Non-commutative martingale VMO-spaces

Abstract

Abstract. We study Banach space properties of non-commutative martingale VMOspaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if M is hyperfinite then the noncommutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of M has property (u).1. Introduction. The space of functions of bounded mean oscillation generally referred to as BMO-space has been instrumental in several aspects of analysis. Its martingale version plays an equally important role in probability.In this paper, we analyze subspaces of BMO-spaces related to non-commutative martingales. Our main motivation comes primarily from a paper by Müller and Schechtman [14] who studied structural properties of closed subspaces of dyadic martingale VMO (vanishing mean oscillation) as Banach spaces. More precisely, they provided, among other things, a version of the classical Kadets-Pełczyński dichotomy theorem for closed subspaces of dyadic martingale VMO-spaces. In order to explain the details, we first recall the celebrated Kadets-Pełczyński dichotomy theorem, which states that every closed subspace of L p (0, 1), 2 < p < ∞, either is isomorphic to a Hilbert space or contains a subspace which is isomorphic to l p . This dichotomy plays a crucial role in the development of L p -spaces and the theory of function spaces in general. Non-commutative analogues of the Kadets-Pełczyński dichotomy has been considered by several authors with the most general result obtained by Raynaud and Xu (see [21]) in the context of Haagerup L p -spaces when 2 ≤ p < ∞. Clearly, the dichotomy does not extend to closed subspaces of L ∞ (0, 1) or any C(K)-spaces in general. As a substitute, the following result was obtained by Müller and Schechtman:

Subprojective Banach spaces

“…contains isomorphic copies of L p for 1 < p 2 [42, Section 3]. On the other hand, VMO is subprojective ( [29], see also [36] for non-commutative generalizations).…”