Martingales in Banach lattices, II

Abstract

This note is a follow-up to Troitsky (Positivity 9(3): [437][438][439][440][441][442][443][444][445][446][447][448][449][450][451][452][453][454][455][456] 2005). We provide several sufficient conditions for the space M of bounded martingale on a Banach lattice F to be a Banach lattice itself. We also present examples in which M is not a Banach lattice. It is shown that if F is a KB-space and the filtration is dense then F is a projection band in M. IntroductionThis short note is a follow-up to [5], where the second author introduced and studied spaces of bounded martingales on Banach lattices. The results of [5] have since been used and extended in several papers by Labuschagne et al, see, e.g., [2]. Let us briefly recall some key definitions from [5]. Throughout this paper, F is a Banach lattice. By a filtration on F we mean a sequence (E n ) of positive contractive projections such that E n E m = E n∧m . A sequence (x n ) in F is said to be a martingale (a submartingale) relative to a filtration (E n ) if E n x m = x n (E n x m ≥ x n , respectively) whenever n ≤ m. A (sub)martingale X = (x n ) is bounded if it has finite martingale norm given by X = sup n x n . We write M = M (F, (E n )) for the space of all bounded

Martingale-like sequences in Banach lattices

“…In [6] and [2] several sufficient conditions are established where the set of bounded martingales M is a Banach lattice. In [2], counter examples are provided where M is not a Banach lattice. So, one may similarly ask when are M X and M E Banach spaces and Banach lattices?…”

“…Moreover, under certain conditions on the Banach lattice, it was shown that the set of bounded martingales forms a Banach lattice with respect to the point-wise order. In 2011, H. Gessesse and V. Troitsky [2] produced several sufficient conditions for the space of bounded martingales on a Banach lattice to be a Banach lattice itself. They also provided examples showing that the space of bounded martingales is not necessarily a vector lattice.…”

Spaces of regular abstract martingales

“…However, we will present a counterexample to the contrary. Our example will be based on [8,Example 6], which we outline here for convenience of the reader.…”

“…We denote with M = M F, (E n ) the space of all martingales on F with respect to the filtration (E n ). The space M equipped with the coordinate-wise order is an ordered vector space and we denote with M + the positive cone of M. There is an extensive literature on abstract martingales on vector and Banach lattices, see, e.g., [5,18,11,15,10,12,8,9,6,7]. For unexplained terminology on ordered vector and Banach spaces we refer the reader to [2,3,13].…”

Unbounded order convergence in dual spaces

“…We now prove (5)⇒(1). Assume (5) We apply this result to establish the following dual version of [4,Theorem 4.3]. Recall that a net (x α ) in a vector lattice X is unbounded order Cauchy (or uo-Cauchy, for short), if the net (x α − x α ′ ) (α,α ′ ) uo-converges to 0 in X. Theorem 2.2.…”