Some non-commutative analogs in lattice theory.
von Neumann's coordinatization theorem is the
non-commutative analogue of Stone's definitional equivalence between Boolean algebras
(complemented distributive lattices) and Boolean rings (associative rings with 1 where all
elements are idempotent).
The Baer - Inaba - Jonnsson - Monk coordinatization theorem gives the non-commutative
analogue of direct products of finite chains (Łukasiewicz propositional logics);
for this one uses not the usual formulation of the theorem
(which coordinatizes primary lattices with modules over a artinian ring where
one-sided ideals are two-sided and form a chain), but a reformulation that gives
a true equivalence between the lattice (of submodules of the module) and the ring
(of endomorphisms of the module); this way one has a true analogue of the (dual)
equivalence between commutative C^*-algebras and compact Hausdorff spaces
(or better, their lattice of open sets).
One has a common generalization of the two cases above: equivalence between lattices of
subobjects and rings of endomorphisms for finitely presented modules
(of geometric dimension at least 3) over a "auxiliary" ring which is WQF
(weakly quasi Frobenius, a.k.a. IF, injectives are flat).
The categories of finitely presented modules over such auxiliary
rings are exactly (up to equivalence) the abelian categories with an object which is
injective, projective and finitely generates and finitely cogenerates every object.
The ultimate generalization is G.Hutchinson's coordinatization theorem,
a correspondence between arbitrary abelian categories and
modular lattices with 0 where each element can be doubled
and intervals are projective (in lattice theory, sense i.e. the classical projective
geometry meaning) to initial intervals. When one looks at this theorem together
with the Freyd - Mitchell embedding theorem, one has that three languages are fully
adequate and equivalent ways to do linear algebra:
(1) the usual language of sums and products (modules over associative rings);
(2) the language of category theory (abelian categories);
(3) the old fashioned language of synthetic geometry of incidence (joins and meets in
suitable modular lattices).
In these equivalences, the lattices are the (pointless, noncommutative) spaces,
and the rings are the rings of coordinates or functions over the space (I am not
considering the distinction between equivalences and dual equivalences because
I am more interested in structures, with their unique concept of isomorphism attached,
rather than more general morphisms, which depend upon the particular way to define a
structure. But the complementarity between the structural and categorical views,
where neither subsumes the other, is another long theme).
Since all these ideas have their origin in von Neumann works about continuous geometries
and rings of operators, I now explain the relation with operator algebras.
First note that the usual definition of pointless topological space as complete
Heyting algebra is not a true generalization of the "topological space" concept:
they are a true generalization of sober spaces,
but to generalize topological spaces one must consider pairs: a complete boolean
algebra (which in the atomic case is the same thing as a set) with a complete
Heyting subalgebra (the lattice of open sets). To obtain the non-commutative
analogue, complete boolean algebras are generalized to meet-continuous geometries,
and algebras of measurable sets are replaced with suitable structures (projection
ortholattices of von Neumann algebras) which are embedded in the meet-continuous
geometries in the same way as a right nonsingular ring is embedded in its maximal
ring of right fractions (a regular right self-injective ring).
A meet continuous geometry is a complete lattice which is modular, complemented
and meet distubutes over increasing joins (not arbitray joins, like Heyting algebras).
These structures were introduced by von Neumann and Halperin in 1939; they are a common
generalization of (possibly reducible) continuous geometries (the subcase where
join distrubutes over decreasing meets) and (possibly reducible and infinite dimensional)
projective geometries (the atomic subcase). For them one has a dimension and decomposition
theory much like the one for continuous geometries and rings of operators (the theory of
S.Maeda, in its last version of 1961, is sufficient; one does not need the 2003 theory
by Wehrung and Goodearl). One can define the components of various types, in particular
I_1 (the boolean component, i.e. classical logic), the I_2 component (the 2-distibutive
component i.e. subdirect product of projective lines; physically these are "spin
factors" and quantum-logically it is the non-classical component which nonetheless has
non-contextual hidden variables), the I_3 nonarguesian component (subdirect product
of projective nonarguesian planes i.e. irreducible projective geometries that cannot
be embedded in larger irreducible projective geometries; quantum-logically this means
that interacion with other components is only possible classically, without superposition).
