# Near inclusions of injective von Neumann algebras

### Liam Dickson (University of Glasgow)

Tuesday 4th February, 2014 16:00-17:00 Maths 416

#### Abstract

Kadison and Kastler introduced a metric on the set of closed linear subspaces  of the bounded operators on a fixed Hilbert space. They conjectured that sufficiently close C*-algebras in this metric are isomorphic. Along with other early progress, the conjecture was established for injective von Neumann algebras by  Christensen. The same author then introduced a one-sided version of this metric-  so called near containment'- and proved that given an injective von Neumann algebra which is nearly contained in an arbitrary von Neumann algebra one may find a unitary close to the identity that implements a genuine containment.

In recent work Roydor showed that given an injective von Neumann algebra that is sufficiently close to a  weak*-closed subalgebra of $\mathbb{B}(\mathcal{H})$ with a normal virtual diagonal, then the two algebras are isomorphic via a similarity. We investigate a one-sided version of this situation- an injective von Neumann algebra which is nearly contained in a weak*-closed subalgebra of $\mathbb{B}(\mathcal{H})$.`