The dual of noncommutative H^1

Abstract

Let M be a von Neumann algebra with a faithful normal tracial state τ and let H ∞ be a finite maximal subdiagonal subalgebra of M. In previous work we defined a harmonic conjugate relative to H ∞ . Let H 1 be the closure of H ∞ in the noncommutative Lebesgue space L 1 (M). By analysing the behaviour of the harmonic conjugate in L 1 (M), we identify the dual space of H 1 as a concrete space of operators.

BMO spaces associated with semigroups of operators

“…[PX97], [JX03], [JM07], [Mus03] and [Pop00], and to [NPTV02], [Mei07] and [BP] for work on operator-or matrix-valued functions. In [Mei08] a first approach towards a H 1 − BM O duality associated with semigroups of operators on von Neumann algebras has been obtained, whereas a duality theory for Averson's subdiagonal algebras is studied in [MW98].…”

Noncommutative Hardy space associated with semi-finite subdiagonal algebras

“…The main objective of this paper is to obtain a generalization of the results in [16,17,19,21], for the semi-finite case.…”

“…They obtained generalizations of several classical results, including a Riesz factorization theorem for H 1 (A), a Riesz-Bochner theorem on the existence and boundedness of harmonic conjugates, a projection from L p (M) to H p (A), the duality between H p (A) and H q (A), and continuity of the Hilbert transform from L 1 (M) into L 1,∞ (M). In [17] the authors identified the dual of H 1 (A) with a noncommutative analogue of the BMO space as in Fefferman's classical result.…”

Characterization of subdiagonal algebras

“…In [1], Arveson introduced the notion of finite, maximal, subdiagonal algebras A of M, as noncommutative analogues of weak-* Dirichlet algebras. Subsequently several authors studied the (noncommutative) H p -spaces associated with such algebras ( [9,11,12,13,14,16,17,18]). Arveson [1] proved a Szegö type factorization theorem.…”