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.
The Wigner theorem, in its Uhlhorn's formulation, states that a bijective transformation of the set of all one-dimensional linear subspaces of a complex Hilbert space which preserves orthogonality is induced by either a unitary or an antiunitary operator. There exist in the literature many Wigner-type theorems and the purpose of this paper is to prove in an algebraic setting a very general Wigner-type theorem for projections (idempotent linear mappings). As corollaries, Wigner-type theorems for projections in real locally convex spaces, infinite dimensional complex normed spaces and Hilbert spaces are obtained.
By using a lattice characterization of continuous projections defined on a topological vector space E arising from a dual pair, we determine the automorphism group of their orthomodular poset Proj(E) by means of automorphisms and anti-automorphisms of the lattice L of all closed subspaces of E. A connection between the automorphism group of the ring of all continuous linear mappings defined on E and the automorphism group of the orthoposet Proj(E) is established. KEY WORDS: orthomodular lattice; symmetric lattice; lattice of closed subspaces; automorphism of poset of projections; dual pair.
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