Topological and bornological characterisations of ideals in von Neumann algebras: I

Abstract

Suppose\(\mathcal{M}\) is a von Neumann algebra on a Hilbert space\(\mathcal{H}\) and\(\mathcal{I}\) is any ideal in\(\mathcal{M}\). We determine a topology\(t(\mathcal{I})\) on\(\mathcal{H}\), for which the members of\(\mathcal{M}\) that are\(t(\mathcal{I})\) to norm continuous are exactly those in\(\mathcal{I}\); and a bornology\(b(\mathcal{I})\) on\(\mathcal{H}\) such that the elements of\(\mathcal{M}\) which map the unit ball to an element of\(b(\mathcal{I})\), equivalently those members of\(\mathcal{M}\) that are norm to\(b(\mathcal{I})\) bounded, are exactly those in\(\mathcal{I}\). This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.