Conditionally expectations and martingales in noncommutative $L_p$-spaces associated with center-valued traces

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.