Once these bad low dimensional components are disregarded, von Neumann coordinatization
theorem gives a equivalence between the meet-continuos geometries and the right
self-injective von Neumann regular rings. So meet-continuous geometries are pointeless
quantum (i.e. non-commutative) sets in the same way as complete Boolean algebras are
(commutative) pointless sets. Regular rings are the coordinate rings of these quantum
sets in the same way as (commutative, regular) rings of step functions (with values in
a field) are the ring equivalent of a boolean algebra (classical propositional logic);
the important new fact is that in the "truly non-commutative case" the regular ring
is uniquely and canonically determined by the lattice (in the distributive case, on the
contrary, it is not: one can use step functions with values in any field, and one can
change the field with the point; commutative [resp. strongly] regular rings are the
subrings of direct products of [skew] fields which are stable for the generalized inverse
operation).
The above "propositional logics" are without the negation operator; on the other hand,
the projection ortholattices of von Neumann algebras are complete orthomodular lattices
with sufficiently many completely additive probability measures and sufficiently many
internal simmetries (von Neumann said that the strict logic of orthocomplementation
and the probability logic of the states uniquely determine each other by means of his
symmetry axioms in his characterization of finite factors as continuos geometries with
a transition probability. One should also note how much more physically meaningful are
von Neumann axioms when compared with the "modern" ones based on Soler's theorem,
but this is another large topic). Using Gleason's theorem (and as always in absence of
the bad low-dimensional components) one obtains an equivalence between von Neumann's
"rings of operators" (i.e. real von Neumann algebras, or their self-adjoint part,
real JBW-algebras) and their projection ortholattices (the normal measures on the
ortholattice give the predual of the ring of operators). One can see these logics
inside a meet-continuos geometry by equipping the geometry with a linear orthogonality
relation which has for each element a maximum orthogonal element
(pseudo-orthocomplementation in part analogous to "external" in a stonean topological
space, in part anti-analogous as it happens with Lowere closure when compared to
Kuratowski closure). The regular ring of the lattice is the ring generated by all
complementary pairs in the lattice (which are the idempotents of the ring: kernel and
image) with the
relations corresponding to the partial operation e+f-ef which is defined on idempotents
whenever fe=0 (at the lattice level this partial operation is implemented with disjoint
join of the images and co-disjoint meet of the kernels); in the case associated to
a "ring of operators", this regular ring is the ring of maximal right quotients of
the von Neumann algebra, and conversely the algebra is recovered from the lattice with
orthogonality by taking the subring generated by orthogonal projections (idempotents
whose kernel and image are orthogonal); by a theorem of Berberian the algebra is ring
generated by its self-adjoint idempotents, and there is clearly at most one involution
on the algebra which fixes such generators.
Hence, in summary, in absence of bad low dimensional components (whose exclusion
is physically meaningful, see their meanings above) one has equivalences
between the following concepts:
(0) right self-injective regular rings with a suitable additional structure
(to associate a orthogonal projection onto the closure of the image to any element)
(1) real von Neumann algebras
(2) real JBW-algebras (the Jordan algebra of self-adjoint operators i.e. observables)
(3) the effect algebra (of operators with spectrum in [0,1]) i.e. unsharp quantum logic
(4) the projection ortholattice (the sharp quantum logic)
(5) the pointeless quantum set (meet-continuous geometry) with a suitable orthogonality
(6) the convex compact set of normal states
Any of the above is a adequate starting point for quantum foundation since all
the other points of view can be canonically recovered.
All this is restricted to the level of non-commutative measure spaces; a locally compact
topological space is something more precise (like a C^* algebra when compared to a
von Neumann algebra), and a (Riemannian) metric space is something still more precise
(and in particular a differentiable structure, which can be seen as an equivalence class
of riemannian structures: note that a isometry for the geodesic metric between complete
Riemannian manifolds is automatically differentiable, so the differentiable structure
must be definable from the geodesic metric, and infact Busemann and Menger had such a
explicit definition of the tangent spaces from the global metric. The topological
structure is then another equivalence class of metrics, for another weaker equivalence).
Given the equivalence (0)--(6) above, one can note that all the concepts which are used
in Connes definition of spectral triples, and analogues structures, can be seen
equivalently from each of the above points of view. In this way, all of the above points
of view are a possible starting point for non-commutative geometry.
[Sorry for the too long post and for my bad pseudo-english language]
