---
abstract: |
    In this paper we prove the existence of conditional expectations in the noncommutative $L_p(M,\Phi)$ spaces associated with center-valued traces. Moreover, their description is also provided. As an application of the obtained results, we establish the norm convergence of the martingales in noncommutative $L_p(M,\Phi)$ spaces. 0.3cm

    [*Keywords:*]{} Noncommutative $L_p(M,\Phi)$-space, Banach-Kantorovich space; measurable; conditional expectation; measurable bundle; martingale\
    [*AMS Subject Classification:*]{} 46L52; 46L53;
address:
- 'Department of Science in Engineering, Faculty of Engineering, International Islamic University Malaysia, P.O. Box 10, 50728 Kuala-Lumpur, Malaysia'
- |
    Farrukh Mukhamedov\
    Department of Computational & Theoretical Sciences\
    Faculty of Science, International Islamic University Malaysia\
    P.O. Box, 141, 25710, Kuantan\
    Pahang, Malaysia
author:
- Inomjon Ganiev
- Farrukh Mukhamedov
title: 'Conditionally expectations and martingales in noncommutative $L_p$-spaces associated with center-valued traces'
---

Introduction
============

One of important nations of probability theory is a conditionally expectation operators, which plays general roll theory martingales, ergodic theory and et al. Existence and basic properties of conditional expectation operators provided in many books of functional analysis and the theory of probability.

To the existence of conditional expectations on operator algebras was first studied by Umegaki’s works [@U1; @U2]. Furthermore, the existence criterion was provided in [@Ta]. Then the martingale convergence theory for conditional expectations in von Neumann algebras has been developed (see for example, [@DN; @La; @Ts]). One of the important arias in the theory of operator algebras, is the integration theory for traces and weights defined on von Neumann algebras (se for example [@S; @N; @Y]). In this directions, analogous of $L_p$-spaces have been investigated. In [@CP; @G; @HT] the martingale convergence in non-commutative $L_p$-spaces was obtained.

On the other hand, development of the theory of integration for measures $\mu$ with the values in Dedekind complete Riesz spaces has inspired the study of (bo)-complete lattice-normed $L_p$-spaces (see, for example, [@K2]). The existence of center-valued traces on finite von Neumann algebras naturally leads to develop the theory of integration for this kind of traces. In [@GaC] non-commutative $L_p$-spaces assiciated with with central-valued traces have been investigated. In [@CK] an abstract characterization of these spaces have been provided. Going further, in [@CZ] more general $L_p$-spaces associated with Maharam traces have been studied.

Therefore, main aim of this paper to investigate the existence of conditional expectations in this $L_p$-spaces. Note that first vector-valued analog existence conditional expectations was given in [@Kus] for commutative von Neumann algebras. It is known [@GaC] that $L_p$-spaces associated with central-valued traces are Banach-Kantorovich spaces. The theory of Banach-Kantorovich spaces is now sufficiently well-developed (for instance, see [@Kus; @K2]). The methods of Boolean valued analysis play an important role in the theory. Using these methods, the Banach-Kantorovich lattices and the corresponding homomorphisms can be interpreted as the Banach lattice and bounded linear operators within a suitable Boolean valued model of set theory [@Kus]. This approach to the theory of Banach-Kantorovich lattices makes ut possible to use the transfer principle [@Kus] for obtaining various properties of Banach-Kantorovich lattices that are analogous of the corresponding properties of classical Banach lattices. Naturally, when using such a method, an additional study is needed for establishing the required interrelation between the objects of 2-valued and Boolean valued models of set theory.

Another important approach to study Banach-Kantorovich spaces is provided by the theory of continuous and measurable Banach bundles [@G0]-[@G2]. In this approach the representation of a Banach-Kantorovich lattice as a space of measurable sections of a measurable Banach bundle makes it possible to obtain the needed properties of the lattice by means of the corresponding stalkwise verification of the properties. As an application of this approach, in [@Ga] it has been given a representation of the conditional expectations as measurable bundle of classical conditional expectations, and proved martingale convergence theorems. Moreover, noncommutative $L_p(M,\Phi)$-spaces associated with center-valued traces are represented as bundle of noncommutative $L_p$-spaces associated with numerical traces [@GaC]. In [@CK] an abstract characterization of these spaces are provided. Certain other extensions of these $L_p$-spaces have been investigated in [@CZ]. For other applications of the mentioned method, we refer the reader to [@AAK; @CGa],[@GM]-[@GM4].

In the present paper, we mainly employ the last mentioned approach to establish the existence of conditional expectations on noncommutative $L_p(M,\Phi)$-spaces associated with center-valued traces. Moreover, we describe such expectations and as an application of the obtained results, we prove the norm convergence of martingales in noncommutative $L_p(M,\Phi)$-spaces. We note that the obtained results open new perspective in the field of martingale convergence.

Preliminaries
=============

In this section we recall necessary notions and facts which will be used in the next sections.

Let $(\Omega,\Sigma,\mu)$ be a measurable space with finite measure $\mu$, and $L_0(\Omega)$ be the algebra of all measurable functions on $\Omega$ (as usual, a.e. equal functions are identified). By ${\mathcal{L}}^{\infty}(\Omega)$ we denote the set of all measurable essentially bounded functions on $\Omega$, and $L^{\infty}(\Omega)$ denote an algebra of equivalence classes of essentially bounded measurable functions.

Let $\mathcal{U}$ be a linear space over the real field $\mathbb{R}$. By $\|\cdot\|$ we denote a $L_0(\Omega)$-valued norm on $U$. Then the pair $(\mathcal{U},\|\cdot\|)$ is called a [*lattice-normed space (LNS) over $L_0(\Omega)$*]{}. An LNS $\mathcal{U}$ is said to be [*$d$-decomposable*]{} if for every $x\in \mathcal{U}$ and the decomposition $\|x\|=f+g$ with $f$ and $g$ disjoint positive elements in $L_0(\Omega)$ there exist $y,z\in \mathcal{U}$ such that $x=y+z$ with $\|y\|=f$, $\|z\|=g$.

Suppose that $(\mathcal{U},\|\cdot\|)$ is an LNS over $L_0(\Omega)$. A net $\{x_\alpha\}$ of elements of $\mathcal{U}$ is said to be [*$(bo)$-converging*]{} to $x\in \mathcal{U}$ (in this case, we write $x=(bo)$-$\lim x_\alpha$), if the net $\{\|x_\alpha - x\|\}$ $(o)$-converges to zero in $L_0(\Omega)$ (written as $(o)$-$\lim \|x_\alpha -x\|=0$). A net $\{x_\alpha\}_{\alpha\in A}$ is called [*$(bo)$-fundamental*]{} if $(x_\alpha-x_\beta)_{(\alpha,\beta)\in A\times A}$ $(bo)$-converges to zero.

An LNS in which every $(bo)$-fundamental net $(bo)$-converges is called [*$(bo)$-complete*]{}. A [*Banach-Kantorovich space (BKS) over $L_0(\Omega)$*]{} is a $(bo)$-complete $d$-decomposable LNS over $L_0(\Omega)$. It is well known [@G1; @G2] that every BKS $\mathcal{U}$ over $L_0(\Omega)$ admits an $L_0(\Omega)$-module structure such that $\|fx\|=|f|\cdot\|x\|$ for every $x\in \mathcal{U},\ f\in L_0(\Omega)$, where $|f|$ is the modulus of a function $f\in L_0(\Omega)$.

