Orthogonal apartments in Hilbert Grassmannians

TL;DR

This paper characterizes transformations of Grassmannians in infinite-dimensional Hilbert spaces that preserve orthogonal apartments and compatibility, showing they extend uniquely to automorphisms of the lattice of all closed subspaces.

Contribution

It proves that bijections preserving orthogonal apartments in Grassmannians extend uniquely to automorphisms of the logic of all closed subspaces.

Findings

Transformations preserving orthogonal apartments are automorphisms of the logic.

Orthogonal apartments are characterized as maximal compatible sets.

Such transformations can be uniquely extended to the entire lattice.

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 ${\mathcal L}(H)$.