Orthogonal apartments in Hilbert Grassmannians

2015/12/03 by Mark Pankov, Pankov, Mark
Mathematics · Computer Science · #Advanced Combinatorial Mathematics #Advanced Graph Theory Research #Advanced Algebra and Logic

paper · pdf · doi:10.48550/arxiv.1512.01007

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).

Related

• Orthogonal apartments in Hilbert Grassmannians. Finite-dimensional case
• Orthogonality preserving transformations of Hilbert Grassmannians
• Preserver problems for the logics associated to Hilbert spaces and related Grassmannians
• Apartments preserving transformations of Grassmannians of infinite-dimensional vector spaces
• Order preserving transformations of the Hilbert grassmannian