A set $B \subset \mathcal{U}$ is called *bounded*, if set $\{\|x\|: x \in B\}$ is order bounded in $ L_{0}$. An operator $T: \mathcal{U} \rightarrow \mathcal{U}$ is called $L_{0}$-linear, if $T(\alpha x + \beta y)= \alpha T(x) + \beta T(y)$ for all $\alpha,\beta \in L_{0}$ and $x,y \in \mathcal{U}$. An $L_{0}$-linear operator $T$ is called $ L_{0}$–bounded, if for any bounded set $B$ in $\mathcal{U}$, the set $T (B)$ bounded in $\mathcal{U}$. For an $L_{0}$-bounded operator $T$ we put $$\|T\|=\sup\{\|T(x)\|: \|x\|\leq \bf{1}\},$$ where $\bf{1}$ - identity element in $L_{0}$

Let $H$ be a Hilbert space, let $B(H)$ be the $*$-algebra of all bounded linear operators on $H,$ and let $I$ be the identity operator on $H.$ Given a von Neumann algebra $M$ acting on $H,$ denote by $Z(M)$ the center of $M$ and by $P(M)$ the lattice of all projections in $M$. Let $P_{fin}(M)$ be the set of all finite projections in $M.$ A densely-defined closed linear operator $x$ (possibly unbounded) affiliated with $M$ is said to be *measurable* if there exists a sequence $\{p_n\}_{n=1}^{\infty}\subset P(M)$ such that $p_n\uparrow
\mathbf{1}$,  $p_n(H)\subset \mathfrak{D}(x)$ and $p_n^\bot=\mathbf{1}-p_n \in P_{fin}(M) $ for every $n=1,2,\ldots$ (here $\mathfrak{D}(x)$ is the domain of $x$). Let us denote by $S(M)$ the set of all measurable operators. If $M$ is a commutative von Neumann algebra, it is $*$-isomorphic to the $*$-algebra $L_\infty(\Omega,\Sigma,\mu)$ of all essentially bounded complex measurable functions with the identification almost everywhere, where $(\Omega,\Sigma,\mu)$ is a measurable space. In addition $S(M)\cong L_0(\Omega,\Sigma,\mu)$ [@S].

Let $x,y$ be measurable operators. Then $x+y,~xy$ and $x^*$ are densely-defined and preclosed. Moreover, the closures $\overline{x+y}$ (strong sum), $\overline{xy}$ (strong product) and $x^*$ are also measurable, and $S(M)$ is a $*$-algebra with respect to the strong sum, strong product, and the adjoint operation (see [@S]). For any subset $E\subset S(M)$ we denote by $E_{sa}$ (resp. $ E_+$ ) the set of all self-adjoint (resp. positive ) operators from $E.$

For $x\in S(M)$ let $x=u|x|$ be the polar decomposition, where $|x|=(x^*x)^{\frac{1}{2}},$ $u$ is a partial isometry in $B(H).$ Then $u\in M$ and $|x|\in S(M).$ If $x\in S_h(M)$ and $\{E_\lambda(x)\}$ are the spectral projections of $x,$ then $\{E_\lambda(x)\}\subset P(M).$

The locally measure topology $t(M)$ on $L_0(\Omega,\Sigma,\mu)$ is by definition the linear (Hausdorff) topology whose fundamental system of neighborhoods of $0$ is given by $$W(B,\varepsilon,\delta)=\{f\in\ L_0(\Omega,\, \Sigma,\, \mu)
\colon \hbox{ there exists a set } \ E\in \Sigma, \mbox{ such
that}$$ $$\ E\subseteq B, \ \mu(B\setminus E)\leq\delta, \
f\chi_E \in L^\infty(\Omega,\Sigma,\mu), \
\|f\chi_E\|_{{L_\infty}(\Omega,\Sigma,\mu)}\leq\varepsilon\}.$$ Here  $\varepsilon, \ \delta $ run over all strictly positive numbers and $B\in\Sigma$,  $\mu(B)<\infty.$ It is known that $(S(M),t(M))$ is a complete topological $*$-algebra. Note that a net $\{f_\alpha\}$ converges locally in measure to $f$ (notation: $f_\alpha \stackrel{t(M)}{\longrightarrow}f$) if and only if $f_\alpha \chi_B $ converges in $\mu$-measure to $f\chi_B$ for each $B\in\Sigma$ with $\mu(B)<\infty$.

Let $M$ be any finite von Neumann algebra, $S(M)$ be the set all measurable operators affiliated to $M$. Let $Z$ be some subalgebra of the center $Z(M)$. Then one may identify $Z$ with $*$-algebra $L_\infty(\Omega,\Sigma,m)$ and do $S(Z)$ with $L_0(\Omega,\Sigma,m)$. Recall that *a center valued (i.e. $Z$-valued) trace* on the von Neumann algebra $M$ is a $Z$-linear mapping $\Phi:M\to Z$ with $\Phi(x^*x)=\Phi(xx^*)\geq 0$ for all $x\in M$. It is clear that $\Phi(M_+)\subset Z_+$. A trace $\Phi$ is said to be *faithful* if the equality $\Phi(x^*x)=0$ implies $x=0$, *normal* if $\Phi(x_{\alpha})\uparrow\Phi(x)$ for every $x_{\alpha},x\in
M_{sa}$, $x_{\alpha}\uparrow x$. Note that the existence of such kind of traces has been studied in [@CZ1].

Let now $M$ be an arbitrary finite von Neumann algebra, $\Phi$ be a center-valued trace on $M$. The locally measure topology $t(M)$ on $S(M)$ is the linear (Hausdorff) topology whose fundamental system of neighborhoods of $0$ is given by $$V(B,\varepsilon,
\delta ) = \{x\in S(M)\colon \ \mbox{there exists } \ p\in P(M),
z\in P(Z(M))$$ $$\mbox{ such that} \ xp\in M,
\|xp\|_{M}\leq\varepsilon, \ z^\bot \in W(B,\varepsilon,\delta), \
\Phi_M(zp^\bot)\leq\varepsilon z\},$$ where $\|\cdot\|_{M}$ is the $C^*$-norm in $M.$ It is known that $(S(M),t(M))$ is a complete topological $*$-algebra [@Y].

From [@Mur_m §3.5] we have the following criterion for convergence in the topology $t(M). $

\[2.1.\] A net $\{x_\alpha\}_{\alpha\in A} \subset S(M)$ converges to zero in the topology $t(M)$ if and only if $\Phi_M(E^\bot_\lambda (|x_\alpha|) \stackrel{t(M)}{\longrightarrow} 0$ for any $\lambda>0.$

Following [@CZ] an operator $x\in S(M)$ is said to be [*$\Phi$-integrable*]{} if there exists a sequence $\{x_n\} \subset M$ such that $x_n \stackrel{t(M)}{\to} x $ and $\|x_n-x_m\|_\Phi
\stackrel{t(Z)}{\longrightarrow} 0$ as $n,m \to \infty.$

Let $x$ be a $\Phi$-integrable operator from $ S(M).$ Then there exists a $\widehat{\Phi}(x)\in S(Z)$ such that $\Phi(x_n)
\stackrel{t(Z)}{\longrightarrow}\widehat{\Phi}(x).$ In addition $\widehat{\Phi}(x)$ does not depend on the choice of a sequence $\{x_n\}\subset M,$ for which $x_n\stackrel{t(M)}{\longrightarrow}x,$ $\Phi(|x_n-x_m|)
\stackrel{t(Z)}{\longrightarrow}0$ [@CZ]. It is clear that each operator $x\in M$ is $\Phi$-integrable and $\widehat{\Phi}
(x)=\Phi(x).$

Denote by $L_1(M,\Phi)$ the set of all $\Phi$-integrable operators from $S(M).$ If $x\in S(M)$ then $x\in L_1(M,\Phi)$ iff $|x|\in
L_1(M,\Phi),$ in addition $|\widehat{\Phi}(x)|\leq
\widehat{\Phi}(|x|)$ [@CZ1]. For any $x\in L_1(M,\Phi),$ set $\|x\|_{1,\Phi}=\widehat{\Phi}(|x|).$ It is known that $L_1(M,\Phi)$ is a linear subspace of $S(M),$ $ML_1(M,\Phi)M
\subset L_1(M,\Phi),$ and $x^*\in L_1(M,\Phi)$ for all $x\in
L_1(M,\Phi)$ [@CZ1]. Let $\Phi$ be a faithful normal trace on $M$ with values in $L_0(\Omega)$ (associated with $Z$). Following [@CZ] for any $p > 1$, we set $L_p(M,\Phi) = \{x \in S(M) :
\|x\|_{p,\Phi} \in L_1(M,\Phi)\}$ and $\|x\|_{p,\Phi} =
\widehat{\Phi}(|x|^p)^{\frac{1}{p}}$.

