There is a problem with the quantum logical significance of Soler's theorem.

Maria Pia Soler's theorem is a deep, difficult and above all extremely elegant purely algebraic characterization of infinite dimensional Hilbert spaces.

If $K$ is a skew field with involution, $V$ a $K$-vector space equipped with an hermitian form, then the following two conditions are equivalent:

(1) there is a infinite orthonormal sequence in $V$, and orthomodularity holds (for any subset $X$ of $V$, the orthogonal $X^\perp$ and the double orthogonal $X^{\perp\perp}$ give a algebraic direct sum decomposition of $V$)

(2) up to isomorphism (a unique one for the self-adjoint part of $K$), the $K$ with involution is one of the usual three (real, complex, quaternions, all with the usual involution) and $V$ with hermitian form is a infinite dimensional Hilbert space.

Among other things, this destroys the false belief that the concept of "Hilbert space" is something "reserved to analysts" with algebraists restricted only to more elementary cases by the lack of interaction with topology.

This mathematical achievement is not sufficiently celebrated. Above all because quantum logicians do celebrate it, but for the *wrong* reason. To understand why, start with a better known and simpler question.

- There are two ways to see why measures are real numbers.

The first: measures form a (open subset of) the positive cone in a totally ordered Archimedean abelian group with fixed unit (of measure),
and any such has exactly one embedding in the real ordered additive group with 1 as unit.
Either such a group is that of integers (with an arbitrary $n>0$ fixed as unit), or there is exactly one completion, the real number system.
Since nothing is lost in the completion, assume real numbers.

The second: measures form a cancellative commutative semigroup, which is also totally ordered. Embed the semigroup in a group,
complete it by Dedekind cuts (every poset has such a completion). You have a Dedekind complete totally ordered group, i.e. the real number system.

Everyone uses the first method (including Bourbaki, introducing real numbers essentially in this way). Why? Because the second *is wrong*.
Yes, you can "complete" a semigroup to obtain a group (and for cancellative abelian ones one has an embedding,
and a compatible total order uniquely extends), and yes, you can complete by cuts any poset (with a total order preserved as total). But the two constructions are
(in general) mutually incompatible: the order completion of a totally ordered semigroup (resp. poset or total order) is again such, but when you combine the two completions
(from semigroups to groups; from posets to complete lattices) you discover a incompatibility (semigroups losing cancellation of infinite elements), unless the order is Archimedean.

- The second question is about quantum logic, formalized as a "AC irreducible orthoposet with free mobility", which corresponds to the foundations of elementary geometry.

"Irreducible AC orthoposet" means poset with a orthocomplementation whose completion by cuts (which always exist and is a orthocomplemented complete lattice) is a complete AC irreducible lattice
as treated in Maeda - Maeda book "theory of symmetric lattices", including the equivalence with the ideal of finite height elements and the essentially unique coordinatization except for nonarguesian planes, lines and points:
finite dimensional subspaces of a vector space equipped with an anisotropic hermitian form, normalized by the existence of a vector of "length" 1 (so that the ortholattice with a fixed atom and the vector space with such a fixed vector
are really equivalent concepts, even in a categorical way).
Free mobility, more generally introduced by von Neumann in quantum logic,
in this context coincides with the one treated by Baer,
first in a paper (solving the linear algebra version of the "Riemann - Helmholtz free mobility problem") and then in the book about linear algebra and projective geometry.
The algebraic equivalent that Baer found is the generalization of the case treated by Hilbert's grundlagen (identity involution):
the skew field with involution is "$*$-pythagorean $*$-formally real" (a sum $xx^*+yy^*$ has again the form $zz^*$ and is zero only when $x=0=y$), and each finite dimensional subspace has a orthonormal basis for the form.
Really von Neumann had already treated a more general case involving $*$-regular rings,
and this was then re-discovered in many works, the most important ones by Vidav, Hendelman, Burke, Ara.

- There are two ways to see why an AC irreducible orthoposet with free mobility and length at least 4
(or equivalently the ortholattice of finite dimensional subspaces of a vector space of dimension at least 4 equipped with an anisotropic hermitian form with Baer condition),
to be also useful in quantum logic, must be (embeddable in) a standard, i.e. Hilbertian, one.

The (historically) second: assume a complete lattice and orthomodularity, then apply Soler's theorem (in infinite length; disregard finite length).

The (historically) first: disregard points (classical logic, no need for quantum), lines (noncontextual hidden variables, no need for quantum), nonarguesian planes (not embeddable in larger irreducibly quantum logics,
they have only classical interaction with other quantum components). Use von Neumann's transition probability primitive additional concepts (with relative axioms and proofs, for the easier type I case only)
to have a Hilbertian embedding (without any need of von Neumann's completeness axioms). Then observe that one can always complete the resulting structure, and in presence of the completeness axioms one has
a orthodox Hilbertian quantum logic.

Everyone uses the second method. Why? Perhaps because they avoid to read von Neumann (too long, too difficult?), but above all because they avoid to see that that the second method is *wrong*,
for the exact same reason that the second method is wrong for the measure problem. The two axioms, orthomodularity and complete lattice, can be separately justified
(any orthoposet has a completion by cuts that is again a orthoposet; any structure propositions - states has an internal logic which is orthomodular, being an orthoalgebra)
but they generally clash when used together (case of pre-Hilbert spaces, incomplete infinite dimensional spaces: the orthomodular poset of splitting subspaces is not a (complete) lattice;
the complete ortholattice of orthoclosed sets is not orthomodular). No completion process is generally known for an orthomodular structure to produce again a orthomodular (and complete lattice) structure.
Except for the metric completion given by von Neumann's method, which needs specific assumptions.

# Complementary notes.

- [A] There is a (not so big) difference between the two cases.

