The invariant subspace structure of nonselfadjoint crossed products

Abstract

Let L \mathcal {L} be the von Neumann algebra crossed product determined by a finite von Neumann algebra M M and a trace preserving ∗ ^{\ast } -automorphism α \alpha of M M . We study the invariant subspace structure of the subalgebra L + {\mathcal {L}_ + } of L \mathcal {L} consisting of those operators whose spectrum with respect to the dual automorphism group on L \mathcal {L} is nonnegative. We investigate the conditions for two invariant subspaces M 1 {\mathcal {M}_1} , and M 2 {\mathcal {M}_2} (with Q 1 , Q 2 {Q_{1}},{Q_2} the corresponding orthogonal projections) to satisfy Q 1 = R υ Q 2 R υ ∗ {Q_1} = {R_\upsilon }\,{Q_2}\,R_\upsilon ^{\ast } for some partial isometry R υ {R_{\upsilon }} in L ′ \mathcal {L}’ . We use this analysis to find the general form of a σ \sigma -weakly closed subalgebra of L \mathcal {L} that contains L + {\mathcal {L}_ + } .