[@CZ1; @CZ] \[1.1\] Let $x,y\in S(M)$. Then the following statements hold:

1.  $ \|x\|_{1,\Phi}=\sup\{|\Phi(xy)|: \|y\|_M\leq 1\}.$

2.  for $p>1$, one has $\|x\|_{p,\Phi}=\sup\{|\Phi(xy)|:
    \|y\|_{q,\Phi}\leq \mathbf{1}\},$ where $\frac{1}{p}+\frac{1}{q}=1.$

In the early 90’s of the last century A.E. Gutman [@G1] introduced the measurable Banach bundles with lifting axioms. It was established that every Banach–Kantorovich space over the ring of measurable functions can be expressed as measurable bundle of Banach spaces. Now let us recall some notions from this approach.

Let $X$ be a mapping which maps every point $\omega\in \Omega$ to some Banach space $(X(\omega),\|\cdot\|_{X(\omega)})$. In what follows, we assume that $X(\omega)\neq \{0\}$ for all $\omega\in \Omega.$ A function $u$ is said to be a *section* of $X$, if it is defined almost everywhere in $\Omega$ and takes its value $u(\omega)\in X(\omega)$ for $\omega\in dom(u),$ where $\omega\in dom(u)$ is the domain of $u.$ Let $L$ be some set of sections.

[@G1]. A pair $(X, L)$ is said to be a [*measurable bundle of Banach spaces*]{} over $\Omega$ if

1.  $\lambda_1 c_1+\lambda_2 c_2\in L$ for all $\lambda_1, \lambda_2\in \mathbb{R}$ and $c_1, c_2\in L,$ where $\lambda_1 c_1+\lambda_2 c_2:\omega\in dom(c_1)\cap
       dom(c_2) \rightarrow \lambda_1 c_1(\omega)+\lambda_2 c_2(\omega);$

2.  the function $||c||:\omega\in  dom(c)\rightarrow
    ||c(\omega)||_{X(\omega)}$ is measurable for all $c\in L;$

3.  for every $\omega\in \Omega$ the set $\{c(\omega): c\in L, \omega\in dom(c)\}$ is dense in $X(\omega).$

A section $s$ is a step-section, if there are pairwise disjoint sets $A_1,A_2,\ldots,A_n\in\Sigma$ and sections $c_1,c_2,\ldots,c_n\in L$ such that $\bigcup\limits_{i=1}^n
 A_i=\Omega$ è $s(\omega)=\sum\limits_{i=1}^n
 \chi_{A_i}(\omega)c_i(\omega)$ for almost all $\omega\in\Omega$.

A section $u$ is measurable, if for any $A\in\Sigma$ there is a sequence $s_n$ of step-sections such that $s_n(\omega)\rightarrow u(\omega)$ for almost all $\omega\in A$.

Let $M(\Omega,X)$ be the set of all measurable sections. By symbol $L_0(\Omega,X)$ we denote factorization of the $M(\Omega,X)$ with respect to almost everywhere equality. Usually, by $\hat{u}$ we denote a class from $L_0(\Omega,X)$, containing a section $u\in M(\Omega,X)$, and by $\|\hat{u}\|$ we denote an element of $L_0(\Omega)$, containing $\|u(\omega)\|_{X(\omega)}$. Let ${\mathcal L^{\infty}}(\Omega,X)=\{u\in
M(\Omega,X):\|u(\omega)\|_{X(\omega)}\in \mathcal
L^{\infty}(\Omega)\}$ and $L^{\infty}(\Omega,X)=\{\widehat{u}\in
L_0(\Omega,X): \|\widehat{u}\|\in L^{\infty}(\Omega)\}.$ One can define the spaces $\mathcal {L^{\infty}}(\Omega,X)$ and $L^{\infty}(\Omega,X)$ with real-valued norms $\|u\|_{\mathcal
L^{\infty}(\Omega,X)}=\sup\limits_{\omega\in
\Omega}|u(\omega)|_{X(\omega)}$ and $\|\widehat{u}\|_{\infty}=\bigg\|\|\widehat{u}\|\bigg\|_{L^{\infty}(\Omega)},$ respectively.

Let $X,Y$ be measurable bundles of Banach spaces. A set linear operators $\{T(\omega) : X(\omega)\rightarrow
 Y(\omega)\}$ is called *measurable bundle of linear operators* if $T(\omega)(u(\omega))$ is measurable section for any measurable section $u$.

Let $(X,L)$ be a measurable bundle of Banach spaces. If each $X(\omega)$ is a noncommutative $L_p$-space, i.e. $X(\omega)=L_p(M(\omega),\tau_\omega)$, associated with finite von Neumann algebras $M(\omega)$ and with strictly normal numerical trace $\tau_\omega$ on $M(\omega)$, then the measurable bundle $(X,L)$ of Banach spaces is called *measurable bundle of noncommutative $L_p$-spaces*.

\[1.2\][@GaC] There exists a measurable bundle $(X,L)$ of noncommutative $L_p$-spaces $L_p(M(\omega),\tau_\omega)$, such that $L_0(\Omega,X)$ is Banach —Kantorovich $*$-algebroid, which is isometrically and order $*$-isomorph to $L_p(M,\Phi)$. Moreover, the isometric and order $*$-isomorphism $H: L_p(M,\Phi)\rightarrow L_0(\Omega,X)$ can be chosen with the following propertires\

1.  $\Phi(x)(\omega) = \tau_\omega(H(x)(\omega))$ for all $x\in M$ and for almost all $\omega\in\Omega$;\

2.  $x\in M$ if and only if $H(x)(\omega)\in
     M(\omega)$ a.e. and there exist positive number $\lambda>0$, that $\|H(x)(\omega)\|_{M(\omega)}\leq\lambda$ for almost all $\omega$;\

3.  $z\in Z$ if and only if $H(z)=(\widehat{z(\omega)\mathbf{1}_\omega)}$ for some $\widehat{z(\omega)}\in L_\infty(\Omega)$, where $\mathbf{1}_\omega$ — unit of algebra $M(\omega)$, in particular $H(\mathbf{1})(\omega)=
     \mathbf{1}_\omega$ for almost all $\omega$.\

4.  the section $(H(x)(\omega))^*$ is measurable for all $x\in L_p(M,\Phi)$.\

5.  the section $H(x)(\omega)\cdot H(y)(\omega)$ is measurable for all $x,y\in M$.

The existence of vector-valued lifting
======================================

In this section we establishes the existence of the lifting in a non-commutative setting. Note that in the case of $C^*$-algebras, the existence of the lifting has been given in [@GC1] (see also [@GM3]).

