $\delta$ is the generator of the automorphism $(\alpha_t)$,
and $L_a$ and $R_a $ are  the left and right multiplication
operators respectively : $L_a x= ax$ , $R_a x= xa$, $x \in \cA$.
\end{thm}
\begin{pf}
(a) This follows as a corollary of Theorem \ref{thm1}, Lemma \ref{lemm3.1} and Propositions \ref{prop3.2} and \ref{prop3.3}. (b) (i) follows from the part (c) stated below, i.e., $G^a (\1)=0$ if $a=a^* \in \cA$. (ii) holds by Theorem \ref{thm1} (b) for any $a \in \cA$. (iii) follows from (i) and the positivity of ($P_t^a$). The positivity and the identity preserving property of $(P_t^a)$ imply the contractivity (iv). See Corollary 3.2.6 of \cite{BR}. Thus it remains to prove the part (c).

(c) The following result is Proposition 1.3 of \cite{LS}. Let $x \in D(\delta^2)$ and  $\varphi \in \ch$. Then
\begin{equation} \label{3.18}
J_{s,t}(x) \varphi = x \varphi + \int^t_s J_{s,\tau} (\delta x) \varphi \, dB_\tau +\frac 12 \int^t_s J_{s, \tau } (\delta^2 x) \varphi \, d\tau.
\end{equation}
By the Ito product formula for Hilbert space valued Ito integrals,
(\ref{3.18}) and Proposition \ref{prop3.2} (a)  combine to give
\begin{eqnarray} \label{3.19}
J_t(x) M_t^a \varphi &=& x \varphi + \int^t_0 J_\tau (\delta x + x a) M_\tau ^a \varphi \, dB_\tau \\
&& + \int^t_0 J_\tau (\frac12 \delta^2 x + (\delta x) a -\frac 12 x a^2) M_\tau^a \varphi \,d\tau. \nonumber
\end{eqnarray}
Here we have used the fact that $\nE_\tau M_\tau^a \varphi =
M_\tau^a \varphi$ ( Proposition \ref{prop3.3} and  Assumption
\ref{ass2} (c)). We use the Ito product formula, Proposition
\ref{prop3.2} (a) and (\ref{3.19})  to obtain that for any
$\varphi_1, \varphi_2 \in \ch$
\begin{eqnarray*}
&& \langle M_t^a\varphi_1,  J_t(x) M_t^a \varphi_2\rangle_\ch \\
&&= \langle \varphi_1, x\varphi_2 \rangle_\ch + \int^t_0 \langle M_\tau^a\varphi_1, J_\tau (\delta x + a^* x + xa) M_\tau^a \varphi_2 \rangle_\ch \, dB_\tau \\
&& \quad  + \int^t_0 \langle M_\tau^a\varphi_1,  J_\tau (\frac 12 \delta^2 x+ a^* \delta x +(\delta x) a + a^* x a -\frac 12 (a^*)^2 x -\frac 12 x a^2 ) M_\tau^a \varphi_2 \rangle_\ch \, d\tau.
\end{eqnarray*}
By taking the Wiener expectation of the above expression,  we have that
\begin{eqnarray*}
P_t^a (x) \varphi &= &x \varphi + \int^t_0 P_\tau^a (G^a(x)) \varphi \, d\tau, \\
G^a(x) &= &\frac 12 \delta^2 (x) + a^* \delta(x) + \delta(x) a + a^* xa - \frac 12 (a^*)^2 x-\frac 12 x a^2,
\end{eqnarray*}
and so the generator $(P_t^a)$ is a weak*-closed extension of
$G^a$. This completes the proof.
\end{pf}


