A characterisation of orthomodular spaces by Sasaki maps

Given a Hilbert space $H$, the set $P(H)$ of one-dimensional subspaces of $H$ becomes an orthoset when equipped with the orthogonality relation $\perp$ induced by the inner product on $H$. Here, an \emph{orthoset} is a pair $(X,\perp)$ of a set $X$ and a symmetric, irreflexive binary relation $\perp$ on $X$. In this contribution, we investigate what conditions on an orthoset $(X,\perp)$ are sufficient to conclude that the orthoset is isomorphic to $(P(H),\perp)$ for some orthomodular space $H$, where \emph{orthomodular spaces} are linear spaces that generalize Hilbert spaces. In order to achieve this goal, we introduce \emph{Sasaki maps} on orthosets, which are strongly related to Sasaki projections on orthomodular lattices. We show that any orthoset $(X,\perp)$ with sufficiently many Sasaki maps is isomorphic to $(P(H),\perp)$ for some orthomodular space, and we give more conditions on $(X,\perp)$ to assure that $H$ is actually a Hilbert space over $\mathbb R$, $\mathbb C$ or $\mathbb H$.


Introduction
In order to obtain a deeper understanding of quantum physics, it is imperative to analyse the structure of complex Hilbert spaces, the objects that are used to represent quantum systems mathematically. For the following, it is useful to dissect the definition of a Hilbert space as follows. Firstly, we call a division ring equipped with an involutorial antiautomorphism (−) ⋆ a ⋆-sfield. Then an (anisotropic) Hermitian space is a linear space over a ⋆-sfield K that is equipped with an anisotropic, symmetric sesquilinear form (−, −) : H × H → K. In case H is a Hermitian space over one of the classical ⋆-sfields, that is, R, C, or H, and if in addition H is complete with respect to the norm induced by the inner product, we call H a (classical) Hilbert space.
One way to analyse the structure of Hilbert spaces, which we aim to follow, consists of trying to find different axioms for Hilbert spaces in terms of simpler structures. For example, the closed subspaces of Hilbert spaces form a complete orthomodular lattice, hence one could choose orthomodular structures as a starting point, and try to add conditions to assure that an orthomodular lattice is isomorphic to the lattice of closed subspaces of a Hilbert space.
Since we aim to disentangle the structure of Hilbert spaces as much as possible, we choose for an even simpler structure. Our starting point is the orthogonality relation ⊥ on a Hilbert space, or more generally, a Hermitian space H: we say that x, y ∈ H are orthogonal and write x ⊥ y if (x, y) = 0. The relation ⊥ induces a symmetric and irreflexive binary relation on the set P (H) of one-dimensional subspaces of H, representing the pure states of the quantum system modelled by H. This induced relation, which we also denote by ⊥, is called the orthogonality relation and keeps track of transitions between pure states that have a probability of zero. The pair (P (H), ⊥) is perhaps the bare minimum of a possible structure necessary for quantum physics, and Foulis proposed to generalize this structure as follows: Definition 1.1. An orthoset (sometimes also called an orthogonality space) is a set X equipped with a symmetric and irreflexive binary relation ⊥, called the orthogonality relation. We call x, y ∈ X orthogonal if x ⊥ y. The rank of X is the supremum of the cardinalities of sets of mutually orthogonal elements.
We refer to [Dac, Wil] for more details on orthosets, and discuss here only the concepts we need for the following. The definition of an orthoset is essentially that of an undirected graph. However, only a few graph-theoretic concepts are relevant for our purposes. For instance a morphism of orthosets is simply defined as a graph homomorphism. Moreover, a subset of mutually orthogonal elements of an orthoset, also called a ⊥-set, is precisely a clique of the orthoset regarded as a graph. Then the rank of the orthoset is the supremum of the cardinalities of its cliques.
