Automorphisms of An Orthomodular Poset of Projections

Abstract

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.

The Many Classical Faces of Quantum Structures

“…For morphisms of active lattices, we refer to [79], but let us point out that thanks to Lemma 5.3 they can be phrased in terms of projections alone, just like the above definition of the active lattice itself. See also [23]. We can now make precise that we can reconstruct an AW*-algebra A from its active lattice AProj(A).…”

Wigner-Type Theorems for Projections

“…The following main result of [3] is a generalization of a theorem of [12] and it gives a description of automorphisms of a projection orthoposet P (L) by means of automorphisms and antiautomorphisms of the lattice L when L is a G-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.…”

“…As L 0 is a G-lattice of height ≥ 4 then, by Theorem 2, there exists an automorphism f of the lattice L 0 such that φ ((a, b)) = (f (a), f (b)), (a, b) ∈ P (L 0 ), if φ is even or there exist an antiautomorphism g of L 0 such that φ ((a, b) g(a)) if φ is odd. By Lemma 1 of [3], f can be extended to an automorphism of L and we define φ for (a, b) ∈ P (L) by φ((a, b)…”

“…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.…”

“…[3]) Let L be a G-lattice of height ≥ 4. For any automorphism φ of the orthoposet P (L) there exists…”