Let $M$ be a von Neumann algebra. Then it can be identify with a linear subspace of $L^\infty(\Omega,X)$ by the isomorphism $H$, since if $x\in M$, then one has $$\|H(x)\|_{L_0(\Omega,X)}=\|x\|_1=\Phi(|x|)\in L^\infty(\Omega)$$

\[1.3\] There exists a mapping $\ell: M(\subset
L^\infty(\Omega,X))\rightarrow \mathcal{L}^\infty(\Omega,X)$ with following properties

1.  for every $x\in M $ one has $\ell(x)\in x,\ {\rm dom}\ \ell (x)=\Omega$;\

2.  if $x_1, x_2\in M$ and $\lambda_1, \lambda_2\in
    \mathbb{R}$, then $\ell(\lambda_1 x_1+ \lambda_2 x_2)=\lambda_1\ell(x_1)+\lambda_2\ell(x_2)$;\

3.  $\|\ell(x)(\omega)\|_{L_p(M(\omega),\tau_\omega)}=p(\|x\|_p)(\omega)$ for all $x\in M$ and for all $\omega\in\Omega$;\

4.  if $x\in M,\ \lambda\in
     L^\infty(\Omega)$, then $\ell(ex)=p(e)\ell(x)$;\

5.  if $x\in M$, then $\ell(x^*)=\ell(x)^*$;\

6.  if $x,y\in M$, then $\ell(xy)=\ell(x)\ell(y)$;\

7.  the set $\{\ell(x)(\omega) : x\in M\}$ is dense in $L_p(M(\omega),\tau_\omega)$ for all $\omega\in\Omega$.

Following [@GaC] for every $x\in M$ we define $$\Phi_0(x)=\Phi(x)(1+\Phi(\id))^{-1}.$$ One can see that $\Phi_0$ is an $L^{\infty}(\Omega)$-valued faithful normal trace on $M$. By $\rho$ we denote the lifting on $L^{\infty}(\Omega)$ (see [@G0]). Now define a finite trace $\varphi_\omega$ on $M$ by $\varphi_\omega(x)=\rho(\Phi_0(x))(\omega)$, where $\omega\in\Omega$. Due to [@Dik Lemma 6.4.1] the function $s_\omega(x,y)=\varphi_\omega(y^*x)$ is a bi-trace on $M$, and therefore, the equality $\langle x,y\rangle_\omega=s_\omega(x,y)$ defines a quasi-inner product on $M$.

Denote $I_\omega=\{x\in M: \ s_\omega(x,x)=0\}$. It is known that $I_\omega$ is a two-sided ideal in $M$, therefore, one considers the quotient space $\Gamma_\omega=M/I_\omega$, by $\pi_\omega:M\to\Gamma_\omega$ we denote the canonical mapping. The involution and multiplication are defined on $\G_\w$ by the usual way, i.e. $\pi_\w(x)^*=\pi_\w(x^*)$ and $\pi_\w(x)\cdot\pi_\w(y)=\pi_\w(xy)$. According to [@Dik Proposition 6.2.3] $\G_\w$ is a Hilbert algebra. By $\ch(\w)$ we denote the Hilbert space which is the completion of $\G_\w$; the inner product in $\ch(\w)$ we denote by the same symbol, i.e. $\langle\cdot,\cdot\rangle_\w$.

The mapping $\pi_\w(x)\to \pi_\w(y)\pi_\w(x)$, $x,y\in M$, can be extended by continuity to a bounded linear operator $T_\w(y)$ on $\ch(\w)$. It is known [@Dik] that $T_\w(x)$ is a representation of $M$ in $\ch(\w)$. Let $M(\w)$ be the von Neumann algebra generated by $T_\w(M)$, i.e. $M(\w)=T_\w(M)''$. By $\m_\w$ we denote the natural trace on $M(\w)$ which is defined by $\m_\w(\pi_\w(x))=\langle\pi_\w(x)\id_\w,\id_\w\rangle_\w$ for all $\pi_\w(x)\in\G_\w$. One can see that $\m_\w$ is a faithful, normal and finite trace on $M(\w)$ (see [@Dik Proposition 6.8.3]). Now let us consider a non-commutative $L_1$-space $L_1(M(\w),\t_w)$, where $\t_\w(\cdot)=(1+\Phi(\id))(\w)\m_\w(\cdot)$. By $i_\w:\G_\w\to M(\w)$ one denotes the canonical embedding, and $j_\w:(M(\w),\t_w)\to L_1(M(\w),\t_\w)$ denotes the natural embedding. Then $\g_\w=j_\w\circ i_\w\circ\pi_\w$ is a linear mapping from $M$ to $L_1(M(\w),\t_\w)$.

Let us define $$\ell(x)(\omega)=\gamma_\omega(x)$$ for any $x\in M$.

\(a) Since any element $x\in M$ is identified with the element $\widehat{\gamma_\omega(x)}$, then one has $\ell(x)\in x$ (see [@GaC]).

\(b) The linearity of $\ell$ is obvious.

\(c) Let $x\in M$. Then $$\begin{aligned}
\|\ell(x)(\omega)\|_{L_1(M(\omega),\tau_\omega)}&=&\|\gamma_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}=\tau_\omega(|\pi_\omega(x)|)\\
&=&\tau_\omega(\pi_\omega(|x|))=\rho(\Phi(|x|))(\omega)\\
&=&\rho(\|x\|_1)(\omega)\end{aligned}$$ for all $\omega\in\Omega$.

\(d) Let $\chi_A\in L^\infty(\Omega)$ and $x\in M$, then $\chi_A\cdot x\in M$. By $\tilde{\Sigma}$ we denote a complete Boolean algebra of equivalent classes w.r.t. a.e. equality, of sets from $\Sigma$. The lifting $\rho :
 L^\infty(\Omega)\rightarrow\mathcal{ L}^\infty(\Omega)$ induces a lifting $\tilde{\rho} : \tilde{\Sigma}\rightarrow\Sigma$ such that $p(\chi_A)=\chi_{\tilde{p}(A)}$. Due to $$\begin{aligned}
 \|\pi_\omega(\chi_A x)\|_{L_1(M(\omega),\tau_\omega)} &=&  \rho(\|\chi_A \cdot
 x\|_1)(\omega)=\rho(\chi_A)(\omega)\cdot
 p(\|x\|_1)(\omega)\\
 &=& \rho(\chi_A)(\omega)\cdot\|\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}\\
 &=&
 \chi_{\tilde{p}(A)}(\omega)\|\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)},
 \end{aligned}$$ we obtain $\pi_\omega(\chi_A\cdot x)=0$, if $\omega\in\tilde{\rho}(A)$. Let $\omega\in\tilde{\rho}(A)$, then $$\begin{aligned}
 \|\pi_\omega(\chi_A\cdot
 x)-\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}&=& \|\pi_\omega(\chi_{\Omega\setminus
 A}\cdot x)\|_{L_1(M(\omega),\tau_\omega)}\\
 & =&\rho(\|\chi_{\Omega\setminus A}\cdot
 x\|_1)(\omega)= \rho(\chi_{\Omega\setminus A})(\omega)\cdot
 p(\|x\|_1)(\omega)\\
 &=& \chi_{\tilde{\rho}(\Omega\setminus
 A)}(\omega)\cdot \|\pi_\omega(x)\|_{L_1(M(\omega),\tau_\omega)}=0.
 \end{aligned}$$ Therefore, we have $\pi_\omega(\chi_A \cdot x)= \chi_{\tilde{\rho}(A)}(\omega)
 \cdot\pi_\omega(x)= p(\chi_A)(\omega)\cdot\pi_\omega(x)$ for all $\omega\in\Omega$.

