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Let M be a σ-finite von Neumann algebra and A a maximal subdiagonal algebra of M with respect to a faithful normal conditional expectation Φ. Based on Haagerup's noncommutative L p space L p (M) associated with M, we give a noncommutative version of H p space relative to A. If h0 is the image of a faithful normal state ϕ in L 1 (M) such that ϕ • Φ = ϕ, then it is shown that the closure of {Ah 1 p 0 } in L p (M) for 1 ≤ p < ∞ is independent of the choice of the state preserving Φ. Moreover, several characterizations for a subalgebra of the von Neumann algebra M to be a maximal subdiagonal algebra are given. Mathematics Subject Classification (2010). Primary 46L52, 47L75; Secondary 46K50, 46J15.

Abstract. We give three kinds of characterizations of the commutativity of C * -algebras. The first is the one from operator monotone property of functions regarded as the nonlinear version of Stinespring theorem, the second one is the characterization of commutativity of local type from expansion formulae of related functions and the third one is of global type from multiple positivity of those nonlinear positive maps induced from functions.