Wigner-Type Theorems for Hilbert Grassmannians

Abstract

The main objects of this chapter are the lattices formed by closed subspaces of infinite-dimensional complex normed spaces. Our first result is the following analogue of the Fundamental Theorem of Projective Geometry: all isomorphisms of such lattices are induced by linear or conjugate-linear homeomorphisms between the corresponding normed spaces (for the finite-dimensional case this fails). This statement is closely connected to the remarkable Kakutani-Mackey theorem [31] which states that every orthomodular lattice consisting of all closed subspaces of an infinite-dimensional complex Banach space is the orthomodular lattice associated to an infinite-dimensional complex Hilbert space. At the end, we consider the partially ordered set formed by all closed subspaces of a complex Hilbert space whose dimension and codimension both are infinite. It will be shown that every isomorphism between such partially ordered sets can be uniquely extended to an isomorphism of the lattices of closed subspaces. Using the same arguments, we obtain an analogue of Chow's theorem for connected components of the Grassmamm graphs associated to infinite-dimensional Hilbert spaces (note that for Grassmannians of infinitedimensional vector spaces there is no result of such kind).

Commutativity preserving transformations on conjugacy classes of finite rank self-adjoint operators

“…If dim H = 2k, then S(C) consists of two elements and the non-identity element transfers every P X to P X ⊥ = Id − P X . Theorem 1 generalizes the following result obtained in [6] (see also [8]).…”

“…Any two operators from an orthogonal apartment commute. Furthermore, the family of all orthogonal apartments of C coincides with the family of all subsets X ⊂ C maximal with respect to the property that elements of X are mutually commuting [8,Proposition 1.15]. Therefore, for a bijective transformation f of C the following two conditions are equivalent:…”

“…By [6] (see also [8,Theorem 5.9]), there is a unitary or anti-unitary operator U such that for every X ∈ G k (H) we have either…”

“…By Proposition 1, h is induced by an invertible linear or conjugate-linear operator. This operator sends orthogonal vectors to orthogonal vectors which implies that it is a scalar multiple of a unitary or anti-unitary operator [8,Proposition 4.2].…”

“…By [8,Theorem 3.17], every bijective transformation of G ∞ (H) preserving the order in both directions is induced by an invertible bounded linear or conjugatelinear operator. The proof of Proposition 1 is based of the same arguments.…”