The main motivation for considering orthosets doesn't lie in graph theory, but in our example of the one-dimensional subspaces of a Hilbert spaces, which can be generalized to Hermitian spaces as follows: We note that also other quantum structures give rise to orthosets, for instance the non-empty closed subspaces of a Hilbert space, representing the propositions of the quantum system represented by H. This example can be generalized to orthocomplemented posets, i.e., bounded posets P equipped with an order-reversing involution (−) ⊥ : P → P such that x ∧ x ⊥ = 0 for each x ∈ P . Example 1.3. Let P be an orthocomplemented poset. Let X = P \ {0}. Then X equipped with relation ⊥ on X defined by x ⊥ y if and only if x y ⊥ is an orthoset.
Since we view orthosets as abstractions of pure state spaces, we are less interested in the orthosets in Example 1.3. Instead, we are interested in finding conditions that assure that a given orthoset (X, ⊥) is isomorphic to the orthoset in Example 1.2 for some Hilbert space H. In particular, this means we aim to achieve the following three goals: (i) finding conditions on (X, ⊥) that assure that X = P (H) for some Hermitian space H over a ⋆-sfield K; (ii) finding conditions on (X, ⊥) that imply that K = C; (iii) finding conditions on (X, ⊥) that guarantee that H is metrically complete with respect to the norm on H induced by the form (−, −).
The last goal can be reformulated as follows. Given a Hermitian space H, for any subset S ⊆ H let S ⊥ be the set of all x ∈ X such that x ⊥ y for each y ∈ S. Then S ⊥ is a subspace of H and any subspace of this form is called closed. We say that H is orthomodular if M + M ⊥ = H for each closed subspace M of H. Now, a Hermitian space H over one of the classical ⋆-sfields, i.e., C, R or H, is orthomodular if and only if H is metrically complete [AmAr], so (iii) can be replaced by: (iii') finding conditions on (X, ⊥) that guarantee that H is an orthomodular space.
We note that (i) and (iii') can be combined into: (i') finding conditions on (X, ⊥) that assure that X = P (H) for some orthomodular space H over a ⋆-sfield K.
This contribution deals with goal (i'). Under the assumption that (i') has been achieved, (ii) already has been dealt with in a satisfactory way by means of automorphisms, which is a natural choice of structure, since for any Hilbert space H, automorphisms on (P (H), ⊥) are related to unitary transformations on H, which describe the symmetries of the physical system represented by H such as time evolution. See also Section 5.
With respect to goal (i'), it has been known for a long time how to characterize orthomodular spaces by lattice-theoretic means. However, the involved properties do not have a straightforward physical meaning. Moreover, this requires associating lattices to orthosets, whereas we prefer to formulate our conditions directly in the framework of orthosets.
More explicitly, given A ⊆ X, we can define the set A ⊥ := {x ∈ X : x ⊥ a for each a ∈ A}. We call A ⊆ X orthoclosed if A = A ⊥⊥ . Then the set C(X, ⊥) of all orthoclosed subsets of X is a complete ortholattice if we order it by inclusion and with (−) ⊥ as orthocomplementation.
Conversely, given an atomistic ortholattice L with orthocomplementation (−) ⊥ , the set At(L) of atoms of L becomes an orthoset when equipped with the relation ⊥ defined by a ⊥ b if and only a b ⊥ .
The next lemma states that these operations yield a one-to-one correspondence between some class of ortholattices and some class of orthosets: Lemma 1.4. [Vet22,Proposition 2] Let (X, ⊥) be a point-closed orthoset, i.e., every singleton subset of X is orthoclosed. Then L := C(X, ⊥) is a complete atomistic ortholattice, and the map X → At(L), x → {x} is an isomorphism of orthosets.
Conversely, given a complete atomistic ortholattice L, let X := At(L). Then (X, ⊥) is a point-closed orthoset, and the map L → C(X, ⊥), p → {x ∈ X : x p} is an isomorphism of ortholattices.
Because we are interested in models of pure state spaces, we will restrict ourselves to point-closed orthosets.
