HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM LOGIC 373 Edited by K. Engesser, D. M. Gabbay and D. Lehmann © 2009 Elsevier B.V. All rights reserved SOLER’S` THEOREM Alexander Prestel Sol`er’s Theorem gives an axiomatic characterization (in algebraic terms) of in- finite dimensional Hilbert spaces over the reals, the complex, and the quaternions. At the same time it gives a characterization of the lattices that are isomorphic to the lattice of closed subspaces of the just mentioned Hilbert spaces. These lattices play an important role in quantum logic (see [Holland, 1995]) and, more gene- rally, in Hilbert space logic (see [Engesser and Gabbay, 2002]). More about the history of Sol`er’sTheorem and its consequences in areas like Baer *-rings, infinite dimensional projective geometry, orthomodular lattices, and the logic of quantum mechanics can be found in S. Holland’s exposition [Holland, 1995]. The aim of this paper is to provide a complete proof of Sol`er’sTheorem for the reader of the ‘Handbook of Quantum Logic and Quantum Structures’.1 Sol`er’s Theorem deals with infinite dimensional hermitian spaces (E,<>)over an arbitrary skew field K which are orthomodular, i.e., every subspace X of E satisfies (1) X =(X⊥)⊥ ⇒ E = X ⊕ X⊥. The theorem can be stated as follows (for definitions see below): Sol`er’s Theorem Let (K,∗ ) be a skew field together with an (anti-)involution ∗,andlet(E,<>) be an infinite dimensional hermitian vector space over (K,∗ ).If (E,<>) is orthomodular and contains an infinite orthonormal sequence (en)n∈N, then (i) K = R, C, or H where in the first case ∗ is the identity and in the case of C and the quaternions H,∗ is the canonical conjugation, (ii) (E,<>) is a Hilbert space over R, C or H,resp. The main part of this theorem actually is (i). Once we know that K is the field of real or complex numbers, or the skew field H of the quaternions, it is not difficult to prove (ii) (cf. the end of Section 2). While the (skew) fields R, C and H carry a canonical metric that makes them complete, there is no mention of any metric on 1The proof presented here is identical with that of [Prestel, 1995]. 374 Alexander Prestel K in the assumption of the theorem. Not even any topology is mentioned. Nevert- heless, the seemingly ‘algebraic’ conditions of orthomodularity and the existence of an infinite orthonormal sequence will lead to the surprising fact that K is R, C or H. It has been a long standing open problem whether orthomodularity of (E,<>) would already force K to be one of R, C or H. In 1980, finally, H. Keller (see [Kel- ler, 1980]) constructed “non-classical Hilbert spaces”, i.e., orthomodular hermitian spaces, not isomorphic to one of the classical Hilbert spaces. Only in 1995, M.P. Sol`er showed in her Ph.D. thesis (cf [Sol`er, 1995]) that adding the existence of an infinite orthonormal sequences expelles all the non-classical Hilbert spaces. In sections 2 to 4 below we shall first treat the commutative case, i.e., we let K be a commutative field. Under this additional assumption we give a complete proof of Sol`er’sTheorem. In Section 4 we deal with the general case, i.e., we let K be an arbitrary skew field. The proof in this case needs a few refinements of the earlier ones which we shall explain. The structure of the proof in the non-commutative case, however, is essentially the same as that in the commutative case. 1 PRELIMINARIES AND STRUCTURE OF THE PROOF Let K be a (commutative) field and ∗ : K → K an involution on K, i.e. ∗ satisfies (α + β)∗ = α∗ + β∗, (αβ)∗ = α∗β∗,α∗∗ = α for all α, β ∈ K.TheidentityonR and the complex conjugation − on C are our standard examples for such an involution. Furthermore, let E be an infinite dimensional K-vector space and <>: E ×E → K a hermitian form on E, i.e. <> satisfies <αx+ βy,z > = α<x,z>+ β<y,z> <z,αx+ βy > = α∗ <z,x>+ β∗ <z,y> <x,z> = <z,x>∗ for all α, β ∈ K and x, y, z ∈ E, The pair (E,<>) is called a hermitian space if <> is a hermitian form on E. The hermitian form <> is called anisotropic if for x ∈ E, <x,x>= 0 implies x =0. As usual, we define orthogonality x ⊥ y by <x,y>= 0 for vectors x, y ∈ E. The orthogonal space U ⊥ to a subset U of E is defined by U ⊥ = {x ∈ E| x ⊥ u for all u ∈ U}. We simply write U ⊥⊥ for (U ⊥)⊥. The