Solèr theorem
Let be a *-field, E a left vector space over \mathcal{K}, and \langle \, .\, ,\, . \, \rangle an orthomodular form on E that has an infinite orthonormal sequence (see below for definitions). Then Solèr's theorem states that \mathcal{K} must be \mathbf{R}, \mathbf{C}, or \bf H, and \{ E , \mathcal{K} , \langle \cdot , \cdot \rangle \} is the corresponding Hilbert space [a7].
Definitions.
A *-field \mathcal{K} is a (commutative or non-commutative) field with involution. (An involution is a mapping \alpha \mapsto \alpha ^ { * } of \mathcal{K} onto itself that satisfies ( \alpha + \beta ) ^ { * } = \alpha ^ { * } + \beta ^ { * }, ( \alpha \beta ) ^ { * } = \beta ^ { * } \alpha ^ { * }, and \alpha ^ { * * } = \alpha for all \alpha , \beta \in \cal{K}.)
The set of real numbers \mathbf{R} with the identity involution, the set of complex numbers \mathbf{C} with complex conjugation as involution, and the set of real quaternions \bf H with the usual quaternionic conjugation as involution are the three classical examples of *-fields.
A Hermitian form \langle \, .\, ,\, . \, \rangle on a left vector space E over a *-field \mathcal{K} is a mapping E \times E \rightarrow \mathcal{K} that associates to every pair of vectors x , y \in E a scalar \langle x , y \rangle \in \mathcal{K} in accordance with the following rules:
i) it is linear in the first variable and conjugate linear with respect to * in the second variable (in short: it is conjugate bilinear);
ii) \langle a , x \rangle = 0 or \langle x , a \rangle = 0 for all x \in E implies a = 0 (in short: it is regular);
iii) \langle x , y \rangle ^ { * } = \langle y , x \rangle for all x , y \in E (the Hermitian property). Two vectors x, y in a Hermitian space \{ E , \mathcal{K} , \langle \cdot , \cdot \rangle \} are orthogonal when \langle x , y \rangle = 0. A sequence \{ e _ { i } : i = 1,2 , \ldots \} of non-zero vectors is called orthogonal when \langle e _ { i } , e _ { j } \rangle = 0 for i \neq j; and it is called orthonormal when also \langle e _ { i } , e _ { i } \rangle = 1, i = 1,2 , \dots.
Given a non-empty subset S of E, the symbol S ^ { \perp } stands for the set of those elements in E that are orthogonal to every element of S:
\begin{equation*} S ^ { \perp } = \{ x \in E : \langle x , s \rangle = 0 \text { for all } s \in S \}. \end{equation*}
A subspace M of E is called closed when M = M ^ { \perp \perp }. A Hermitian space is orthomodular when M + M ^ { \perp } = E for every closed subspace M; in symbols: \emptyset \neq M \subseteq E and M = M ^ { \perp \perp } imply M + M ^ { \perp } = E.
A Hilbert space is a Hermitian space over \mathbf{R}, \mathbf{C}, or \bf H whose form is positive definite, i.e., x \neq 0 implies \langle x , x \rangle > 0, and which is complete with respect to the metric \rho ( x , y ) = \langle x - y , x - y \rangle ^ { 1 / 2 } derived from that form. The well-known projection theorem asserts that a Hilbert space is orthomodular.
History.
There are two conclusions in Solèr's theorem:
1) the underlying *-field is \mathbf{R}, \mathbf{C} or \bf H; and
2) the resulting space is metrically complete. The first conclusion, which materializes \mathbf{R}, \mathbf{C}, and \bf H out of all possible *-fields, is Solèr's contribution. Of the two conclusions, the first is by far the most difficult to establish, the most striking in its appearance, and the most far-reaching in its consequences. The metric completeness was actually surmised earlier by C. Piron [a5], then proved by I. Amemiya and H. Araki [a1]. Keller's example of a "non-classical" Hilbert space shows that Solèr's result no longer holds if only orthomodularity is assumed [a3]. Prior to Solèr's definitive result, a vital contribution was made by W.J. Wilbur [a8].
M.P. Solèr is a student of the late Professor H. Gross. This result is her 1994 doctoral thesis at the University of Zürich. The volume [a4] is a paean to the legacy of Gross. Reference [a4], Article by Keller–Künzi–Solèr, discusses the orthomodular axiom in depth, and contains a detailed proof of Solèr's theorem in the case where \mathcal{K} is commutative with the identity involution. Another proof of Solèr's theorem in the general case has been provided by A. Prestel [a6]. Solèr's theorem has applications to Baer *-rings, infinite-dimensional projective geometries, orthomodular lattices, and quantum logic [a2].
References
[a1] | I. Amemiya, H. Araki, "A remark on Piron's paper" Publ. Res. Inst. Math. Sci. , A2 (1966/67) pp. 423–427 |
[a2] | S.S. Holland Jr., "Orthomodularity in infinite dimensions: a theorem of M. Solèr" Bull. Amer. Math. Soc. , 32 (1995) pp. 205–234 |
[a3] | H.A. Keller, "Ein nicht-klassischer Hilbertscher Raum" Math. Z. , 172 (1980) pp. 41–49 |
[a4] | "Orthogonal geometry in infinite dimensional spaces" H.A. Keller (ed.) U.-M. Künzi (ed.) M. Wild (ed.) , Bayreuth. Math. Schrift. , 53 (1998) |
[a5] | C. Piron, "Axiomatique quantique" Helv. Phys. Acta , 37 (1964) pp. 439–468 |
[a6] | A. Prestel, "On Solèr's characterization of Hilbert spaces" Manuscripta Math. , 86 (1995) pp. 225–238 |
[a7] | M.P. Solèr, "Characterization of Hilbert spaces with orthomodular spaces" Commun. Algebra , 23 (1995) pp. 219–243 |
[a8] | W. John Wilbur, "On characterizing the standard quantum logics" Trans. Amer. Math. Soc. , 233 (1977) pp. 265–292 |
Solèr theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sol%C3%A8r_theorem&oldid=49915