A trick for investigation of near-martingales in quantum probability spaces. (English) Zbl 1458.46052

Sadeghi, Ghadir; Talebi, Ali

Adv. Oper. Theory 4, No. 4, 784-792 (2019).

Summary: In this paper, we introduce near-martingales in the setting of quantum probability spaces and present a trick for investigating some of their properties. For instance, we give a near-martingale analogous result of the fact that the space of all bounded \(L^p\)-martingales, equipped with the norm \(\|\cdot\|_p\), is isometric to \(L^p(\mathfrak{M})\) for \(p>1\). We also present Doob and Riesz decompositions for the near-submartingale and provide Gundy’s decomposition for \(L^1\)-bounded near-martingales. In addition, the interrelation between near-martingales and the instantly independence is studied.

MSC:

46L53	Noncommutative probability and statistics
46L10	General theory of von Neumann algebras
60E15	Inequalities; stochastic orderings
47A30	Norms (inequalities, more than one norm, etc.) of linear operators
60G42	Martingales with discrete parameter

Keywords:

quantum probability space; Gundy decomposition; noncommutative near-martingale; Doob decomposition; Riesz decomposition