space U ⊥⊥ is called the closure of U,andU is called closed if U ⊥⊥ = U. In case (E,<>) is anisotropic, every finite dimensional Sol`er’sTheorem 375 subspace of E is closed. Since U ⊥⊥⊥ = U ⊥, U ⊥ is also closed. Thus, in particular, orthomodularity of (E,<>) may be equivalently expressed by (1) E = U ⊥ ⊕ U ⊥⊥ for all subspaces U of E. Two important consequences of orthomodularity of a hermitian space are (2) if U and V are orthogonal subspaces of E then (U + V )⊥⊥ = U ⊥⊥ + V ⊥⊥ and U ⊥⊥ ⊥ V ⊥⊥, (3) if U is a closed subspace of E, then U together with the restriction of <> to U × U is an orthomodular space, too. Applying (1) to U = {x} we see that orthomodular spaces are anisotropic. A finite dimensional hermitian space is orthomodular if and only if it is anisotropic. It is also well-known that any R-Hilbert space and any C-Hilbert space is orthomo- dular (w.r.t. its defining inner product). The content of the above theorem is that there are no other examples of hermitian spaces (E,<>) which are orthomodular and contain an infinite sequence (en)n∈N which is orthonormal, i.e. for all n, m ∈ N we have: (4) <en,en >=1anden ⊥ em for n = m. As H. Keller has shown in [Keller, 1980], infinite dimensional orthomodular spaces exist which do not contain any orthonormal sequence, hence cannot be real or complex Hilbert spaces. Now let us explain the three steps into which we will divide the proof of the theorem. In Step 1 we will show that <> is ‘positive definite’, i.e. we will show that the fixed field F = {α ∈ K| α∗ = α} of ‘symmetric’ elements admits an ordering ≤ such that <x,x>≥ 0 for all x ∈ E. (Note that <x,x>is symmetric.) In fact, the ordering on F will be given by (5) α ≤ β iff β − α ∈ P, where P = {<x,x>| x ∈ E} is the set of ‘lengths’ of vectors x ∈ E. In Step 2 we will show first that ≤ is archimedean on F , i.e. (6) to every α ∈ P there exists n ∈ N such that α ≤ n. At this point we make use of the fact that, in order to prove that (5) defines an ordering, it suffices to know here that ≤ is a semi-ordering (as introduced in 376 Alexander Prestel [Prestel, 1984] by the author), i.e. ≤ linearly orders F such that in addition we have for α, β ∈ F : 0 ≤ α, 0 ≤ β =⇒ 0 ≤ α + β (7) 0 ≤ α =⇒ 0 ≤ αβ2 0 ≤ 1 In fact, an archimedean semi-ordering is already an ordering, i.e., it satisfies in addition 0 ≤ α, 0 ≤ β =⇒ 0 ≤ α · β for all α, β ∈ F ([Prestel, 1984], Theorem 1.20). Thus, in Step 1 it therefore suffices to show that (5) defines a semi-ordering on F . The linearity of ≤ is obtained from Sol`er’s main Lemma 5, which also gives the archimedeanity (Lemma 6). Actually, the linearity of ≤, i.e. P ∪−P = F , is not important in the commutative case, since it may be simply obtained by maximalizing a subset P of F satisfying 2 P + P ⊂ P, PF ⊂ P, P ∩−P = {0} and 1 ∈ P by Zorn’s Lemma to some P0, w.r.t. to these properties. The maximal object P0 then satisfies P0 ∪−P0 = F (see [Prestel, 1984], Lemma 1.13). In the non-commutative case, however, this method may not work (see the remarks at the end of Section 6). As it is well-known, an archimedean ordered field (F, ≤) contains (an isomor- phic) copy of the rational number field Q as a dense subfield. Thus, it suffices to prove that every Dedekind cut is realized in (F, ≤) in order to find that F is iso- morphic to R. This is done in Lemma 7. Since F is the fixed field of the involution ∗,wehave[K : F ] ≤ 2. Hence K = R and ∗=id or K = C and ∗ is the complex conjugation. NowweknowthatK is R or C and <> is positive definite, i.e., (E,<>)isa pre-Hilbert space. Thus, in Step 3 it remains to prove that every orthomodular pre-Hilbert space is complete. This, however, can already be found in the literature (see [Maeda and Maeda, 1970], Theorem 34.9). The argument runs as follows. Let Eˆ be the completion of E w.r.t. the metric induced by <>.Givena ∈ Eˆ we have to show that a already belongs to E.Choosingc ∈ E suitably such that <a,c>= 0, we find b ∈ Eˆ such that a ⊥ b and c = a + b. By standard arguments we then find sequences (an)n∈N and (bn)n∈N in E converging to a and b resp. such that an ⊥ bm for all n, m ∈ N.Ifwethentake ⊥ A = {bn| n ∈ N} we see that A⊥⊥ = A in E and thus the modularity (1) of E gives us a decomposition ⊥ c = c1 + c2 with c1 ∈ A and c2 ∈ A .