Tree Martingales in Noncommutative Probability Spaces
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Complex Analysis and Operator Theory
Tree Martingales in Noncommutative Probability Spaces Ghadir Sadeghi1,2 Received: 6 February 2020 / Accepted: 29 July 2020 © Springer Nature Switzerland AG 2020
Abstract We introduce the notion of a tree martingale in a noncommutative probability space and prove the Burkholder–Gundy inequalities for tree martingales in symmetric operator spaces. In particular, we establish some inequalities in this setting via an approach based on the concept of generalized singular valued functions of noncommutative random valuables. Keywords Noncommutative probability space · Symmetric operator spaces · Tree martingale · Burkholder–Gundy inequalities Mathematics Subject Classification Primary 46L53; Secondary 46L10 · 47A30 · 60G46
1 Introduction and Preliminaries Tree martingales have been studied by Fridli, Schipp, Weisz, and other mathematicians; see [1,9,10]. They investigated maximal inequalities with respect to tree martingales and showed that Burkhoder–Gundy’s inequalities hold if 2 < p < ∞; cf. [11, Chapter 4]. Furthermore, this result was extended to all 1 < p < ∞ for a regular tree stochastic basis [11, Chapter 4]. Moreover, using some results on tree martingales, Schipp [10] and Gosselin [5] proved that for an arbitrary Vilenkin system and for f ∈ L p , the
Communicated by Marek Bozejko. This article is part of the topical collection “Infinite-dimensional Analysis and Non-commutative Theory (Marek Bozejko, Palle Jorgensen and Yuri Kondratiev”.
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Ghadir Sadeghi [email protected]; [email protected]
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Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran
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Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran 0123456789().: V,-vol
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Vilenkin-Fourier series of f converges to itself in L p -norm. As noticed by [11, Chapter 4], in the theory tree martingales there are two sources of difficulties that are both related to the fact that the tree set cannot be well ordered in a useful way. The first problem is that there is no sensible way to uniquely define tree martingale’s stopping times. The second is that there is a partial ordering-nonlinear. In the sequel, we collect some basic facts and introduce some notation related to τ –measurable operators. Let H be a Hilbert space, B(H) be the algebra of all bounded linear operators on H, and M ⊆ B(H) be a noncommutative probability space with a normal faithful trace τ ; cf. [6–8,13]. The identity in M is denoted by 1 and we denote by P(M) the complete lattice of all self–adjoint projections in M. A linear operator x : D(x) → H with domain D(x) ⊆ H is called affiliated with M, if ux = xu for all unitaries u in the commutant M of M. This is denoted by xηM. Note that the equality ux = xu involves the equality of the domains of ux and xu, that is, D(x) = u −1 (D(x)). If x is in the algebra B(H), then x is affiliated with M if and only if x ∈ M. If x is a self–adjoint operator in B(H) affiliated wi
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