The Douglas lemma for von Neumann algebras and some applications

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 (1981), 63--78) pertaining to $C^*$-convex subsets of $\mathcal{B}(\mathscr{H})$ to the context of $\mathscr{R}$-bimodules.