Solèr theorem

Let $\mathcal{K}$ 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