A Noncommutative Version of H p and Characterizations of Subdiagonal Algebras

Abstract

Let \({\mathcal M}\) be a σ-finite von Neumann algebra and \({\mathfrak A}\) a maximal subdiagonal algebra of \({\mathcal M}\) with respect to a faithful normal conditional expectation \({\Phi}\) . Based on Haagerup’s noncommutative Lp space \({L^p(\mathcal M)}\) associated with \({\mathcal M}\) , we give a noncommutative version of Hp space relative to \({\mathfrak A}\) . If h0 is the image of a faithful normal state \({\varphi}\) in \({L^1(\mathcal M)}\) such that \({\varphi\circ \Phi=\varphi}\) , then it is shown that the closure of \({\{\mathfrak Ah_0^{\frac1p}\}}\) in \({L^p(\mathcal M)}\) for 1 ≤ p < ∞ is independent of the choice of the state preserving \({\Phi}\) . Moreover, several characterizations for a subalgebra of the von Neumann algebra \({\mathcal M}\) to be a maximal subdiagonal algebra are given.