The Orthomodular Poset of Projections of A Symmetric Lattices

Abstract

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P (L) of L × L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P (L) correspond to continuous projections and we prove that automorphisms of P (L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces.

Wigner-Type Theorems for Projections

“…Finally, by Proposition 6 of [4], φ is an automorphism of the orthomodular poset P (L) and the proof is complete.…”

“…Proof The proof is easy by using the previous proposition, Proposition 6 of [4] saying that any order automorphism f of P (L) satisfies f (p ⊥ ) = f (p) ⊥ and the fact that, in any orthocomplemented lattice T ,…”

“…An irreducible DAC-lattice L is called a G-lattice [4] if L is complete or if L is modular and complemented. Typical examples of G-lattices are obtained by considering a Hilbert space H : the lattice of all closed subspaces of H is a G-lattice as a complete irreducible DAC-lattice and its sublattice of finite or cofinite dimensional elements is a G-lattice as a complemented modular irreducible DAC-lattice.…”

“…Moreover, it is proved in [4] that there are exactly two kinds of automorphisms on an orthoposet of projections: the so-called even automorphisms which transform projections with the same image into projections with the same image and the odd automorphisms which transform projections with the same image into projections with the same kernel. This fact generalizes a theorem of [12].…”

“…We specify this approach in Sect. 2 where we recall some results of [3,4]. In particular, to any lattice L with properties of lattices of closed subspaces is associated a subset P (L) of L × L, called its orthoposet of projections.…”