Non-Commutative Bounded Vilenkin Systems

Abstract

We consider orthonormal systems in spaces of measurable operators associated with a finite von Neumann algebra which contain the classical bounded Vilenkin systems. We show that they form Schauder bases in all reflexive non-commutative Lp-spaces when taken in the lexicographic order. This is a non-commutative analogue of a theorem of Paley. The principal theme of this paper finds its source in the classical theorems of Paley [Pa]. In the terminology of the theory of Banach spaces, these theorems assert that the classical Walsh system, taken in the Walsh-Paley ordering (respectively, partitioned into dyadic blocks) forms a (Schauder) basis (respectively, an unconditional decomposition) in each of the spaces Lpâ0;1a; 1 < p <1. The classical Walsh system may be identified with the dual group of the familiar dyadic group. From this point of view, the notion of a Vilenkin system introduced in [Vi] generalizes that of the Walsh system. Watari [Wa] extended Paley’s theorems to bounded Vilenkin systems and showed that such systems when enumerated lexicographically not only form a Schauder basis in each of the reflexive Lp-spaces, but also permit a finer unconditional decomposition than that given by the dyadic decomposition associated with the Walsh system. The study of orthogonal series in the setting of non-commutative Lpspaces was initiated in [SF1,2]. In this setting, the classical Walsh system is replaced by a system of eigenoperators arising from the action of a compact Abelian group on the underlying von Neumann algebra. The methods of [SF1,2] are based on the fact that non-commutative Lp-spaces have the unconditionality property for martingale difference sequences and this approach permits an extension of Paley’s theorems to the non-commutative setting. Our intention in this paper is to study orthogonal systems in spaces of measurable operators affiliated with a finite von Neumann algebra, which