The invariant subspace structure of nonselfadjoint crossed products

Abstract:

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