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arXiv:1512.01007

Orthogonal apartments in Hilbert Grassmannians

Mark Pankov

Published Dec 3, 2015• 0 citations
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Abstract

Let $H$ be an infinite-dimensional complex Hilbert space and let ${\mathcal L}(H)$ be the logic formed by all closed subspaces of $H$. For every natural $k$ we denote by ${\mathcal G}_{k}(H)$ the Grassmannian consisting of $k$-dimensional subspaces. An orthogonal apartment of ${\mathcal G}_{k}(H)$ is the set consisting of all $k$-dimensional subspaces spanned by subsets of a certain orthogonal base of $H$. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of ${\mathcal G}_{k}(H)$. We show that every bijective transformation $f$ of ${\mathcal G}_{k}(H)$ such that $f$ and $f^{-1}$ send orthogonal apartments to orthogonal apartments (in other words, $f$ preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of ${\mathcal L}(H)$.

DOI
10.48550/arxiv.1512.01007

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Let HHH be an infinite-dimensional complex Hilbert space and let L(H){\mathcal L}(H)L(H) be the logic formed by all closed subspaces of HHH. For every natural kkk we denote by Gk(H){\mathcal G}_{k}(H)Gk​(H) the Grassmannian consisting of kkk-dimensional subspaces. An orthogonal apartment of Gk(H){\mathcal G}_{k}(H)Gk​(H) is the set consisting of all kkk-dimensional subspaces spanned by subsets of a certain orthogonal base of HHH. Orthogonal apartments can be characterized as maximal sets of mutually compatible elements of Gk(H){\mathcal G}_{k}(H)Gk​(H). We show that every bijective transformation fff of Gk(H){\mathcal G}_{k}(H)Gk​(H) such that fff and f−1f^{-1}f−1 send orthogonal apartments to orthogonal apartments (in other words, fff preserves the compatibility relation in both directions) can be uniquely extended to an automorphism of L(H){\mathcal L}(H)L(H).
We consider the standard quantum logic L(H){\mathcal L}(H)L(H) associated to a complex Hilbert space HHH, i.e. the lattice of closed subspaces of HHH together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic L(H){\mathcal L}(H)L(H)) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of L(H){\mathcal L}(H)L(H) and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.
Abstract The collection of one-dimensional subspaces of an anisotropic Hermitian space is naturally endowed with an orthogonality relation and represents the typical example of what is called an orthogonality space: a set endowed with a symmetric, irreflexive binary relation. We investigate in this paper symmetry properties of orthogonality spaces. We show that two conditions concerning the existence of automorphisms of orthogonality spaces are essentially sufficient to characterise the basic model of quantum physics, the countably infinite dimensional complex Hilbert space.
Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.
We prove rigidity results describing contextually-constrained maps defined on Grassmannians and manifolds of ordered independent line tuples in finite-dimensional vector or Hilbert spaces. One statement in the spirit of the Fundamental Theorem of Projective Geometry classifies maps between full Grassmannians of two nnn-dimensional Hilbert spaces, n≥3n\ge 3n≥3, preserving dimension and lattice operations for pairs with commuting orthogonal projections, as precisely those induced by semilinear injections unique up to scaling. In a different but related direction, denote the manifolds of ordered orthogonal (linearly-independent) nnn-tuples of lines in an nnn-dimensional Hilbert space VVV by F⊥(V)\mathbb{F}^{\perp}(V)F⊥(V) (respectively F(V)\mathbb{F}(V)F(V)) and, for partitions πππ of the set {1..n}\{1..n\}{1..n}, call two tuples πππ-linked if the spans along πππ-blocks agree. A Wigner-style rigidity theorem proves that the symmetric maps F⊥(Cn)→F(Cn)\mathbb{F}^{\perp}(\mathbb{C}^n)\to \mathbb{F}(\mathbb{C}^n)F⊥(Cn)→F(Cn), n≥3n\ge 3n≥3 respecting πππ-linkage are precisely those induced by semilinear injections, hence by linear or conjugate-linear maps if also assumed measurable. On the other hand, in the F(Cn)\mathbb{F}(\mathbb{C}^n)F(Cn)-defined analogue the only other possibility is a qualitatively new type of purely-contextual-global symmetry transforming a tuple (ℓi)i(\ell_i)_i(ℓi​)i​ of lines into ((⨁j≠iℓj)⊥)i\left(\left(\bigoplus_{j\ne i}\ell_j\right)^{\perp}\right)_i((⨁j=i​ℓj​)⊥)i​.
Let H be a Hilbert space and E ( H ) the effect algebra on H . A bijective map is called an ortho-order automorphism of E ( H ) if for every we have and . The classical theorem of Ludwig states that every such ϕ is of the form , , for some unitary or antiunitary operator U . It is also known that each bijective map on E ( H ) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.