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] 