"... a von Neumann algebra is a *-algebra of bounded operators on some Hilbert space of countable dimension - that is, a bunch of bounded operators closed under addition, multiplication, scalar multiplication, and taking adjoints: that's the * business. However, to be a von Neumann algebra, our *-algebra needs one extra property! ... given any self-adjoint operator A in our von Neumann algebra and any measurable function f: R -> R, we want there to be a self-adjoint operator f(A) that again lies in our von Neumann algebra. ... Every von Neumann algebra can be built from so-called "simple" ones as a direct sum, or more generally a "direct integral", which is a kind of continuous version of a direct sum. As usual in algebra, the "simple" von Neumann algebras are defined to be those without any nontrivial ideals. This turns out to be equivalent to saying that only scalar multiples of the identity commute with everything in the von Neumann algebra. People call simple von Neumann algebras "factors" for short. Anyway, the point is that we just need to classify the factors: the process of sticking these together to get the other von Neumann algebras is not tricky. The first step in classifying factors was done by von Neumann and Murray, who divided them into types I, II, and III. ... We say a factor is type II1 if it admits a trace whose values on projections are all the numbers in the unit interval [0,1]. ... Playing with type II factors amounts to letting dimension be a continuous rather than discrete parameter! ... Weird as this seems, it's easy to construct a type II1 factor. Start with the algebra of 1 x 1 matrices, and stuff it into the algebra of 2 x 2 matrices as follows:( x 0 ) x |-> ( ) ( 0 x ) This doubles the trace, so define a new trace on the algebra of 2 x 2 matrices which is half the usual one. Now keep doing this, doubling the dimension each time, using the above formula to define a map from the 2^n x 2^n matrices into the 2^(n+1) x 2^(n+1) matrices, and normalizing the trace on each of these matrix algebras so that all the maps are trace-preserving. Then take the union of all these algebras... and finally, with a little work, complete this and get a von Neumann algebra! One can show this von Neumann algebra is a factor. It's pretty obvious that the trace of a projection can be any fraction in the interval [0,1] whose denominator is a power of two. But actually, any number from 0 to 1 is the trace of some projection in this algebra - so we've got our paws on a type II1 factor. This is ... the only II1 factor ... that contains a sequence of finite-dimensional von Neumann algebras whose union is dense in the weak topology. A von Neumann algebra like that is called "hyperfinite", so this guy is called "the hyperfinite II1 factor". ... the algebra of 2^n x 2^n matrices is a Clifford algebra, so the hyperfinite II1 factor is a kind of infinite-dimensional Clifford algebra. But the Clifford algebra of 2^n x 2^n matrices is secretly just another name for the algebra generated by creation and annihilation operators on the fermionic Fock space over C^(2n). Pondering this a bit, you can show that the hyperfinite II1 factor is the smallest von Neumann algebra containing the creation and annihilation operators on a fermionic Fock space of countably infinite dimension. ...".
David Finkelstein's Chronon Model, in which "... The dynamics is composed of a finite number of elementary dynamical processes, "chronons," with a finite time scale tav and Clifford statistics. ...", is Fundamentally Finite,
and my HyperDiamond Feynman Checkerboard formulation of my D4-D5-E6-E7-E8 VoDou Physics model is also Fundamentally Finite,
so that in David's Quantum Relativity Workshop in January 2002 we discussed (among other things) the question: if Physics is really Fundamentally Finite, what is the point of taking the union of all 2n x 2n matrix Clifford algebras and completing that to get the hyperfinite II1 factor as an infinite-dimensional algebra? In short,
My impression of our discussion is that:
- although a Finite Structure may be Fundamental,
- we don't know exactly how big our Finite Structure has to be to describe Everything
- so we might have to always be readjusting the size of our Finite Structure according to circumstances,
- while an Infinite Limit Structure can be chosen once-and-for-all.
Furthermore,
- the existence of an Infinite Limit in the mathematical sense implies that going from one level of finiteness N to another larger level of finiteness M is a consistent process so that no Bad Surprises will happen if as the level of finiteness increases (compare the Epsilon-Delta formulation of differential calculus, and the existence of Infinitesimals in Surreal Numbers and NonStandard Analysis), and
- an Infinite Limit Structure allows us to make Continuous Structures and related things for which calculation can be much simpler than calculations using large Finite Structures (for example, my calculations of particle masses and force strength constants use Continuous Structures).
