The Douglas lemma for von Neumann algebras and some applications

TL;DR

This paper explores applications of the Douglas factorization lemma within von Neumann algebras, establishing new results on ideals and generalizing previous findings to the setting of imodules, thus advancing operator algebra theory.

Contribution

It introduces new results on left ideals of von Neumann algebras and extends a known theorem on C*-convex sets to imodules, broadening the lemma's applicability.

Findings

Every left ideal of a von Neumann algebra can be expressed as an intersection with (H).

Generalization of Loebl and Paulsen's result to imodules.

Enhanced understanding of the structure of ideals in von Neumann algebras.

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