For the embedding of a semigroup in the reals, the non-trivial axiom, to be added to commutative cancellative and torsion free, is clear: add the new concept of order,
and postulate Archimedean (even to express the postulate "orderable" in not enough). An easy necessary and sufficient condition, and one physically "ideally falsifiable" by a "Gedankenexperiment"
(with any pair of homogeneous magnitudes, one can compare multiples on one with the other; for lengths, with subatomic scale vs. cosmological scale, this is not practical but at least ideally possible.
With arbitrary subsets, searching for Sup and Inf is not even ideally feasible. A equivalent form of completability is somewhat physically OK, completeness itself is not directly so).

For (the finite dimensional ideal of) a AC irreducible orthoposet with free mobility
to be embedded in a Hilbert space, again von Neumann's condition (add the new concept of state, or that of transition probability, and the relative axioms) is necessary and sufficient, but this time
it is not so easy to understand (and, in particular, is is not clear how much it can be formally weakened and still remain equivalent to standard embedability).
But it is still a Archimedean axiom with a noncommutative twist, because probabilities take values in the Archimedean $[0,1]$, that can be replaced by a unit interval in an
Archimedean totally ordered group with strong unit.

[One can also look at Chaqui work about "qualitative" foundations of probability using a binary relation "at least as probable as" between elements of a Boolean algebra; since for the type I$_n$ factors Boolean subalgebras are finite, one can apply Chaqui "countably additive" theory without philosophical problems.]

Once Popper criticized a specific point of
von Neumann - Birkhoff quantum logic,
but his argument was based on a plain mathematical error (or at least a misunderstanding leading to that error).
Only one review had the ardor, at the times, to say that there was something wrong with the argument.
But for the rest, total silence (both from Popper and from others) in printed sources.
Only many years after Popper left this world someone else has the courage to refer to the episode.
https://duckduckgo.com/?q=popper+quantum+logic+birkoff
Mathematics is not philosophy. One should not be afraid "by authority" to say when a mathematical argument (about quantum logic or whatever) is wrong.

- [C] The path leading to Soler.

Before Soler's theorem there were many attempts to characterize Hilbert spaces in a "quantum logic" way.

Before purely algebraically attempts, there were partially topological ones [Zierler; Cirelli Cotta-Ramusino Novati; Szambien]. Three steps: (1) use MacLaren - Piron theorem (1963, but it is a very easy and standard
extension of the finite dimensional setting of von Neumann and Birkhoff, 1936, using the fact that the ideal of finite dimensional elements completely determines the structure) to reach a vector space with an anisotropic hermitian form.
(2) Use the well known (since the 1930 at least) characterizations of the three standard $*$-sfields as the only non totally disconnected T$_0$ locally (countably) compact topological rings which are sfields,
and the (von Staudt) correspondence between the sfield operations and incidence geometry operations between points and line in a plane.
(3) Use Amemya - Araki theorem (over the classical $*$-sfields, a infinite dimensional pre-Hilbert space with orthomodularity is Hilbert); this last step is not needed if one only wants embedability and so show that a
orthomodular completion is possible.

[It seems to me that this partially topological approach is physically sensible, since the topological conditions can be seen to have sense looking at the need of a continuous evolution in time
of the system (pure states bijectively correspond with atoms [seen as proposition answering Mackey's yes-no questions "is the system in such a pure state?"], and evolution in time gives continuous paths;
(pre)compactness is required for pure states which are superpositions of a finite number of pure states).
However, for reasons that I cannot understand, this method (with its merit of not requiring completeness + orthomodularity, and showing the non obvious result that a orthomodular completion exists)
was not considered interesting by quantum logicians, who instead wanted a non-topological method using a complete orthomodular lattice. Note: I consider instead completely justified the mathematicians
who searched for a solution of a purely mathematical problem.]

Then Morash extended Amemiya - Araki to $*$-subfields of the quaternions and some pre-Hilbert spaces over them, Gross and coauthors (Kunzi, Keller) had quite general results
(among them, Keller's rightly celebrated example showing that orthomodularity alone does not force the $*$-field to be one of the three usual ones),
and one of the best results for many years was Wilbur's theorem (and Holland's re-working of its methods using $*$-ordered $*$-sfields).
The mayor problem with this is that only commutative and quaternional cases are covered; nothing can be said for non quaternional noncommutative $*$-sfields.

- [D] Personal recollections.

To apply Solers's theorem in quantum logic one has not only to justify a complete orthomodular lattice (a thing that is not as obvious as quantum logicians suppose, as noted above), but also the infinite orthonormal basis.
Even disregard the problem with the "infinite" part (one can dismiss logics with non-contextual hidden variables, and also non-arguesian planes since they have no embedding in any larger purely quantum logic and
so they admit obly classical interactions with other parts of the system. But why can one disregard all other finite dimensional irreducible components, or equivalently say that they must have an embedding in an infinite dimensional one?)
One still has the problem of justifying "orthonormal", and every attempt at this that I know boils down to a "free mobility" property of the orthogeometry, and so to von Neumann's own discussion of such a property
in quantum logic (a brief discussion in his "continuous geometries with a transition probability", but a fuller discussion in other manuscripts of his reviewed in his collected papers).

Immediately after Soler's result, a (mathematical, not physical) paper appeared, with a (correct) celebration of Soler's achievement, and a final section with the application to quantum logic. A section with all the above problems
(including a "ample unitary group" axiom without reference to such a "free mobility" property being relevant in quantum logic going back to von Neumann, and all other things as above).
I wrote to the author that, in view of his recent paper, he should also consider von Neumann's old results.

It's better avoiding discussing his reply.

After almost 28 years I am still patiently confident that one day even quantum logicians will see von Neumann's light.

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