My conclusion is that an Infinite Limit Structure is an Ideal Structure that is Practically Useful.
Irving Segal, in his review (Bull. AMS 33 (1996) 459-465) of the book Noncommutative Geometry (Academic Press 1994) by Alain Connes, said:
"... The W*-algebras ... investigated by Murray and von Neumann (henceforth, MvN) ... and ... C*-algebras ... investigated by the Gelfand school were formally similar and in fact essentially identical in the finite-dimensional case. But they differ in topology in the infinite-dimensional case, which ... in fact ... rather fundamentally changes the character of the theory.The algebras of MvN were closed in the weak operator topology,
while those of Gelfand were closed in the uniform (operator bound norm) topology. ...
... von Neumann ... was particularly fond of the "approximately finite" factor of type II1, which in fact plays a basic role in the representation of fermion fields. ...
... the Clifford algebra over a Hilbert space ... is the simplest of the type II W*-algebras ... the Clifford algebra is a central simple algebra that is altogether different from the algebra of all bounded operators on Hilbert space ... this algebra plays a fundamental role in the analysis of free fermionic quantum fields. ... it is the algebra generated by the canonical fermionic Q's ... The Clifford algebra is the simplest of the factors that are direct limits of matrix algebras ...".
Vaughan Jones and Henri Moscovici, in their review (Notices of the AMS 44 (August 1997) 792-799) of the book Noncommutative Geometry (Academic Press 1994) by Alain Connes, said:.
"... If functions on [0,1] are represented on L^2([0,1],dx) as multiplication operators, the continuous functions form a C*-algebra and the L^infinity functions form a von Neumann algebra [also called a W*-[algebra] ... [a factor of a von Neumann algebra is a] von Neumann algebra ... whose center contains only scalar multiples of the identity ...... von Neumann wrote a paper reducing any von Neumann algebra to a factor ... by a "direct integral" - a continuous analogue of the direct sum of algebras. ...
... There is, up to isomorphism, a single II1 factor with the "hyperfinite" property; i.e., any finite set of elements can be arbitrarily well approximated by elements in a finite-dimensional subalgebra. ...
... Connes began a penetrating study of automorphisms of type II factors ... The first step was was to classify periodic automorphisms of the hyperfinite II1 factor R ... Connes ...[introduced]... a new approach to hyperfiniteness: injectivity of the von Neumann algebra M, meaning that M has a Banach space complement in B(H). ... The result [was]
(injective) <=> (hyperfinite) ... If one had to find a single word to sum up ... the work of Connes ... it would be the word "automorphisms". ...".
Goodman, de la Harpe, and Jones, in their book Coxeter Graphs and Towers of Algebras (Springer 1989), say (at pages 31, 49-52, 185-186, 222-231):
"... We realize the hyperfinite II1 factor R as the completion, with respect to the unique tracial state tr, of the infinite tensor product of Mat2(C),R = ( x(infinite) Mat2(C)) )-. Any closed subgroup G of SU(2) acts on x(infinite) Mat2(C) by the infinite tensor product of its action by conjugation on Mat2(C). The action preserves the trace, so extends to an action on R. ...
We can now use the McKay correspondence between finite subgroups of SU(2) and affine Coxeter graphs ... to calculate the Bratteli diagrams or principal graphs when G is finite. ...
... It is useful to describe a pair of multi-matrix algebras N in M by its Bratteli diagram B(NinM),which is a bicolored weighted multigraph ... These diagrams were first introduced in order to study inductive limit systems of finite dimensional C*-algebras ... Example ... let S3 be the group of permutations of {1,2,3} and let S2 be that of {1,2}. ... the Bratteli diagram ...[is]...
1 2 1 * * * \ / \ / * * 1 1 Consider similarly S3 as a subgroup of the group S4 of permutations of {1,2,3,4} ...
1 3 2 3 1 * * * * * \ | \ | / | / * * * 1 2 1 ... As ... we always draw Bratteli diagrams ... with the upper level representing the larger algebra ... the coloring of the vertices is actually superfluous, since the ... types of vertices are labelled by their level ...
... the derived tower is the chain of relative commutants ( N' n Mk )(k>0) , where ( Mk )(k>0) is the tower for the pair N in M. ...