\section{Properties of Noncommutative Elliptic Operators}

Let $\delta$ be a densely defined *-derivation on $\cA$   such
that $\1 \in D(\delta)$ and $\delta(\1)=0$. For given self-adjoint
element $a $ of $\cA$ ($a=a^*\in\cA$), define and operator $G^a $
of $\cA$ by
\begin{eqnarray}
D(G^a)& = &D(\delta^2), \nonumber \\
G^a(x) &=& \frac12 \delta^2(x) + a\delta(x) +\delta(x) a - \frac 12 [\,a,[a, x]\,], \quad x \in D(G^a). \label{4.1}
\end{eqnarray}
Recall that for $a =a^* \in \cA$, the pre-generator (\ref{3.17})
of the  semigroup $(P_t^a)$ has the above expression.  Since $G^a$
can be considered as a noncommutative version of classical
elliptic operator $\frac 12 \Delta + \beta\circ \nabla$, where
$\nabla$ and $\Delta$ are the gradient and Laplacian operators on
$L^2(\R^n)$ respectively, and $\beta$ is a $\R^n$- valued function
on $\R^n$, it would be worth to study detail properties of $G^a$.

\begin{thm}\label{thm4.1}
Let $\delta$ be a densely defined *-derivation on $\cA$ such that
$\1\in D(\delta)$ and $\delta(\1) =0$.  For given $a=a^* \in \cA$,
let $G^a$ be the operator on $\cA$ defined as in (\ref{4.1}). Then
the following properties hold:
\begin{enumerate}
\item[(a)] $\1 \in D(G^a)$ and $G^a (\1) =0$,
\item[(b)] $G^a(x^*) = (G^a(x))^*$, $ x\in D(G^a)$,
\item[(c)] complete dissipationity : the matrix inequality
$$
[G^a (x_i^* x_j) ] \ge [ G^a (x_i^*) x_j + x_i^* G^a(x_j)]
$$
holds for any finite sequence $\{ x_1, x_2, \cdots , x_n\}$ in $ D(G^a)$.
\end{enumerate}
\end{thm}
\begin{pf}
It is clear that the properties (a) and (b) hold. Let us prove (c). Put
$$
\delta_a (x) := i [a,x], \quad x \in \cA.
$$
Then $\delta_a$ is a *-derivation on $\cA$ since $a=a^* \in \cA$. $G^a(x) $ can be written as
\begin{equation} \label{4.2}
G^a(x) = \frac12 \delta^2 (x) + a\delta(x) + \delta(x) a + \frac 12 \delta^2_a (x).
\end{equation}
Since $\delta$ and $\delta_a$ are derivations on $\cA$, we get
that for any finite  sequence $\{x_1, x_2, \cdots , x_n \}$ in $
D(G^a)$
\begin{eqnarray*}
\delta(x_i^* x_j) & =& \delta(x_i^*) x_j + x_i^* \delta(x_j), \\
\delta_a(x_i^* x_j) & =& \delta_a(x_i^*) x_j + x_i^* \delta_a(x_j),
\end{eqnarray*}
and
\begin{eqnarray*}
\delta^2(x_i^* x_j) &=& \delta^2 (x_i^*) x_j + 2 \delta(x_i^*) \delta(x_j) + x_i^* \delta^2(x_j), \\
\delta^2_a(x_i^* x_j) &=& \delta^2_a (x_i^*) x_j + 2 \delta_a(x_i^*) \delta_a(x_j) + x_i^* \delta^2_a(x_j),
\end{eqnarray*}
and so it follows from (\ref{4.2}) and the above equalities that
\begin{eqnarray*}
&&G^a(x_i^* x_j) - G^a(x_i^*) x_j - x_i^* G^a(x_j)  \\
&& \quad = \delta(x_i^*) [x_j , a] + [a,x_i^*]\delta(x_j) + \delta(x_i^*)\delta(x_j) + \delta_a(x_i^*) \delta_a(x_j) \\
&& \quad = \{ \delta(x_i^*) -i \delta_a(x_i^*)\} \{ \delta(x_j) + i\delta_a(x_j) \}\\
&& \quad = (\delta(x_i ) + i\delta_a(x_i))^* (\delta(x_j) + i \delta_a(x_j)).
\end{eqnarray*}
The complete dissipationity of $G^a$ follows from the above equality.
\end{pf}

