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

TL;DR

This paper establishes the existence and description of conditional expectations in noncommutative Lp-spaces with center-valued traces and demonstrates the norm convergence of martingales within these spaces.

Contribution

It provides the first comprehensive analysis of conditional expectations and martingale convergence in noncommutative Lp-spaces associated with center-valued traces.

Findings

Existence of conditional expectations in noncommutative Lp-spaces.

Explicit description of these conditional expectations.

Proof of norm convergence of martingales in these spaces.

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.