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Does there exist any theorem which permits, under suitable hypotheses, to represent a particular complete orthomodular lattice as the projection lattice of a Type III von Neumann factor?

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2 Answers 2

I seem to recall that work of Wright and others contains constructions of type III AW* factors that are not W*. The lattice of projections of the former is also a complete orthomodular lattice, and these probably (?) characterize the AW* algebras (i.e., if $A$ and $B$ are AW*-factors, and their projection lattices are isomorphic, then presumably $A$ and $B$ would be isomorphic; I don't know whether this is true, but it looks plausible). So if there were a condition for representation in W*-factors, it would have to include some additional constraint referring to normal states, which would not be very natural.

Perhaps this is not the most concise analogy, but consider what happens in the commutative case. Commutative AW* algebras are $C(X)$ where $X$ is extremally disconnected, but commutative W* algebras are $C(X)$ where $X$ is hyperstonean, a rather artificial** concept, because it requires a separating family of normal states.

**Not the right word; I mean that it cannot be easily made more abstract.

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A characterization of projectioon ortholattices of von Neumann algebras
(and more generally JBW algebras) with no type I$_2$ component was given
by Bunce and J.D.M. Wright, Comm. mat. Phys; on projecteuclid.org they
are euclid.cmp/1104114067 and euclid.cmp/1103941854

You obtain an answer since "factorial" and "type III" are expressed
in ortholattice terms (see the Loomis - Maeda dimension theory, in
particoular the last version [Maeda, 1961] where equidimensionality is
identified with lattice semi-projectivity).

As expected by professor Handelman, a big role in such a
characterization is played by (the faces of) the convex set of normal
states (exclusion of I$_2$ components is needed to use Gleason's theorem
to identifiy states with completely additive probability measures on the
projection ortholattice. It is also needed since, as well known from
projective geometry, not every (ortho)lattice automorphisms of a
projective (ortho)line (i.e. a arbitray permutaion of the points, or
half of them in the orthocomplemented case) is semilinearly induced. In
particoular, order two matrices over real, complex or quanternion
numbers all give the same projection ortholattice (as it trivially
happens also with order 1 matrices); by artificially restricting to the
complex case one has unicity, but only up to a noncanonical
isomorphism).

A much better (from the quantum logic point of view) characterization of
the projection ortholattices of (real or complex, always excluding type
I$_2$ cases) finite factors as "continuous geometries with transition
probability" is due to von Neumann (and then unfortunately well
forgotten by modern quantum logicians). Form this, two generalizations
are obtainable with standard methods: (1) to the decomposable case,
using Boolean valued analysis (a decomposable case is the same as
indecomposable object of a boolean valued universe); (2) to the
semifinite (instead of finite) case, using the fact that in the
semifinite case the join-dense ideal of finite elements completely
determines $L$ (a standard method "to adjoin 1 to a generalized
orthomodular lattice", due to Janowitz, produces the lattice of all
finite and cofinite elements; then the Dedekind completion produces
$L$).

[Digression. In particoular, this gives a characteriazion of Hilbertian
logics (of type I factors) that is physically much better than the
characterization that modern quantum logicians deduce from Soler's
theorem (which is however mathematically wonderful). The modern theorem
must exclude all finite dimensional factors (why a finite dimensional
irreducibly quantum logic should be automatically embeddable in a
infinite dimensional one? von Neumann's method instead excludes only the
"spin factors", which are not really quantum since they are the only
factors with nonclassical logic but with "noncontextual hidden variables",
and the nonarguesian planes, which cannot be embedded in any larger
logic except by direct product, which means that these exceptional
components can have only classical, not quantum, interactions with the
other components) and must presuppose together a complete lattice and
orthomodularity without physical reasons (orthomodularity is justified
by restricting only to certain propositions, and the "complete
lattice" property is justified by enlarging using completions, like
Dedekind completions. Unfortunately this only produces two possibly
different structures, a restricted orthomodular one and a a larger
complete lattice; almost no known mathematical theorem produces
automatically a orthomodular completion. The only exception is precisely
von Neumann's method when applied to type I cases (and, analogoulsy, the metric completion of pre-Hilbert spaces): the only completeness
axiom which is not trivially satisfied in the finite dimensional case is
used only in the last step, to show that an already constructed
Hilbertian representation is surjective; so, were this last completeness
axiom not satisfied, one can always take as completion the bicommutant
of the algebra in the Hilbertian representation: one has proved that a
completion exists, a conceptual case analogous to the well known proof
that, assuming the archimedean axiom for the measures of physical
quantities, then one can assume that the measures are real numbers: the
archimedean axiom, involving two magnitudes and a simple arithmetic
progression, is experimentally falsifiable at least ideally, but the
completeness axions for real numbers, with arbitrary infinite sets, is
physically hopeless). End digression.]