The crucial lattice-theoretic condition one usually imposes on C(X, ⊥) is the covering property, which an ortholattice L possesses if for each x ∈ L and atom a ∈ L such that a ∧ x = 0 (or equivalently, such that a x), we have that x ∨ a covers x. Hence, this is a property that has to be stated in the framework of ortholattices instead of in that of orthosets. Moreover, at first sight it is not clear what the physical interpretation of the covering property is.
We aim to solve all these issues by introducing so-called Sasaki maps on orthosets. These maps are closely related to Sasaki projections, which explains our choice of terminology, and are natural to use because their definition reminds us to that of a projection operator, since the first defining conditions expresses idempotency, and the second self-adjointness.
Definition 1.5. Let A be an orthoclosed subset of an orthoset (X, ⊥). A map is called a Sasaki map to A if the following conditions hold: (S1) ϕ(e) = e for all e ∈ A.
Here ∁S denotes the set-theoretic complement of a subset S of X. We call (X, ⊥) a Sasaki space if for any orthoclosed subset A of X, there exists a Sasaki map ϕ A to A.
An orthoset (X, ⊥) is called irreducible if X is not the disjoint union of two non-empty set A and B such that e ⊥ f for any e ∈ A and f ∈ B. Moreover, we note that given a Hermitian space H, all orthoclosed subsets of P (H) are of the form P (S), where S is a closed subspace S of H, hence a Hermitian space itself.
We are now able to formulate our main theorem, which implies that Sasaki maps do not only provide an alternative for the covering property, but they also provide a condition that is strong enough to assure that an orthoset is induced by a Hermitian space, even an orthomodular space.
Under this isomorphism, the Sasaki maps are, for any closed subspace S of H, given by The latter expression for ϕ P (S) indeed shows the relation between Sasaki maps and Sasaki projections.
We give an outline for the rest of the article. In Section 2, we give some examples of Sasaki spaces. In Section 3, we describe the relation between Sasaki spaces and orthomodular spaces, culminating in the proof of our main theorem. In Section 4, we show that Sasaki maps of a Sasaki space (X, ⊥) induce a full Sasaki set of projections on C(X, ⊥) in the sense of Finch. Finally, in the last section we give a characterization of infinite-dimensional classical Hilbert spaces in terms of Sasaki spaces.

Examples of Sasaki spaces
In this section, we explore some examples of Sasaki spaces. The most important example is provided by our guiding example: Proposition 2.1. Let H be an orthomodular space. Then (P (H), ⊥) is a Sasaki space.
Proof. Since H is orthomodular, it follows that for any closed subspace S of H the orthogonal projection P S associated to S exists. This map P S is an idempotent, selfadjoint linear operator. For any Before we identify some examples of Sasaki spaces that are not associated to some orthomodular space, we need one lemma.
Lemma 2.2. Let (X, ⊥) be an orthoset and let A be an orthoclosed subset of X. Then the each of the following conditions are sufficient to assure the existence of a Sasaki map for A are: Proof. For (a), we note that A ⊥ = ∁A implies ∁A ⊥ = A, whence we can take the identity on A. For (b), if ∁A ⊥ is a ⊥-set, then A ⊥ = ∁A, hence this case reduces to (a). Finally, given (c), i.e., if A is a singleton, say A = {a} for some a ∈ X, then the only possible map ϕ : ∁A ⊥ → A is the map that is constant a. Since ϕ clearly satisfies (S1), we only have to verify (S2). For each e ∈ ∁A ⊥ , we have e / ∈ A ⊥ , so e ⊥ a. Hence for each e, f ∈ ∁A ⊥ , we have ϕ(e) = a ⊥ f and e ⊥ a = ϕ(f ), so ϕ is a Sasaki map.
Example 2.3. Let X be a set and let ⊥ be the relation =, or equivalently (X, ⊥) is the complete undirected graph on |X| vertices. Then C(X, ⊥) = P(X), the power set of corresponding to the cyclic undirected graph on four vertices as depicted below: The name 'Sasaki map' suggests a connection with Sasaki projections on an orthomodular lattice. We include a definition: Definition 2.6. Let L be an orthomodular lattice. Then the map π x : L → L, y → x ∧ (x ⊥ ∨ y) is called the Sasaki projection to x ∈ L.