Theorem \ref{thm4.1} implies that, if $G^a$ is closable in an
appropriate topology and if a closed extension of $G^a$ generates
a (positive preserving) semigroup $(P_t^a)$, then $(P_t^a)$ is
completely positive, and preserves identity and reality
\cite{Brat, BR}. For instance. if $\delta$ is a bounded
*-derivation on $\cA$, then $G^a$ is bounded and so $G^a$
generates a strongly continuous semigroup on $\cA$ and the
properties mentioned above hold. See Corollary 3.2.1 of
\cite{Brat}. The main result (Theorem \ref{thm3.1}) in Section 3
conformed that if $\delta$ generates a weak*-continuous
one-parameter group of *-automorphisms of $\cA$, a closed
extension of $G^a$ generates a weak*-continuous semigroup. It
would be interest to know conditions on $\delta$ and $a \in \cA$
under which $G^a$ is closable and a closed extension generates a
semigroup \cite{BR,Kat}.

\vspace{0.2cm} \noindent {\bf Acknowledgment} : This work was
supported in  part by Korean Research Foundation (99-005-D00010),
Korean Ministry of Education.

\begin{thebibliography}{99}
\bibitem[AFL]{AFL}  L.  Accardi, A. Frigerio, and J.T. Lewis,  Quantum stochastic processes, {\it Publ. RIMS} {\bf 18}, 97-133 (1982).
\bibitem[AMV]{AMV} L. Accardi, A. Mohari and C. V. Volterra, On the structure of classical and quantum flows, {\it J. Funct. Anal.} {\bf 135}, 421-455 (1996).
\bibitem[BR]{BR}  O.  Bratteli and D.W. Robinson,
 {\it  Operator algebras  and quantum  statistical mechanics},
 Springer-Verlag, New  York-Heidelberg-Berlin, Vol I (1979), Vol. II (1981).
\bibitem[Bra]{Brat} O. Bratteli, Derivations, Dissipations and Group Actions on $C^*$-algebras, {\it Lecture Notes in Mathematics}, Vol. 1229,  Springer-verlag, (1986).
\bibitem[DL]{DL}  E.B.  Davies and J.M. Lindsay,
Non commutative symmetric Markov semigroups, {\it Math. Zeit.}
 {\bf 210}, 379-411 (1992).
\bibitem[Fag]{Fag} F. Fagnola, Quantum stochastic differential equations and Feynmann-Kac pertubation of quantum evolution in {\it Quantum probability and related topics VI} (L. Accardi Ed) World Scientific, Singapore, 267-284 (1991).
\bibitem[HIP]{HIP}  R. L. Hudson, P. D. F. Ion and K. R. Parthasarathy, Time orthogonal unitary dilations and non-commutative Feynmann-Kac formula, {\it Comm. Math. Phys.} {\bf 83}, 261-280 (1982).
\bibitem[Kat]{Kat} T. Kato, {\it Perturbation theory for linear operators}, Springer-Verlag, Berlin-New York, (1980).
\bibitem[LS]{LS} J. M. Lindsay and K. B. Sinha,  Feynman-Kac representation of some noncommutative elliptic operators, {\it J. Funct. Anal. } {\bf 147}, 400-419 (1997).
\bibitem[LW]{LW} J. M. Lindsay and S. J. Wills, Markovian cocycles on operator algebras adapted to a Fock filtration, {\it J. Funct. Anal.} {\bf 178}, 269-305 (2000).
\bibitem[Mey]{Mey} P.-A. Meyer, {\it Quantum probability for probabilists}, Lecture Notes in Mathematics, vol. 1538, Springer-Verlag, Berlin, New York, (1993).
\bibitem[Par]{Par}  K.R.  Parthasarathy,  {\it An Introduction to Quantum
Stochastic Calculus}, Birkh\"auser, Basel (1992).
\bibitem[PS]{PS} K. R. Parthasarathy and K. B. Sinha, A stochastic Dyson series expansion {\it Lecture Notes in Control and Inform. Sci.} Vol. 49, 227-232 (1983).
\bibitem[RS]{RS}  M.  Reed and B. Simon,  {\it Method of modern mathematical
physics I, II: Fourier analysis, Self-adjointness}, Academic Press, New York (1975).
\end{thebibliography}

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