Orthogonal apartments in Hilbert Grassmannians

Abstract

Let $H$ be an infinite-dimensional complex Hilbert space and let $\mathcal{L}(H)$ be the logic formed by all closed subspaces of $H$. For every natural $k$ we denote by ${\mathcal{G}}_k(H)$ the Grassmannian consisting of $k$-dimensional subspaces. An orthogonal apartment of ${\mathcal{G}}_k(H)$ is the set consisting of all $k$-dimensional subspaces spanned by subsets of a certain orthogonal base of $H$. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of ${\mathcal{G}}_k(H)$. We show that every bijective transformation $f$ of ${\mathcal{G}}_k(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments (in other words, $f$ preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of $\mathcal{L}(H)$.