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 $L^{\infty}(\Omega)$ 