Indeed, in Proposition 2.8 below we will show that any Sasaki projection on a complete orthomodular lattice induces a Sasaki map on an orthoset associated to the lattice as in Example 1.3. Before we show this, we need the following facts about Sasaki projections: Lemma 2.7. Let L be an orthomodular lattice, and let x ∈ L. Then for each y, z ∈ L we have (a) y x if and only if π x (y) = y; Proof. Properties (a) and (b) follow from the paragraph between Lemma 5 and Theorem 6 in [Fou60]. Property (c) follows from [Fou62,Lemma 1]. For (d), assume that π x (y) ⊥ z, i.e., z π x (y) ⊥ . Since π x is clearly monotone, we have π x (z) π x (π x (y) ⊥ )), which implies π x (z) y ⊥ by (b). Thus y ⊥ π x (z). Since ⊥ is a symmetric relation, the implication in the other direction holds as well.
Proposition 2.8. Let L be a complete orthomodular lattice, and let (X, ⊥) be the orthoset obtained from L as in Example 1.3, so X = L \ {0}. Then (X, ⊥) is a Sasaki space with C(X, ⊥) = {X ∩ ↓ x : x ∈ L}, which is isomorphic to L as an orthomodular lattice. For any A = X ∩ ↓ x in C(X, ⊥), restricting and corestricting the Sasaki projection π x : L → L to a map ϕ A : ∁A ⊥ → A yields a Sasaki map to A.
Proof. Since 0 is the only element in L that is orthogonal to itself, it is clear that (X, ⊥) is an orthoset. Let x ∈ X. Then Moreover, for each non-empty S ⊆ X, we have Since any A ∈ C(X, ⊥) is of the form S ⊥ for some S ⊆ X, it follows that each A ∈ C(X, ⊥) is of the form X ∩ ↓ x for some x ∈ L. It is easy to verify that the resulting bijection ψ : L → C(X, ⊥), x → X ∩ ↓ x is an isomorphism of orthomodular lattices. Now for A ∈ C(X, ⊥), with A = X ∩ ↓ x for some x ∈ L, let ϕ A : ∁A ⊥ → A be the restriction and the corestriction of the Sasaki projection π x : L → L. We show that ϕ A is well defined by showing that π x (y) = 0 for each y ∈ ∁A ⊥ . Indeed, we have It now follows from (c) of Lemma 2.7 that for each y ∈ X, we have π x (y) = 0 if and only if y x ⊥ , i.e., if and only if y ∈ ∁A ⊥ . From (a) and (d) of the same lemma follows that ϕ A is a Sasaki map.

Orthomodular spaces
In this section, we aim to characterise orthomodular spaces by means of their associated Sasaki spaces.
We start by showing that the definition of a Sasaki space can be weakened slightly. We first need one lemma.
Lemma 3.1. Let (X, ⊥) be an orthoset, let A be an orthoclosed subspace of X, and let D be a maximal ⊥-set contained in A. If there exists a Sasaki map ϕ D ⊥ to D ⊥ , then A = D ⊥⊥ .
Proof. Since D ⊆ A, and A is orthoclosed, we have D ⊥⊥ ⊆ A. Assume that there is some e ∈ A \ D ⊥⊥ . Then e ⊥ D ⊥ , and since A ⊥ ⊆ D ⊥ , we have x ⊥ D ⊥ for each x ∈ A ⊥ . Hence e ⊥ x = ϕ D ⊥ (x) implies ϕ D ⊥ (e) ⊥ x for any x ∈ A ⊥ . This means that ϕ D ⊥ (e) ∈ A ⊥⊥ = A, and since ϕ D ⊥ (e) ⊥ D by definition of a Sasaki map for D ⊥ , we obtain a contradiction with the maximality of D.

