Maximality of analytic operator algebras

Abstract

LetM be a σ-finite von Neumann algebra andα be an action ofR onM. LetH ∞(α) be the associated analytic subalgebra; i.e.H ∞(α)={X ∈M: sp∞(X) [0, ∞]}. We prove that every σ-weakly closed subalgebra ofM that containsH ∞(α) isH ∞(γ) for some actionγ ofR onM. Also we show that (assumingZ(M)∩M α = Ci)H ∞(α) is a maximal σ-weakly closed subalgebra ofM if and only if eitherH ∞(α)={A ∈M: (I−F)xF=0} for some projectionF ∈M, or sp(α)=Γ(α).