Let $\lambda=\sum\limits_{i=1}^n r_i\chi_{A_i}\in
 L^\infty(\Omega)$ be a simple function. Then $$\begin{aligned}
\pi_\omega(\lambda
 x)&=& \pi_\omega\bigg(\sum\limits_{i=1}^n r_i \chi_{A_i} x\bigg)=
 \sum\limits_{i=1}^n \pi_\omega(r_i \chi_{A_i} x)\\
 &=&
 \sum\limits_{i=1}^n r_i \rho(\chi_{A_i})(\omega)\pi_\omega(x) =
\rho(\lambda)(\omega) \pi_\omega(u).
 \end{aligned}$$

The density argument implies that for any $\lambda\in
L^\infty(\Omega)$ there exists a sequence of simple functions $\{\lambda_n\}$ such that $\|\lambda_n-\lambda\|_{L^\infty(\Omega)}\rightarrow0$ as $n\rightarrow\infty$. From $$\begin{aligned}
 \|\pi_\omega(\lambda_n x)-\pi_\omega(\lambda
 x)\|_{L_1(M(\omega),\tau_\omega)}&=& \|\pi_\omega((\lambda_n - \lambda)x)\|_{L_1(M(\omega),\tau_\omega)} =
\rho(\|(\lambda_n-\lambda)x\|_1)(\omega)\\[2mm]
 &=& \rho(|\lambda_n -
 \lambda|)(\omega)\cdot \rho(\|x\|_1)(\omega)\\[2mm]
 &\leq&
 \|\rho(\lambda_n - \lambda)\|_{\mathcal{ L}^\infty(\Omega)}\cdot
 \|\rho(\|x\|_1)\|_{\mathcal{ L}^\infty(\Omega)}\\[2mm]
 &=&
 \|\lambda_n-\lambda\|_{L^\infty(\Omega)}\cdot \|\|x\|_1 \|_{L^\infty(\Omega)}
 \end{aligned}$$ one gets $\pi_\omega(\lambda
 x)=\lim\limits_{n\rightarrow\infty}\pi_\omega(\lambda_n x)$ for all $\omega\in\Omega$. So, $$\pi_\omega(\lambda
 x)=\lim\limits_{n\to\infty}\pi_\omega(\lambda_n x)=
 \lim\limits_{n\rightarrow\infty} \rho(\lambda_n)(\omega)\pi_\omega(x)=
\rho(\lambda)(\omega)\pi_\omega(u)$$ for all $\omega\in\Omega$.

Hence, $$i_\omega
 (\pi_\omega(\lambda x)=
 i_\omega(\rho(\lambda)(\omega)\pi_\omega(x))=
 \rho(\lambda)(\omega)i_\omega(\pi_\omega(x))= \rho(\lambda)(\omega)
 i_\omega (\pi_\omega(x))$$ and $j_\omega(\lambda x))=\rho(\lambda)(\omega)
 j_\omega (x)$ for all $\omega\in\Omega$. These mean that $\ell(\lambda x)=\rho(\lambda)\ell(x)$ for any $x\in M$ and $\lambda\in
 L^\infty(\Omega)$.

\(e) According to $\gamma_\omega(x^*)=\gamma_\omega(x)^*$ for any $x\in M$ we get $\ell(x^*)=\ell(x)^*$.

\(f) From $\gamma_\omega(xy)=\gamma_\omega(x)\gamma_\omega(y)$ for any $x,y\in
M$ it follows that $\ell(xy)=\ell(x)\ell(y)$ for any $x,y\in M$.

\(g) By the construction of $\gamma_\omega$ the set $\{\gamma_\omega(x): x\in M\}$ is dense in $L_1(M(\omega),\tau_\omega)$ for any $\omega\in\Omega$. Therefore, the set $\{\ell(x)(\omega) : x\in M\}$ is dense in $L_1(M(\omega),\tau_\omega)$ for all $\omega\in\Omega$.

The proof is complete.

The defined map $\ell$ in Theorem \[1.3\] is called *a noncommutative vector-valued lifting associated with the lifting $\rho$*.

Conditional expectations in $L_p(M,\Phi)$
=========================================

In this section we prove the existence of a conditional expectation in noncommutative $L_p(M,\Phi)$-spaces associated with center-valued traces, and give representation such operators as measurable bundle of expectation operators in usual noncommutative $L_p$-spaces associated with numerical traces.

Let $M$ be any finite von Neumann algebra, with unit element $\id$, and $Z$ be some subalgebra of $Z(M)$. Then one can identify $Z=L_\infty(\Omega,\Sigma,m)$ and $S(Z) = L^0(\Omega,\Sigma,m)$. Let $\Phi$ be a strictly normal trace on $M$ with values in $L^0(\Omega)$, associated with $Z$. Let $p\geq 1$ and $L_p(M,\Phi)$ be the noncommutative $L_p$-space, associated with von Neumann algebra $M$ and the trace $\Phi$.

\[Exp1\] Let $M$ and $\Phi$ be as above. Let $N$ be a von Neumann subalgebra of $M$, and the restriction $\Phi$ to $N$ also denoted by $\Phi$. Then there exists a unique map $\mathcal{E}:
M\rightarrow N$ satisfying the following conditions:

1.  $\mathcal{E}$ is a normal positive contraction from $M$ onto $N$;

2.  $\mathcal{E}(axb)=a\mathcal{E}(x)b$ for any $x\in M$ and $a,b\in N;$

3.  $\Phi(\mathcal{E}(x))=\Phi(x)$ for any $x\in M$.

Without loss of generality, we may assume $\Phi(x)\in L^\infty(\Omega)$ (otherwise, we put $\Phi_0(x) =
 \Phi(x)(\id + \Phi(\id))^{-1}$.) Let $\nu$ be a faithful normal trace on $L^\infty(\Omega)$. Then $\tau(x)=\nu(\Phi(x))$ is also a faithful normal numerical trace on $M$. Hence, we can define a faithful normal numerical trace $\tau$ on $M$. Then the existence of $\mathcal{E}$ and properties (i) and (ii) follow from [@P Proposition 2.1].

Let us show (iii). Let $x\in M$. Then by Proposition 2.1. (iii) [@P] we get $\tau(\mathcal{E}(x))=\tau(x).$ By definition of $\tau$ one has $\nu(\Phi((\mathcal{E}(x))=\nu(\Phi(x))$ for any $x\in M$.

Let $\Phi((\mathcal{E}(x))-\Phi(x)=v|\Phi((\mathcal{E}(x))-\Phi(x)|$ be a polar decomposition of $\Phi((\mathcal{E}(x))-\Phi(x)$, for some partial isometry $v\in L^\infty(\Omega)$. Then one finds $$0=\nu(\Phi((\mathcal{E}(vx))-\nu(\Phi(vx))=\nu(v(\Phi((\mathcal{E}(x))-\Phi(x))=\nu(|\Phi(\mathcal{E}(x))-\Phi(x)|).$$

The faithfulness of $\nu$ implies that $\Phi(\mathcal{E}(x))=\Phi(x)$ for any $x\in M$. Similarly, we can show $\Phi(\mathcal{E}(x)y)=\Phi(xy)$ for any $x\in M, y\in N$.

\[Exp2\] The operator $\mathcal{E}$ (from Proposition \[Exp1\]) can be extended to a positive $L_p(M,\Phi)$–contraction from $L_p(M,\Phi)$ onto $L_p(N,\Phi)$, $1\leq p<\infty$. Moreover, if $a,b\in N$ and $x\in L_p(M,\Phi)$ then $$\mathcal{E}(axb)=a\mathcal{E}(x)b.$$

Let $x\in M, y\in N$. Then $|\Phi(\mathcal{\mathcal{E}}(x)y)|=|\Phi(xy)|\leq \Phi(|xy|)
\leq \|y\|_N\|x\|_1$. Hence, according Proposition \[1.1\] (i) we have $$\|\mathcal{E}(x)\|_1=\sup\{|\Phi(\mathcal{E}(x)y)|: \|y\|_M\leq
1\}\leq \|x\|_1.$$

Let $p>1$. By Proposition \[1.1\] (ii) one gets $$|\Phi(\mathcal{\mathcal{E}}(x)y)|=|\Phi(xy)|\leq \Phi(|xy|)\leq \|y\|_q\|x\|_p,$$ where $\frac{1}{p}+\frac{1}{q}=1.$ Then one finds $$\|\mathcal{E}(x)\|_p=\sup\{|\Phi(\mathcal{E}(x)y)|: \|y\|_q\leq
\mathbf{1}\}\leq\|x\|_p.$$

Hence $\|\mathcal{E}(x)\|_p\leq\|x\|_p$, for any $x\in M$, i.e. $\mathcal{E}$ is an $L_p(M,\Phi)$–contraction. Since $M$ is $(bo)$-dense in $L_p(M,\Phi)$, so we can continue $\mathcal{E}$ until $L_p(M,\Phi)$–contraction to $L_p(M,\Phi)$, which is denoted by the same symbol $\mathcal{E}$ (i.e. $\mathcal{E}(x)=(bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(x_n)$, if $ x_n\in M$ and $\|x_n-x\|_p\stackrel{(o)}\rightarrow0.$) It is clear that $\mathcal{E}(\id)=\id$ and $\|\mathcal{E}\|=\mathbf{1}$.

