Noncommutative Hardy space associated with semi-finite subdiagonal algebras

Abstract

Let M be a von Neumann algebra with a faithful normal semi-finite trace τ , and let A be maximal subdiagonal algebra of M. Then for 0 < p < ∞, we define the noncommutative H p -space. We obtain that the conjugation and Herglotz maps are bounded linear maps from L p (M) into L p (M) for 1 < p < ∞, and continuous map from L 1 (M) into L 1,∞ (M). We also give the dual space of H p (A) for 1 ≤ p < ∞, and extend Pisier's interpolation theorem to this case.

A Beurling-Blecher-Labuschagne Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

“…Remark 2.8. It was shown in [38] and [21] that such a subalgebra H ∞ with respect to (M, Φ) is maximal among semifinite subdigonal subalgebras satisfying (1), (2), (3) and (4). From this fact, it follows that…”

Complex Interpolation of Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras

“…On the other hand, (i) and (iii) of Lemma 6.5 in [2], we have that (H 1 (B), H r (B)) is K-closed with respect to (L 1 (N ), L r (N )) and (H q (B), B) is K-closed with respect to (L q (N ), N ). Hence by Theorem 1.2 in [22], we obtain (H 1 (B), B) is K-closed with respect to (L 1 (N ), N ).…”

“…and let E e be the restriction of E to M e . Using Lemma 3.1 in [2] we obtain that A e is a subdiagonal algebra of M e with respect to E e (or D e ).…”

“…The first-named author [2], using Pisier's method, proved that the real case of Peter Jones' theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras holds (also see [32]). The main purpose of the present paper is to prove the complex case of Peter Jones' theorem for noncommutative Hardy spaces associated with semifinite von Neumann algebras, i.e.…”

On products of noncommutative symmetric quasi Banach spaces and applications

“…In [35], Pisier and Xu obtained noncommutative version of P. Jones' theorem for noncommutative Hardy spaces associated with a finite subdiagonal algebra in Arveson's sense [1]. The first named author [4] extended the Pisier's theorem to noncommutative Hardy spaces associated semifinite von Neumann algebras (also see [39]). We use the noncommutative symmetric quasi Hardy space's analogue of (1.1) and Pisier's method to prove the real case of Peter Jones' theorem for noncommutative symmetric quasi Hardy spaces.…”