Proposition 3.2. An orthoset (X, ⊥) is a Sasaki space if and only if for each
Proof. The 'only if' direction is trivial. For the other direction, let A be an orthoclosed subset of X and let D be a maximal ⊥-set contained in A ⊥ . Assume that there exists a Sasaki map ϕ D ⊥ to D ⊥ . It follows from Lemma 3.1 that A ⊥ = D ⊥⊥ . Since A is orthoclosed, we obtain A = A ⊥⊥ = D ⊥⊥⊥ = D ⊥ , which implies that ϕ D ⊥ is a Sasaki map to A.
An orthoset (X, ⊥) is called a Dacey space if its associated ortholattice C(X, ⊥) is orthomodular. The next example shows that not every orthoset is a Dacey space. The following useful criterion for this property is due to Dacey [Dac]. Proof. Let D be a maximal ⊥-set contained in an orthoclosed subset A of X. Since D ⊥ is orthoclosed, there exists a Sasaki map ϕ D ⊥ to D ⊥ , hence A = D ⊥⊥ by Lemma 3.1. Thus, it follows from Lemma 3.4 that (X, ⊥) is indeed a Dacey space.
From the last theorem it follows that the orthoset in Example 3.3 cannot be a Sasaki space, since it is not even a Dacey space.
Next, we present an example of a Dacey space that is not a Sasaki space.
Example 3.6. Let A = {0, a, a ⊥ , 1} be a Boolean algebra of four elements, and let Then L is an orthomodular lattice without the covering property: we have a ∧ b = 0, but a ∨ b = 1, which does not cover b. Next we investigate what conditions a Dacey space has to satisfy in order to be a Sasaki space, for which we need a lemma, which was stated in [Wil,4.4] without a proof. For the convenience of the reader, we include a proof. We note that with basic elements of a bounded poset, we mean elements that are either atoms or the least element of the poset. This concept was already introduced in [HHLN], and it allows for a more elegant formulation of the lemma.
Lemma 3.7. An orthomodular lattice L has the covering property if and only if all its Sasaki projections send basic elements to basic elements.
Proof. Assume L has the covering property. Let x ∈ L and let a ∈ L be an atom. Assume first that a x ⊥ . Then π x (a) = 0 by (c) of Lemma 2.7. Now assume that a x ⊥ . By orthomodularity of L, we have that x ⊥ ∨ a = x ⊥ ∨ y for some y ∈ L such that x ⊥ ⊥ y. By the covering property, x ⊥ is covered by x ⊥ ∨ a = x ⊥ ∨ y, whence y is an atom of L, and π Conversely, assume that all Sasaki projections on L send basic elements to basic elements. Let a ∈ At(L) and x ∈ L such that a x. Since L is orthomodular, there is a y ∈ L such that x ⊥ y and a ∨ x = x ∨ y. Note that y = 0, because otherwise a ∨ x = x contradicting that a x. Then π x ⊥ (a) = x ⊥ ∧ (x ∨ a) = x ⊥ ∧ (x ∨ y) = y by orthomodularity, which we can apply since x ⊥ y. Thus y must be a basic element of L, so an atom of L. Since y is an atom that is orthogonal to x, it follows that x is covered by x ∨ y = x ∨ a.
Lemma 3.8. Let (X, ⊥) be a point-closed Dacey space such that C(X, ⊥) has the covering property. Then (X, ⊥) is a Sasaki space.
Proof. Since (X, ⊥) is point-closed, {x} is an element of C(X, ⊥) for each x ∈ X.
