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)]