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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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A characterization of projection 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 in two papers: [1] and [2] (full references at end).

You obtain an answer since "factorial" and "type III" are expressed in ortholattice terms (see the Loomis - Maeda dimension theory, in particular the last version [3] 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 identify 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 automorphism of a projective (ortho)line (i.e. a arbitrary permutation of the points, or half of them in the orthocomplemented case) is semilinearly induced. In particular, order two matrices over real, complex or quaternion 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 [4] (and then unfortunately well forgotten by modern quantum logicians). From 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 particular, 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 axioms 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 details 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 Dye's theorem [5]: 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 for von Neumann algebras, but Yen in 1957 [6] and Berberian in 1982 [7] remarked that the proof works also for AW$^*$-algebras. (Recent interest in Dye's theorem appears in C. Heunen, M. L. Reyes [8]; these authors seem unaware of the concept of orthosymmetric ortholattices introduced by Mayet [9].)

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 reconstruct 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 $A$); in fact, $A$ 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 a 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 an 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 [10]). In the von Neumann algebra case, a (external) lattice description of "nonasymptoticy" 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 orthosymmetric structure on $L$ is unique since for each element $e$ of $L$ there is only one involutory automorphism 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)]


Full references:

[1] Bunce, L. J.; Wright, J. D. Maitland. Quantum logics and convex geometry. Comm. Math. Phys. 101 (1985), no. 1, 87--96.

[2] Bunce, L. J.; Wright, J. D. Maitland. Quantum logic, state space geometry and operator algebras. Comm. Math. Phys. 96 (1984), no. 3, 345--348.

[3]Maeda, Shûichirô. Dimension theory on relatively semi-orthocomplemented complete lattices. J. Sci. Hiroshima Univ. Ser. A-I Math. 25 (1961), no. 2, 369--404. doi:10.32917/hmj/1206139804.

[4] Von Neumann, John. Continuous geometries with a transition probability. Vol. 252. American Mathematical Soc., 1981.

[5] Dye, H. A. “On the Geometry of Projections in Certain Operator Algebras.” Annals of Mathematics, vol. 61, no. 1, 1955, pp. 73–89. JSTOR

[6] Yen, Ti. Isomorphism of AW∗-algebras. Proc. Amer. Math. Soc. 8 (1957), 345–349.

[7] Berberian, S.K. The maximal ring of quotients of a finite Von Neumann algebra. Rocky Mountain J. Math. 12 (1982), no. 1, 149--164. doi:10.1216/RMJ-1982-12-1-149.

[8] Heunen, Chris, and Manuel L. Reyes. "Active lattices determine AW*-algebras." Journal of Mathematical Analysis and Applications 416.1 (2014): 289-313.

[9] Mayet, R. “Orthosymmetric Ortholattices.” Proceedings of the American Mathematical Society, vol. 114, no. 2, 1992, pp. 295–306. JSTOR

[10] Anoussis, M., Aristides Katavolos, and Ivan G. Todorov. "Angles in C*-algebras." arXiv preprint math/0601003 (2005).

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  • $\begingroup$ Updated the version on my site with minor corrections and more references $\endgroup$ – user46855 Mar 8 '14 at 6:31
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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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