Operator algebras with the partial factorization property

Abstract

A subalgebra A \mathfrak A in a σ \sigma -finite von Neumann algebra M \mathcal M has the left (resp. right) partial factorization property if for any invertible operator S ∈ M S\in \mathcal M , there exists an isometry (resp. a co-isometry) U ∈ M U\in \mathcal M such that U ∗ S , S − 1 U ∈ A U^*S, S^{-1}U\in \mathfrak A . We show that the relative invariant subspace lattice L a t M A {Lat}_{\mathcal M}\mathfrak A consisting of those projections in M \mathcal M whose ranges are invariant under A \mathfrak A is commutative. Moreover if M \mathcal M has a separable predual, then L a t M A {Lat}_{\mathcal M}\mathfrak A is generated by a nest together with the lattice of projections in the center of M \mathcal M . In particular, if A \mathfrak A is a von Neumann subalgebra, then A = M \mathfrak A=\mathcal M .