I know no attempts to concretize the details of a last, third step in
the extensions of von Neumann's characterization: using Tomita -
Takesaki modular theory to obtain a generic type III factor starting
from a type II infinite factor with a suitable automorphism, one has
that, in principle, the projection ortholattice $L$ of a type III
factor, being equivalent to the factor itself, is somehow obtained from
a type II factor with a given automorphism, which is equivalent to a
projection ortholattice, with fixed automorphism, of a type II factor. I
hope that someone one day will write down the deatails of this method.

Concerning the other remark of professor Handelman:

The possibility of characterization of complex AW$^*$-algebras with no
type I$_2$ components by their projection otholattices follows from's
Dye's theorem: each projection ortholattice isomorphism among them
extends to one and only one (necessarilly real linear) $*$-ring
isomorphism (or equivalently a unique complex linear Jordan isomorphism;
however, since there are type II finite factors not anti-isomorphic to
themselves, there are cases where a complex linear $*$-ring isomorphism
is impossible).

Dye proved his theorem in 1955 (on jstor.org it is number 1969620) for
von Neumann algebras, but Yen in 1957 (proc. ams. S0002-9939-1957-0084123-X ) and Berberian in 1982 (on projecteuclid.org
see euclid.rmjm/1250128413 ) remarked that the proof works also for
AW$^*$-algebras. [Recent interest in Dye's theorem appares in C.
Heunen, M. L. Reyes pdf/1212.5778 on arxiv.org; these authors seem
unaware of the concept of orthosymetric otholattices introduced by
Mayet]

Really, the theorem (but not Dye's own proof) also holds for real AW$^*$
algebras with no abelian or type I$_2$ component (or even more generally
for Rickart real $C^*$-algebras of matrix order at least 3 and
$C^*$-direct sums of such algebras). An explicit reconstruction of the
Rickart $C^*$-algebra $A$ from its projection ortholattice $L$ is the
following:

first note that it is sufficient to recontruct the $*$-ring
$M$ of "affiliated locally measurable operators" (defined by Berberian
and Saito using "strongly dense domains" in $L$, but algebraically it
is the ring of classical quotients of $A$); infact, $A$ is the subring
of $M$ generated by its projections (or also the $*$-subring of bounded
elements in the algebraic sense first used by von Neumann).

Then $M$,
being a direct product of matrix rings of order at least 3, is generated
as ring by its idempotents $e,f,\dots$ using (besides idempotence) the
relations given by a restriction of the classical ``circle operation''
to a partial operation on idempotents: $e\circ f=e+f-ef$ is idempotent
when $fe=0$; moreover, these generators and relations depend only by
lattice theory: idempotents are identified with complementary ordered
pairs $(K,I)$ (kernel and image of the idempotent) in the lattice of
right ideals of $M$, and the partial circle operation becomes
$(K,I)\oplus(K',I')=(K\wedge K',I\vee I')$ when $I\subseteq K'$
(moreover, the join is a independent join and dually for the meet). (All
this follows from the easy part of von Neumann's coordinatization, in
any ring even without regularity conditions).

Finally: the above pairs
$(K,I)$ and the circle partial operation on them only depends upon the
lattice $L$ (which is the same for $A$ and $M$; it is the lattice
associated to these Rickart rings): these are exactly the complementary
and modular pairs in the lattice (by $O$-symmetry of such ortholattices,
all known reasonable modularity conditions for pairs of elements are
equivalent), with join and meet computed in $L$; lastly, the projections
(as opposed to generic idempotents) are the pairs $(K,I)$ which are
orthocomplementary (as opposed to only modular complementary) in $L$;
the involution in $A$ is the only one that makes such projections (that
ring generate $A$) self-adjoint (and then the involution is also unique
on the classical quotient ring $M$).

Note that $A$ contains only some of
the idempotents of $M$; precisely, the idempotents corresponding to
pairs $(K,I)$ which are "nonasymptotic" (for this classical concept
see Topping, Bures [with improvements by S. Maeda in the interaction
with lattice theory], and more recently M. Anoussis, A. Katavolos, I. G.
Todorov math/0601003v2 on arxiv.org). In the von Neumann algebra case, a
(external) lattice description of "nonasimptoticy" is "absolute
modularity": for one normal embedding of $L$ in a Hilbert lattice (as
its own bicommutant; note that the commuting of projections is
ortholattice definable), the pair is modular in the larger lattice (then
the same happens for each normal embedding of $L$ in any projection
ortholattice of a von Neumann algebra).

[Since the ortholattice $L$
determines $A$, it also determines the orthosymmetric structure that $A$
defines on $L$; really, Mayet's othosymmetric structure on $L$ is unique
since for each element $e$ of $L$ there is only one involutory
automorphisms of $L$ i.e. of $A$ that fixes exactly the projections that
commute with $e$ (see for example lemma 2.4 in euclid.cmp/1103859692 on projecteuclid.org ; it is sufficient the even more folklore case of
factors: then apply a subdirect decomposition into factors for the
general case)]

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Updated the version on my site with minor corrections and more references –  user46855 Mar 8 at 6:31

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