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In this article, we discuss some applications of the well-known Douglasfactorization lemma in the context of von Neumann algebras. Let$\mathcal{B}(\mathscr{H})$ denote the set of bounded operators on a complexHilbert space $\mathscr{H}$, and $\mathscr{R}$ be a von Neumann algebra actingon $\mathscr{H}$. We prove some new results about left (or, one-sided) idealsof 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 byLoebl 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.

The Douglas lemma for von Neumann algebras and some applications