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

Abstract

We proved a complex interpolation theorem of noncommutative Hardy spaces associated with semi-finite von Neumann algebras and extend the Riesz type factorization to the semi-finite case.2010 Mathematics Subject Classification. Primary 46L52; Secondary 47L05.

Paley's inequality for nonabelian groups

“…It is proved in [21] and [2] that the dual of ReH 1 can be identified as a BMO space and the complex interpolation between this BMO space and ReH 1 (N ) is L p (N ) for 1 < p < ∞. We then get Corollary 3.4.…”

“…Proof. The 1 < p < ∞ case follows from the aforementioned duality and interpolation results proved in [21], [2]. The 0 < p < 1 case follows from [27, Corollary 2.2] and [3, Theorem 2.6].…”

“…The classical factorization, interpolation and duality results extend to the noncommutative Hardy spaces H p (N ) (see [26], section 8). We will need the following factorization theorem (Theorem 4.3 of [20], Theorem 3.2 of [2]), Lemma 3.1. Given any x ∈ H 1 (N ) and ε > 0, there exist y, z ∈ H 2 (N ) such that x = yz and y 2 z 2 ≤ x 1 + ε.…”

Interpolation of Haagerup noncommutative Hardy spaces

“…with equivalent norms for all n ∈ N (refer to the remark following Lemma 8.5 in [28] or Theorem 1.1 in [2]). By virtue of (1) of Theorem 2.6, we assert that…”