... Let A be the infinite tensor product of Mat2(C) ... It is well known that A has a unique C*-norm and that A-, the C*-completion of A, is a simple C*-algebra. ... Let N be the weak closure of A- ... There is an obvious shift endomorphism on A ... which ... extends to N ... The circle group acts on A by the infinite tensor product of its action ... on Mat2(C); this action ... extends to an action on N. ... We shall ...[let]... M = N x Mat2(C) ...[and]... G the maximal torus T ... in order to compute the derived tower for Rbeta in R ...[where beta is a modulus of a Markov trace]... a Markov trace of modulus beta on N in M ...[is defined to be]... a faithful trace tr on M with faithful restriction to N for which there exists a (necessarily unique) trace Tr on L = End(r,N)(M) such that Tr(x) = tr (x) [and] beta Tr(xE) = tr(x) for all x in M ...
... when beta > 4 ... the tower for the pair Rbeta in R can be identified with
N^T in ( N x Mat2(C) )^T in ( N x Mat2(C) )^T x ( N x Mat2(C) )^T in ... ... the infinite symmetric group Sinfinite acts ergodically on N (by permutation of the tensorands), and this action is implemented by unitarikes in N^T ... That is, ther derived tower is precisely the sequence of fixed-point algebras for the tensor product action of T on x(k) Mat2(C) (k > 0). The Bratteli diagram is Pascal's triangle ...
. . . 1* 3* 3* 1* \ / \ / \ / 1* 2* 1* \ / \ / 1* 1* \ / 1* ... and the principal graph ...[which is obtained from]... the Bratteli diagram of the derived tower ...[by deleting]... on each level the vertices corresponding to the old stuff, and the edges emanating from them ... is
... * * * * * * ... Ainfinite,infinite = \ / \ / \ / \ / \ / * * * * * ... Note that it differs from the graph for beta = 4 ...[which is]...
* * * Ainfinite = / \ / \ / * * * ... ...".
It seems to me that the tensor product of factors of the form Mat2(C) must be based on the complex-Clifford-periodicity of order 2, in which the tensor factorization is
Since Cl(2n,C) = Mat2^n(C) and Cl(2,C) = Mat2(C), it can be written as
That leads me to think that my D4-D5-E6-E7-E8 VoDou Physics model is basically similarto hyperfinite II1, but with real instead of complex structure, as follows:
My D4-D5-E6-E7-E8 VoDou Physics model uses the real-Clifford-periodicity of order 8, in which the tensor factorization is
Cl(8,R) = Mat16(R) and has graded structure
and total dimension 2^8 = 256 with full spinor dimension sqrt(256) = 16 and two mirror image half-spinors, each 8-dimensional. Note that the 52-dimensional exceptional Lie algebra F4 corresponds to the ( 8 + 28 +16 )-dimensional vectors plus bivectors plus full spinors.
Since Cl(8n,R) = Mat16^n(R) and Cl(8,R) = Mat16(R), it can be written as
My D4-D5-E6-E7-E8 VoDou Physics model realizes a real-hyperfinite II1 factor Rreal as the completion of the infinite tensor product of Mat16(R),
Similar to the action of SU(2) on Mat2(C), there is a natural action of Spin(16) on Mat16(R), so that any closed subgroup G of Spin(16) acts on x(infinite) Mat16(R) by the infinite tensor product of its action by conjugation on Mat16(R).
SU(2) = Spin(3) is a nice subgroup of Spin(16), so it seems that the subgroups of Spin(3) would extend to give subgroups of Spin(16), thus giving the McKay results of A-D-E classification for the subgroups of Spin(16).
Adrian Ocneanu, in his article Quantized Groups, String Algebras and Galois Theory for Algebras, at pages 119-172 in Operator Algebras and Applications, Volume 2, edited by David E. Evans and Masamichi Takesaki (Cambridge 1988), said:"... We introduce a Galois type invariant for the position of s subalgebra inside an algebra, called a paragroup, which has a group-like structure. Paragroups are the natural quantization of (finite) groups. ... harmonic analysis for the paragroup corresponding to the group Z2 is done in the Ising model ...... The algebra ... is ... R, or the hyperfinite II1 factor ... also called ... the elementary von Neumann algebra. The algebra R ... is the weak closure of the Clifford algebra of the real separable Hilbert space, is a factor ... which has very many symmetries ... A ... theorem of Connes implies that any closed subalgebra of R which is a factor ... is isomorphic either to Matn(C) or to R itself. Thus any finite index subfactor N of R is isomorphic to R, and all the information in the inclusion N in R comes from the relative position of N in R and not from the structure of N. ... in our context this guarantees that the closure of all finite dimensional constructions done below will us back to R. ... for subfactors of finite Jones index, finite depth and scalar centralizer of ... R ... In index less than 4 ... the conjugacy classes are rigid: ... axioms eliminate one connection for each Dn and the pair of connections on E7 [because they are not geometrically flat]. Thus there is one subfactor for each diagram An, one for each diagram D2n, and a pair of opposite conjugate but nonconjugate subfactors for each diagram E6 and E8. ... there is a crystal-like rigidity of the position of subfactors of R. ...
