Noncommutative dyadic martingales and Walsh–Fourier series

Let R be the hyperfinite II1 ‐factor. For 0 < p < 1 , we show that the dual space of the Hardy spaceh p r ( R )is the Lipschitz spaceLip 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 inH 1 ( R ) . The latter result demonstrates a very substantial difference from its commutative counterpart due to Müller–Schechtman.