Order preserving transformations of the Hilbert grassmannian: complex case

Abstract

Let $H$ be a separable complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion relation. We show that every continuous order preserving bijective transformation of ${\mathcal G}_{\infty}(H)$ is induced by an invertible bounded semi-linear operator.