Let us show $\mathcal{E}$ is positive. Let $0\leq x\in L_p(M,\Phi)$. We will chouse $0\leq x_n\in M$ such that $x_n$ is increasing and $\|x_n-x\|_p\stackrel{(o)}\rightarrow0.$ Then the sequence $\{\mathcal{E}(x_n)\}$ is increasing as well, and one has $\mathcal{E}(x)=(bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(x_n)$. Since $\mathcal{E}(x_n)\geq0$ and thw cone $K$ of positive elements of $L_p(M,\Phi)$ is monotone closed (see[@GaC]), we get $\mathcal{E}(x)\geq 0.$

Let $y\in L_p(N,\Phi)$ and $ y_n\in N$ and $\|y_n-y\|_p\stackrel{(o)}\rightarrow0.$ Then $\mathcal{E}(y)=(bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(y_n)=(bo) -
\lim\limits_{n\rightarrow\infty}y_n=y$, i.e. $\mathcal{E}$ is onto.

Now let $a,b\in N$, $x\in L_p(M,\Phi)$ and $x_n\in M$ which $\|x_n-x\|_p\stackrel{(o)}\rightarrow0.$ Then $$\begin{aligned}
\mathcal{E}(axb)&=&(bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(ax_nb)=(bo) -
\lim\limits_{n\rightarrow\infty}a\mathcal{E}(x_n)b\\[2mm]
&=&a((bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(x_n))b=a\mathcal{E}(x)b.\end{aligned}$$

The defined operator $\mathcal{E}: L_p(M,\Phi)\rightarrow
L_p(N,\Phi)$ in Theorem \[Exp2\] is called a *conditional expectation*.

\[2.3\] Let $\mathcal{E}: L_p(M,\Phi)\rightarrow L_p(N,\Phi)$ be the conditional expectation. Then there exists a measurable bundle of conditional expectations $\mathcal{E}_\omega: L_p(M(\omega),\tau_\omega)\rightarrow L_p(N(\omega),\tau_\omega)$ such that $$\mathcal{E}(x)(\omega)= \mathcal{E}_\omega(x(\omega))$$ for all $x\in L_p(M,\Phi)$ and for almost all $\omega\in\Omega$.

Let $\ell: M\rightarrow \mathcal{L}^\infty(\Omega,X)$ be the noncommutative vector-valued lifting associated with lifting $\rho$. Since $\mathcal{E}(M)\subset N$ can define a linear operator $\varphi_\omega: \{\ell(x)(\omega): x\in M\}\rightarrow
 L_p(N(\omega),\tau_\omega)$ by $$\varphi_\omega(\ell(x)(\omega))=\ell(\mathcal{E}(x))(\omega).$$ From the relations $$\begin{aligned}
\|\varphi_\omega(\ell(x)(\omega))\|_{L_p(N(\omega),\tau_\omega)}&=&
 \|\ell(\mathcal{E}(x))(\omega)\|_{L_p(N(\omega),\tau_\omega)}=\rho(\|\mathcal{E}(x)\|_{p})(\omega)\\[2mm]
 &\leq&\rho(\|x\|_p)(\omega)=\|\ell(x)(\omega)\|_{L_p(M(\omega),\tau_\omega)}
 \end{aligned}$$ follow that $\varphi_\omega$ is a well-defined contraction.

Let $\ell(x)(\omega)\geq0$. From the equality $|\ell(x)|=\ell(|x|)$ we get $\ell(x)(\omega)=\ell(|x|)(\omega)$, which yields $$\varphi_\omega(\ell(x)(\omega))=\varphi_\omega(\ell(|x|)(\omega))=\ell(\mathcal{E}(|x|))(\omega)\geq0,$$ i.e. $\varphi_\omega$ is positive.

The density of $\{\ell(x)(\omega): x\in M\}$ in $L_p(M(\omega),\tau_\omega)$ implies that we can continue $\varphi_\omega$ until a linear positive contraction $T_\omega: L_p(M(\omega),\tau_\omega)\rightarrow
  L_p(N(\omega),\tau_\omega)$ by $$T_\omega(x(\omega))=\lim\limits_{n\rightarrow\infty}\varphi_\omega(\ell(x_n)(\omega))=\lim\limits_{n\rightarrow\infty}\ell(\mathcal{E}(x_n))(\omega).$$

Let $(X,L)$ be a measurable bundle of the noncommutative $L_p(M(\omega),\tau_\omega)$-spaces and $(Y,L)$ be a measurable bundle of the noncommutative $L_p(N(\omega),\tau_\omega)$-spaces. Since $\varphi_\omega(\ell(x)(\omega))\in
\mathcal{L}^\infty(\Omega,Y)$, for any $x\in M$, we obtain that $T_\omega(x(\omega))\in M(\Omega, Y)$ for any $x\in M(\Omega, X)$. Therefore, $\{T_\omega\}$ is a measurable bundle of positive contractions.

It is clear that $\|T_\omega\|_{L_p(M(\omega),\tau_\omega)\rightarrow
L_p(N(\omega),\tau_\omega)}=\|\mathcal{E}\|(\omega)$. From $\|\mathcal{E}\|=\mathbf{1}$, one has $\|T_\omega\|_{L_p(M(\omega),\tau_\omega)\rightarrow
L_p(N(\omega),\tau_\omega)}=1$.

Let us show that $T_\omega$ is a conditional expectation from $M(\omega)$ onto $N(\omega)$.

\(i) Let $x(\omega)\in M(\omega)_{sa}(=\{x(\omega)\in M(\omega):
x(\omega)^*=x(\omega)\}).$ Then $|x(\omega)|\leq
\|x(\omega)\|_{M(\omega)}\id(\omega)$ and $$\begin{aligned}
|T_\omega(x(\omega))|&\leq&
T_\omega(|x(\omega)|)\leq\|x(\omega)\|_{M(\omega)}T_\omega(\id(\omega))\\[2mm]
&=&\|x(\omega)\|_{M(\omega)}\id(\omega),\end{aligned}$$ which means $T_\omega(M(\omega))\subset N(\omega)$ and $T_\omega$ is contraction from $M(\omega)$ to $N(\omega)$, here $\id(\omega)=\ell(\id)(\omega)$.