Since (X, ⊥) is a Dacey space, C(X, ⊥) is an orthomodular lattice. Let A ∈ C(X, ⊥). Since C(X, ⊥) has the covering property, it follows from Lemma 3.7 that the Sasaki projection π A sends each atom of C(X, ⊥) to either the least element or another atom of C(X, ⊥), i.e., π A ({x}) is either empty or a singleton for each x ∈ X. By (c) of Lemma 2.7 we have π A ({x}) = ∅ if and only if {x} ⊆ A ⊥ , hence π A ({x}) = ∅ if and only if x ∈ ∁A ⊥ . By the covering property it then follows then that there is a unique ϕ(x) ∈ A such that π A ({x}) = {ϕ(x)}, which defines a map ϕ : ∁A ⊥ → A. Let x ∈ A. By Lemma 3.9. Let (X, ⊥) be a Sasaki space.
For any A ∈ C(X, ⊥) and e / ∈ A, we have Proof. Let x ⊥ A. Then e, x / ∈ A, hence e and x are in the domain of ϕ A ⊥ . Since ϕ A ⊥ (x) = x, we have that x ⊥ e is equivalent to x ⊥ ϕ A ⊥ (e). We conclude that The assertion now follows from the fact that (S ∪ T ) ⊥⊥ = S ⊥⊥ ∨ T ⊥⊥ for each S, T ⊆ X.
In the following, an AC lattice is meant to be an atomistic lattice with the covering property.
Proof. Since X is point-closed, C(X, ⊥) is atomistic, the atoms being {e}, e ∈ X.
To show the covering property, let A ∈ C(X, ⊥) and let e ∈ X be such that {e} A, that is, e / ∈ A. By Lemma 3.9, A ∨ {e} = A ∨ {ϕ A ⊥ (e)}. By Theorem 3.5, C(X, ⊥) is orthomodular and since {ϕ A ⊥ (e)} is an atom, we conclude that A ∨ {e} covers A.
We collect our results, and under the assumption of point-closedness, we further conclude that the Sasaki maps are uniquely determined.
Proposition 3.11. Any point-closed orthoset (X, ⊥) is a Sasaki space if and only if it is a Dacey space such that C(X, ⊥) is AC, in which case for any A ∈ C(X, ⊥), the Sasaki map to A is given by Proof. The equivalence follows from Lemmas 1.4, 3.8 and 3.10, and Theorem 3.5. By Lemma 3.9 we have that A ⊥ ∨ {e} = A ⊥ ∨ {ϕ A (e)}. The assertion follows now from orthomodularity.
For what follows, we need the characterization of orthomodular spacces by their associated subspace lattices. Furthermore, we call a lattice is irreducible if it is not isomorphic to the direct product of two lattices with at least two elements. The proof of the following theorem can be found in [MaMa,34.5].
Theorem 3.12. Let E be a orthomodular space. Then C(E), the ortholattice of orthoclosed subspaces of E, is a complete, irreducible, AC orthomodular lattice.
Conversely, let L be complete, irreducible, AC orthomodular lattice of height 4. Then there is an orthomodular space E such that L is isomorphic to C(E).
Under the further assumption of irreducibility, we can now show that Sasaki spaces arise from orthomodular spaces.
Theorem 3.13. Let (X, ⊥) be an irreducible, point-closed Sasaki space of rank 4. Then there is an orthomodular space H such that (X, ⊥) is isomorphic to (P (H), ⊥).
Under this isomorphism, the Sasaki maps are, for any closed subspace S of H, given by Proof. By Theorem 3.5 and Lemma 3.10, C(X, ⊥) is a complete AC orthomodular lattice of length 4. Furthermore, as (X, ⊥) is irreducible, so is C(X, ⊥) [Vet21,Lemma 3.6]. Hence, by Theorem 3.12, the first part follows. The second part is clear by Proposition 3.11.

Sasaki projections
In Theorem 3.5, we showed that the associated ortholattice C(X, ⊥) of a Sasaki space (X, ⊥) is an orthomodular lattice. In [Fin], Finch gave conditions on an orthocomplemented poset P that imply that P is an orthomodular lattice. Namely, he defined a Sasaki set of projections on P as a set S of maps P → P such that (i) Each π ∈ S is monotone; (ii) For each π 1 , π 2 ∈ S, we have π 1 (1) π 2 (1) implies π 1 • π 2 = π 1 ; (iii) π(π(x) ⊥ ) x ⊥ for each π ∈ S and each x ∈ P .
