---
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 