Let $y(\omega)\in N(\omega).$ Then there exists $y_n\in N$ such that $y(\omega)=\lim\limits_{n\rightarrow\infty}\ell(y_n)(\omega).$ Due to $\mathcal{E}(y_n)=y_n$ we get $$\begin{aligned}
T_\omega(y(\omega))&=&\lim\limits_{n\rightarrow\infty}T_\omega(\ell(y_n)(\omega))=
\lim\limits_{n\rightarrow\infty}\ell(\mathcal{E}(y_n))(\omega)\\[2mm]
&=&\lim\limits_{n\rightarrow\infty}\ell(y_n)(\omega))=y(\omega).\end{aligned}$$ This means $T_\omega(M(\omega))=N(\omega)$, i.e. $T_\omega$ maps $M(\omega)$ onto $N(\omega).$

\(ii) Let $a(\omega), b(\omega)\in N(\omega)$ and $x(\omega)\in
M(\omega)$. Then $$T_\omega
(a(\omega)x(\omega)b(\omega))=\lim\limits_{n\rightarrow\infty}\ell(\mathcal{E}(a_nx_nb_n))(\omega)$$ for some $a_n, b_n\in N$ and $x_n\in M$. Due to $\mathcal{E}(a_nx_nb_n))=a_n\mathcal{E}(x_n)b_n$ we find $$\begin{aligned}
T_\omega
(a(\omega)x(\omega)b(\omega))&=&\lim\limits_{n\rightarrow\infty}\ell(a_n)(\omega)\ell(\mathcal{E}(x_n))(\omega)\ell(b_n)(\omega)\\[2mm]
&=&a(\omega)T_\omega (x(\omega))b(\omega).\end{aligned}$$

\(iii) Let $x(\omega)\in M(\omega)$. Then $$\tau_\omega(T_\omega
(x(\omega))=\tau_\omega(\lim\limits_{n\rightarrow\infty}\ell(\mathcal{E}(x_n)(\omega)=\lim\limits_{n\rightarrow\infty}p(\Phi(\mathcal{E}(x_n))(\omega)$$ for some $x_n\in M$. From $\Phi(\mathcal{E}(x_n))=\Phi(x_n)$ one gets $$\tau_\omega(T_\omega
(x(\omega))=\lim\limits_{n\rightarrow\infty}p(\Phi(x_n)(\omega)=\lim\limits_{n\rightarrow\infty}
\tau_\omega(\ell(x_n)(\omega))=\tau_\omega(x(\omega)).$$

Hence, $T_\omega$ is a conditional expectation from $M(\omega)$ onto $N(\omega)$. So, it can be extended to $T_\omega:
L_p(M(\omega),\tau_\omega)\rightarrow L_p(N(\omega),\tau_\omega)$, which is a conditional expectation from $L_p(M(\omega),\tau_\omega)$ onto $L_p(N(\omega),\tau_\omega)$, i.e. $T_\omega=\mathcal{E}_\omega.$

We show that $\mathcal{E}(x)(\omega)= \mathcal{E}_\omega(x(\omega))$ for all $x\in L_p(M,\Phi)$ and for almost all $\omega\in\Omega$. It is clear that $\mathcal{E}_\omega x(\omega) =
 \mathcal{E}(x)(\omega)$ for $x\in M$ and all $\omega\in\Omega$.

Let $x\in L_p(M,\Phi)$. As $M$ $(bo)$-dense in $L_p(M,\Phi)$, then there exists a sequence $\{x_n\}\subset M$ such that $\|x_n-x\|_p\stackrel{(o)}\rightarrow0.$ Then $\|x_n(\omega) -
 x(\omega)\|_{L_p(M(\omega),\tau_\omega)} \rightarrow0$ for almost all $\omega\in \Omega$. From the equality $\mathcal{E}(x)=(bo) -
\lim\limits_{n\rightarrow\infty}\mathcal{E}(x_n)$ we get $$\|\mathcal{E}_\omega(x_n(\omega)) -
 (\mathcal{E}x)(\omega)\|_{L_p(M(\omega),\tau_\omega)} \rightarrow0$$ for almost all $\omega\in \Omega$, i.e. $\lim\limits_{n\rightarrow\infty}\mathcal{E}_\omega(x_n(\omega))
 =
 (\mathcal{E}x)(\omega)$ for almost all $\omega\in \Omega$.

On the other hand, from the continuity of $\mathcal{E}_\omega$ we obtain $\lim\limits_n \mathcal{E}_\omega (x_n(\omega)) =
\mathcal{E}_\omega
 x(\omega)$ a.e. Hence $\mathcal{E}(x)(\omega)= \mathcal{E}_\omega(x(\omega))$ for all $x\in L_p(M,\Phi)$ and for almost all $\omega\in\Omega$. This completes the proof.

Martingales theorems in noncommutative $L_p(M,\Phi)$ spaces
===========================================================

In this section, as an application of the results of the previous section, we are going to prove the norm-convergence of martingales in the noncommutative $L_p(M,\Phi)$ spaces.

\[3.1\][@P] Let $\{M_n\}_{n\geq1}$ be a sequence of von Neumann subalgebras of $M$ such that $M_1\subset M_2\subset
\dots$. If one has $\overline{\bigcup\limits_{n}M_n}^{w^*}=M$ then the sequence $\{M_n\}_{n\geq1}$ is called a *filtration* of $M$.

Let $M$ be as in Section 4, and $\{M_n\}$ be a filtration of $M$. Let $\mathcal{E}(\cdot|M_n)$ be a conditional expectation from $L_p(M,\Phi)$ to $L_p(M_n,\Phi)$.

\[3.2\]A sequence $\{x_n\}\subset
L_p(M,\Phi)$ is called an *$L_p(M,\Phi)$-martingale* with respect to $\{M_n\}_{n\geq1}$ if one has $\mathcal{E}(x_{n+1}|M_n)=x_n$ for every $n\geq 1$.

\[3.3\] If $\{x_n\}\subset
L_1(M,\Phi)$ is an $L_1(M,\Phi)$-martingale with respect to $\{M_n\}_{n\geq1}$ then $\{x_n(\omega)\}\subset
L_1(M(\omega),\tau_\omega)$ is also martingale with respect to filtration $\{M_n(\omega)\}_{n\geq1}$ of $M_\infty(\omega):=\overline{\bigcup\limits_{n}M_n(\omega)}^{w^*}$ for almost all $\omega\in \Omega$.

Let $\mathcal{E}(x_{n+1}|M_n)=x_n$ for every $n\geq1$. According Theorem \[2.3\] we have $$\mathcal{E}_\omega(x_{n+1}(\omega)|M_n(\omega))=\mathcal{E}(x_{n+1}|M_n)(\omega)=x_n(\omega)$$ for almost all $\omega\in \Omega$, i.e. $\{x_n(\omega)\}\subset
L_1(M(\omega),\tau_\omega)$ is a martingale with respect to the filtration $(M_n(\omega))_{n\geq1}$ of $\overline{\bigcup\limits_{n}M_n(\omega)}^{w^*}$.

If for a *$L_p(M,\Phi)$-martingale* $\{x_n\}$ one has $\sup\limits_{n\geq 1}\|x_n\|_p \in L^0(\Omega)$ then $\{x_n\}$ is called a *bounded $L_p(M,\Phi)$-martingale*.

\[3.5\] Let $p>1$. The following statements hold:

1.  Let $x\in L_p(M,\Phi)$, $p\geq1$ and define $x_n=\mathcal{E}(x|M_n)$. Then $\{x_n\}$ is a bounded $L_p(M,\Phi)$-martingale, and $x_n\rightarrow x$ in $L_p(M,\Phi)$.

2.  Let $\{x_n\}$ be a bounded $L_p(M,\Phi)$-martingale. Then there exists $x\in L_p(M,\Phi)$ such that $x_n=\mathcal{E}(x|M_n)$.

