View source for Solèr theorem

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201801.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201802.png" />-field, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201803.png" /> a left [[Vector space|vector space]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201804.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201805.png" /> an orthomodular form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201806.png" /> that has an infinite orthonormal sequence (see below for definitions). Then Solèr's theorem states that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201807.png" /> must be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s1201809.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018011.png" /> is the corresponding [[Hilbert space|Hilbert space]] [[#References|[a7]]].

==Definitions.==
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018013.png" />-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018014.png" /> is a (commutative or non-commutative) [[Field|field]] with involution. (An involution is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018015.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018016.png" /> onto itself that satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018018.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018020.png" />.)

The set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018021.png" /> with the identity involution, the set of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018022.png" /> with complex conjugation as involution, and the set of real quaternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018023.png" /> with the usual quaternionic conjugation as involution are the three classical examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018024.png" />-fields.

A Hermitian form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018025.png" /> on a left [[Vector space|vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018026.png" /> over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018027.png" />-field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018028.png" /> is a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018029.png" /> that associates to every pair of vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018030.png" /> a scalar <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018031.png" /> in accordance with the following rules:

i) it is linear in the first variable and conjugate linear with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018032.png" /> in the second variable (in short: it is conjugate bilinear);

ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018033.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018034.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018035.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018036.png" /> (in short: it is regular);

iii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018038.png" /> (the Hermitian property). Two vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018040.png" /> in a Hermitian space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018041.png" /> are orthogonal when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018042.png" />. A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018043.png" /> of non-zero vectors is called orthogonal when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018044.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018045.png" />; and it is called orthonormal when also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018047.png" />.

Given a non-empty subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018048.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018049.png" />, the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018050.png" /> stands for the set of those elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018051.png" /> that are orthogonal to every element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018052.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018053.png" /></td> </tr></table>

A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018054.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018055.png" /> is called closed when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018056.png" />. A Hermitian space is orthomodular when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018057.png" /> for every closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018058.png" />; in symbols: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018059.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018060.png" /> imply <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018061.png" />.

A Hilbert space is a Hermitian space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018063.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018064.png" /> whose form is positive definite, i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018065.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018066.png" />, and which is complete with respect to the metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018067.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018068.png" />-field is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018070.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018071.png" />; and

2) the resulting space is metrically complete. The first conclusion, which materializes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018073.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018074.png" /> out of all possible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018075.png" />-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 [[#References|[a5]]], then proved by I. Amemiya and H. Araki [[#References|[a1]]]. Keller's example of a  "non-classical"  Hilbert space shows that Solèr's result no longer holds if only orthomodularity is assumed [[#References|[a3]]]. Prior to Solèr's definitive result, a vital contribution was made by W.J. Wilbur [[#References|[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 [[#References|[a4]]] is a paean to the legacy of Gross. Reference [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018076.png" /> is commutative with the identity involution. Another proof of Solèr's theorem in the general case has been provided by A. Prestel [[#References|[a6]]]. Solèr's theorem has applications to Baer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s120/s120180/s12018078.png" />-rings, infinite-dimensional projective geometries, orthomodular lattices, and quantum logic [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I. Amemiya,   H. Araki,   "A remark on Piron's paper"  ''Publ. Res. Inst. Math. Sci.'' , '''A2'''  (1966/67)  pp. 423–427</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.S. Holland Jr.,   "Orthomodularity in infinite dimensions: a theorem of M. Solèr"  ''Bull. Amer. Math. Soc.'' , '''32'''  (1995)  pp. 205–234</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.A. Keller,   "Ein nicht-klassischer Hilbertscher Raum"  ''Math. Z.'' , '''172'''  (1980)  pp. 41–49</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  "Orthogonal geometry in infinite dimensional spaces"  H.A. Keller (ed.)  U.-M. Künzi (ed.)  M. Wild (ed.) , ''Bayreuth. Math. Schrift.'' , '''53'''  (1998)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C. Piron,   "Axiomatique quantique"  ''Helv. Phys. Acta'' , '''37'''  (1964)  pp. 439–468</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Prestel,   "On Solèr's characterization of Hilbert spaces"  ''Manuscripta Math.'' , '''86'''  (1995)  pp. 225–238</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M.P. Solèr,   "Characterization of Hilbert spaces with orthomodular spaces"  ''Commun. Algebra'' , '''23'''  (1995)  pp. 219–243</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W. John Wilbur,   "On characterizing the standard quantum logics"  ''Trans. Amer. Math. Soc.'' , '''233'''  (1977)  pp. 265–292</TD></TR></table>