Furthermore, he called a Sasaki set of projection on P full if for each x ∈ P , there is some π x ∈ S such that π x (1) = x, and showed that the existence of a full Sasaki set of projections on P implies that P is an orthomodular lattice, and that the projection in S are actually Sasaki projections in the sense of Definition 2.6.
In this section, let (X, ⊥) be a Sasaki space, which is neither necessarily point-closed nor irreducible. We show that the Sasaki maps of (X, ⊥) induce a full Sasaki set of projections on C(X, ⊥).
In order to do so, for any A ∈ C(X, ⊥), we choose a Sasaki map ϕ A to A and we definē In the case that (X, ⊥) is point-closed, we may identify the elements of X with the singletons and then we may viewφ A as an extension of ϕ A . Indeed, for any e ∈ X we haveφ Furthermore, the next lemmas show thatφ A is determined by ϕ A by its property of being join-preserving.
Lemma 4.1. For any A, B, C ∈ C(X, ⊥), we havē and each c ∈ C \ A ⊥ , which is equivalent to saying that B ⊥φ A (C). By symmetry, the assertion follows.
Proof. Let B ι ∈ C(X, ⊥), ι ∈ I. By Lemma 4.1, we have for any C ∈ C(X, ⊥) that Note that the mapsφ A , A ∈ C(X, ⊥), are in a one-to-one correspondence with C(X, ⊥ ), asφ A (X) = A. These maps determine the order of C(X, ⊥) as follows.
By similar reasoning, we conclude thatφ B It is now evident that the mapsφ A , A ∈ C(X, ⊥), are, in Finch's sense, a full Sasaki set of projections for the involutive poset C(X, ⊥). Indeed, this means that the three properties in the following proposition hold.
Proposition 4.4. For any A, B ∈ C(X, ⊥), the following holds.
(i)φ A is order-preserving.

Classical Hilbert spaces
We have characterised orthomodular spaces, sometimes also called generalised Hilbert spaces, as Sasaki spaces. We conclude by pointing out that, in the case of infinite rank, we may employ known conditions to characterise the Hilbert spaces over the real or complex number field. The tool is Solèr's theorem [Sol]: Theorem 5.1. Let H be an orthomodular space over a ⋆-field K containing an infinite set of mutually orthogonal vectors of equal length. Then K is one of R, C, or H, that is, H is a classical Hilbert space.
Our postulate will again concern the existence of maps, which in this case, however, are symmetries. The following postulate was proposed, in similar form, e.g. in [AeSt] and [Vet19].
Definition 5.2. We call an orthoset (X, ⊥) transitive if, for any e, f ∈ X, there is an automorphism τ of X such that τ (e) = f and τ (x) = x for any x ⊥ e, f . Theorem 5.3. Let (X, ⊥) be a transitive, irreducible, point-closed Sasaki space of infinite rank. Then there is a Hilbert space H over R, C, or H such that (X, ⊥) is isomorphic to (P (H), ⊥).
Proof. By Theorem 3.13, there is an orthogonal space H over some ⋆-sfield K such that (X, ⊥) is isomorphic to (P (H), ⊥). As X has infinite rank, H is infinite-dimensional and hence possesses an infinite set e 1 , . . . of mutually orthogonal vectors. By transitivity, there is, for each i 1, an automorphism τ of (P (H), ⊥) such that τ ([e i ]) = [e i+1 ] and τ ([x]) = [x] for any x ⊥ e i , e i+1 . By Wigner's Theorem as formulated in [May, Lemma 1], τ is induced by a unitary operator on H. We conclude that there is an infinite set of mutually orthogonal vectors of equal length. By Theorem 5.1, the assertion follows.
To single out the field of real or the one of complex numbers, also automorphisms can used. We refer the reader to [May], see also [Vet19].