Adjointable maps between linear orthosets

TL;DR

This paper explores the structure-preserving maps, especially adjointable maps, between Hermitian spaces and their associated linear orthosets, establishing conditions under which these maps are induced by quasilinear or quasiunitary maps, with implications for Wigner-type theorems.

Contribution

It characterizes adjointable maps between linear orthosets as induced by quasilinear maps and connects these to classical results like Wigner's theorem, extending understanding of structure-preserving transformations.

Findings

Adjointable maps are induced by quasilinear maps under mild conditions.

Orthoisomorphisms in 3-dimensional Hermitian spaces are induced by quasiunitary maps.

Orthomodular spaces of dimension ≥4 are characterized by irreducible Fréchet orthosets with adjointable inclusions.

Abstract

Given an (anisotropic) Hermitian space $H$, the collection $P(H)$ of at most one-dimensional subspaces of $H$, equipped with the orthogonal relation $\perp$ and the zero linear subspace $\{0\}$, is a linear orthoset and up to orthoisomorphism any linear orthoset of rank $\geq 4$ arises in this way. We investigate in this paper the correspondence of structure-preserving maps between Hermitian spaces on the one hand and between the associated linear orthosets on the other hand. Our particular focus is on adjointable maps. We show that, under a mild assumption, adjointable maps between linear orthosets are induced by quasilinear maps between Hermitian spaces and if the latter are linear, they are adjointable as well. Specialised versions of this correlation lead to Wigner-type theorems; we see, for instance, that orthoisomorphisms between the orthosets associated with at least $3$-dimensional Hermitian spaces are induced by quasiunitary maps. In addition, we point out that orthomodular spaces of dimension $\geq 4$ can be characterised as irreducible Fr\'echet orthosets such that the inclusion map of any subspace is adjointable. Together with a transitivity condition, we may in this way describe the infinite-dimensional classical Hilbert spaces.