The Douglas lemma for von Neumann algebras and some applications (Q2025786)

In the present paper we find a discussion of some applications of the Douglas factorization lemma in the context of von Neumann algebras. The author give a constructive proof of this lemma and some new results about left ideals of von Neumann algebras. Let \(\mathcal{H}\)\ be a complex Hilbert space, \(\mathcal{B}(\mathcal{H})\) the set of bounded operators on \(\mathcal{H}\) and \(\mathcal{R}\) be a von Neumann algebra acting on \(\mathcal{H}\). It showing that every left ideal of \(\mathcal{R}\) can be realized as the intersection of a left ideal of \( \mathcal{B}(\mathcal{H})\) with \(\mathcal{R}\). Also the author generalize a result by Loebl and Paulsen pertaining to \(C^{\ast }\)-convex subsets of \( \mathcal{B}(\mathcal{H})\) to the context of \(\mathcal{R}\)-bimodules.