Solèr theorem

Let be a -field, a left vector space over , and an orthomodular form on that has an infinite orthonormal sequence (see below for definitions). Then Solèr's theorem states that must be , , or , and is the corresponding Hilbert space [a7].

Definitions.

A -field is a (commutative or non-commutative) field with involution. (An involution is a mapping of onto itself that satisfies , , and for all .)

The set of real numbers with the identity involution, the set of complex numbers with complex conjugation as involution, and the set of real quaternions with the usual quaternionic conjugation as involution are the three classical examples of -fields.

A Hermitian form on a left vector space over a -field is a mapping that associates to every pair of vectors a scalar 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) or for all implies (in short: it is regular);

iii) for all (the Hermitian property). Two vectors , in a Hermitian space are orthogonal when . A sequence of non-zero vectors is called orthogonal when for ; and it is called orthonormal when also , .

Given a non-empty subset of , the symbol stands for the set of those elements in that are orthogonal to every element of :

A subspace of is called closed when . A Hermitian space is orthomodular when for every closed subspace ; in symbols: and imply .

A Hilbert space is a Hermitian space over , , or whose form is positive definite, i.e., implies , and which is complete with respect to the metric 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 , or ; and

2) the resulting space is metrically complete. The first conclusion, which materializes , , and 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 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