Noncommutative dyadic martingales and Walsh–Fourier series

Abstract

Let R be the hyperfinite II 1 -factor. For 0 < p < 1 , we show that the dual space of the Hardy space h p r ( R ) is the Lipschitz space Lip α c ( R ) , α = 1 p − 1 , which partially answers [Bekjan, Chen, Perrin and Yin, J. Funct. Anal. 258 (2010) 2483–2505, Problem 4]. If S is an operator of taking partial sum of the noncommutative Walsh–Fourier series of x ∈ L 1 ( R ) , then ∥ S ( x ) ∥ L 1 , ∞ ( R ) ⩽ c a b s ∥ x ∥ L 1 ( R ) . Furthermore, we show the closed subspace spanned by noncommutative Walsh system of multiplicity 2 is isomorphic to l 2 and is complemented in H 1 ( R ) . The latter result demonstrates a very substantial difference from its commutative counterpart due to Müller–Schechtman.