The Douglas lemma for von Neumann algebras and some applications

Abstract

In this article, we discuss some applications of the well-known Douglas factorization lemma in the context of von Neumann algebras. Let \({\mathcal {B}}({\mathscr {H}})\) denote the set of bounded operators on a complex Hilbert space \({\mathscr {H}}\), and \({\mathscr {R}}\) be a von Neumann algebra acting on \({\mathscr {H}}\). We prove some new results about left (or, one-sided) ideals of von Neumann algebras; for instance, we show that every left ideal of \({\mathscr {R}}\) can be realized as the intersection of a left ideal of \({\mathcal {B}}({\mathscr {H}})\) with \({\mathscr {R}}\). We also generalize a result by Loebl and Paulsen (Linear Algebra Appl 35:63–78, 1981) pertaining to \(C^*\)-convex subsets of \({\mathcal {B}}({\mathscr {H}})\) to the context of \({\mathscr {R}}\)-bimodules.