A representation of noncommutative [formula omitted] spaces

Abstract

It is known that the noncommutative Hardy spaces H1(M) and H1max(M) do not coincide, in general. In fact, it may happen that H1(M)⊈H1max(M). It is an interesting question whether the reverse inclusion holds or not. In this note, motivated by this question, we prove that the validity of inequality ‖x‖H1C(M)≤c‖x‖H1max(M) for some c and all martingales x∈H1max(M) gives rise to a certain representation of positive elements of the noncommutative space BMO. Conversely, existence of such a representation for elements of BMO yields the inequality above for positive martingales of H1max(M).