(i). Due to $\|x_n\|_p=\|\mathcal{E}(x|M_n)\|_p\leq
\|x\|_p$ we get $$\sup\limits_{n\geq 1}\|x_n\|_p\leq \|x\|_p\in
L^0(\Omega).$$ According Theorem \[2.3\] one has $$x_n(\omega)=\mathcal{E}(x|M_n)(\omega)=\mathcal{E}_\omega(x(\omega)|M_n(\omega))$$ for almost all $\omega\in \Omega$. Then using Proposition 2.10 (i) [@P] we obtain $$\|x_n(\omega)-x(\omega)\|_{L_p(M(\omega),\tau_\omega)}\rightarrow
 0$$ for almost all $\omega\in \Omega$. From $$\|x_n-x\|_p(\omega)=\|x_n(\omega)-x(\omega)\|_{L_p(M(\omega),\tau_\omega)}$$ it follows that $\|x_n-x\|_p\stackrel{(o)}\rightarrow0$ i.e. $x_n\rightarrow x$ in $L_p(M,\Phi)$.

(ii). Let $x_n\in L_p(M,\Phi)$ be an $L_p(M,\Phi)$-martingale, i.e. $a:=\sup\limits_{n\geq 1}\|x_n\|_p \in
L^0(\Omega)$. Then $$\sup\limits_{n\geq
1}\|x_n(\omega)\|_{L_p(M(\omega),\tau_\omega)} =\sup\limits_{n\geq
1}\|x_n\|_p(\omega)=a(\omega)<\infty$$ for almost all $\omega\in
\Omega$. According Proposition 2.10 (ii) [@P] there exists $x(\omega)\in L_p(M(\omega),\tau_\omega)$, such that $x_n(\omega)=\mathcal{E}_\omega(x(\omega)|M_n(\omega))$ for almost all $\omega\in \Omega$. Again by Proposition 2.10 (i) [@P] we have $\|x_n(\omega)-x(\omega)\|_{L_p(M(\omega),\tau_\omega)}\rightarrow0$ for almost all $\omega\in \Omega$. This mean $x\in M(\Omega, X)$. Therefore, $x\in L_p(M,\Phi)$. So $$x_n=\widehat{\mathcal{E}_\omega(x(\omega)|M_n(\omega))}=\mathcal{E}(x|M_n).$$ This completes the proof.

\[3.6\] Let $p>1$. The following statements hold:

1.  Let $x_n\rightarrow x$ in $L_p(M,\Phi)$, then $\mathcal{E}(x_n|N)\rightarrow
    \mathcal{E}(x|N)$ in $L_p(M,\Phi)$ for any von Neumann subalgebra $N$ of $M$.

2.  If $x_n\rightarrow x$ in $L_p(M,\Phi)$ and $\overline{\bigcup\limits_{m}M_m}^{w^*}=M$, then $\mathcal{E}(x_n|M_m)\rightarrow x$ in $L_p(M,\Phi)$ as $n,m\rightarrow\infty$.

\(i) follows from Theorem \[2.3\].

(ii). It is clear that $$\begin{aligned}
\label{36-1}\|\mathcal{E}(x_n|M_m)-x\|_p&=&\|\mathcal{E}(x_n|M_m)-\mathcal{E}(x|M_m)+\mathcal{E}(x|M_m)-x\|_p\nonumber\\[2mm]
&\leq& \|x_n-x\|_p+\|\mathcal{E}(x|M_m)- x\|_p.\end{aligned}$$ By condition we have $\|x_n-x\|_p\stackrel{(o)}\rightarrow0$, and Theorem \[3.5\] (i) implies $\|\mathcal{E}(x|M_m)-
x\|_p\stackrel{(o)}\rightarrow0$. Hence, from one finds $\mathcal{E}(x_n|M_m)\rightarrow x$ in $L_p(M,\Phi)$ as $n,m\rightarrow\infty$.

Now we are going to prove the convergence of weighted averages of martingales in noncommutative $L_p(M,\Phi)$ spaces.

Let $\mathbf{x}=\{x_n\}$ be an $L_p(M,\Phi)$-martingale and $\{w_n\}_{n\geq1}$ be a sequence of positive numbers such that $W_n=\sum\limits_{k=1}^nw_k\rightarrow\infty$ as $n\rightarrow\infty$. *The weighted average* (see [@ki1],[@ZH]) $\sigma_n(\mathbf{x})$ of the martingale $x=\{x_n\}$ is given by $$\sigma_n(\mathbf{x})=\frac{1}{W_n}\sum\limits_{k=1}^nw_k x_k.$$

\[3.8\] Let $\mathbf{x}=\{x_n\}$ be an $L_p(M,\Phi)$-martingale and $\sigma_n(\mathbf{x})$ be a weighted average of $\mathbf{x}$. Then one has $$\sup\limits_{n\geq 1}\|x_n\|_p=\sup\limits_{n\geq 1}\|\sigma_n(\mathbf{x})\|_p.$$

Due to $$(\sup\limits_{n\geq 1}\|x_n\|_p)(\omega)=\sup\limits_{n\geq
1}\|x_n(\omega)\|_{L_p(M(\omega),\tau_\omega)}$$ for almost all $\omega\in \Omega$, from Theorem 3.2 [@ZH] we obtain $$\begin{aligned}
(\sup\limits_{n\geq 1}\|x_n\|_p)(\omega)&=&\sup\limits_{n\geq
1}\|x_n(\omega)\|_{L_p(M(\omega),\tau_\omega)}\\[2mm]
&=&\sup\limits_{n\geq
1}\|\sigma_n(\mathbf{x}(\omega))\|_{L_p(M(\omega),\tau_\omega)}\\[2mm]
&=&\sup\limits_{n\geq 1}\|\sigma_n(\mathbf{x})\|_p(\omega)\end{aligned}$$ for almost all $\omega\in \Omega$, here we have used the notation $\mathbf{x}(\omega))=\{x_n(\w)\}$. Hence, we obtain the required equality.

\[3.9\] Let $\mathbf{x}=\{x_n\}$ be an $L_p(M,\Phi)$-martingale and $\sigma_n(x)$ be a weighted average of $\mathbf{x}$. Then one has $\{x_n\}_{n\geq1}$ converges in $L_p(M,\Phi)$ if and only if $\sigma_n(\mathbf{x})$ converges in $L_p(M,\Phi)$.

Let $x_n\rightarrow y$ in $L_p(M,\Phi)$. Then $x_n(\omega)\rightarrow y(\omega)$ in $L_p(M(\omega),\tau_\omega)$ for almost all $\omega\in
\Omega$. According to Theorem 3.7 [@ZH] $\sigma_n(\mathbf{x}(\omega))\rightarrow y(\omega)$ for almost all $\omega\in \Omega$. Then $\sigma_n(\mathbf{x})(\omega)=\sigma_n(\mathbf{x}(\omega))\rightarrow
y(\omega)$ in $L_p(M(\omega),\tau_\omega)$ for almost all $\omega\in
\Omega$. Hence $$\|\sigma_n(\mathbf{x})-y\|_p(\omega)=\|\sigma_n(\mathbf{x}(\omega))-y(\omega)\|_{L_p(M(\omega),\tau_\omega)}\rightarrow
0$$ for almost all $\omega\in \Omega$, which means $\sigma_n(\mathbf{x})$ converges in $L_p(M,\Phi)$.

Let $\sigma_n(\mathbf{x})\rightarrow y$ in $L_p(M,\Phi)$. Then $\sigma_n(\mathbf{x}(\omega))\rightarrow y(\omega)$ in $L_p(M(\omega),\tau_\omega)$ for almost all $\omega\in \Omega$. Again by Theorem 3.7 [@ZH] $x_n(\omega)\rightarrow y(\omega)$ in $L_p(M(\omega),\tau_\omega)$ for almost all $\omega\in \Omega$. Hence $\|x_n-y\|_p\stackrel{(o)}\rightarrow0.$ This completes the proof.

Acknowledgments {#acknowledgments .unnumbered}
===============

The first author (I.G) acknowledges the MOHE Grant FRGS13-071-0312.

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