... the golden ratio subfactor N in M ... of the hyperfinite II1 factor ... is the fixed point algebra of a homomorphism s: M -> Mat2 x M which ... behaves like a group (or rather like a quantum group or Kac algebra.) ...".
Y. Ito and I. Nakamura, in their paper at http://www.maths.warwick.ac.uk/~miles/Warwick_EConf/Draft2/Nakamura.ps , say:
"... The ADE Dynkin diagrams provide a classification of the following types of objects (among others):
- (a) simple singularities (rational double points) of complex surfaces ... ,
- (b) finite subgroups of SL(2,C),
- (c) simple Lie groups and simple Lie algebras ... ,
- (d) quivers of finite type ... ,
- (e) modular invariant partition functions in two dimensions [which I guess would include such physical things as the Ising model, etc] ... ,
- (f) pairs of von Neumann algebras of type II1 ...".
Note that the book of Goodman, de la Harpe, and Jones list, on page 224, a full A-D-E correspondence, including all D and E7, whereas Ocneanu, in his article Quantized Groups, String Algebras and Galois Theory for Algebras, in Operator Algebras and Applications, Volume 2, edited by David E. Evans and Masamichi Takesaki (Cambridge 1988), says that R, or hyperfinite II1, contains subfactors for each of the A-D-E classification except odd D and E7, because odd D and E7 are not geometrically flat. Perhaps Ocneanu's result was written after the book of Goodman, de la Harpe, and Jones had been mostly (but not completely) finished, because Goodman, de la Harpe, and Jones say on page 218 "... Ocneanu ... can show that there are only finitely many subfactors of the hyperfinite I1-factor (of index < 4) for each Coxeter graph, up to conjugation by automorphisms, and he determines which Coxetrer graphs are allowed. ...".
For more work by Ocneanu, see his 1990 lecture n Tokyo titled Quantum Symmetry, Differential Geometry of Finite Graphs and Classification of Subfactors, available as notes taken by Yasuyuki Kawahigashi.
The Spin(16) symmetry of the Mat16(R) components of
has several interesting features:
- the 128-dim full-spinors of Spin(15) and Cl(15,R);
- to the 128-dim full spinors of Spin(14) and Cl(14,R), which are the 64-dim +half-spinors of Spin(14) and Cl(14,R) plus the 64-dim -half-spinors of Spin(14) and Cl(14,R);
- to 2 copies of the 64-dim full spinors of Spin(12) and Cl(12,R);
- to 4 copies of the 32-dim full spinors of Spin(10) and Cl(10,R); and
- to 8 copies of the 16-dim full spinors of Spin(8) and Cl(8,R), which are 8 copies of each of the 8-dim half-spinors of Spin(8) and Cl(8,R).
Therefore,
I visualize each of the Cl(8) = Mat16(R) components of Rreal = ( x(infinite) Mat16(R)) )- of my D4-D5-E6-E7-E8 VoDou Physics model as describing local physics in one small-point-neighborhood-node of what might be called a pregeometric foam.
Marcel Berger says in his book (28 December 2001 version being proof-read by Benjamin McKay) Riemannian Geometry Today Introduction and Panorama:
"... The compact manifold being M, let us denote by RM(M) ... the space of all Riemannian metrics on M and the set of all Riemannian structures RS(M) ... its quotient by the group of all diffeomorphisms of M ... The idea is to consider some map (called a functional) F: RS(M) -> R defined on the Riemannian structures of any compact Riemannian manifold M, and to look for its infimum inf _RS(M) (F) where the infimum is taken over the metrics on M ....... Since 1995, Nabutovsky has been unveiling "awful" properties of RS(M) for compact manifolds M of dimension five or more. We will sketch here some results of this kind, which for us are fascinating both in themselves and in his method of proof. Posterity will judge their value to mathematics, but the least one can say is that they are disturbing. ...
... On a compact surface M, thanks to the conformal representation theorem, and the results at the end of [section 11.4.3]... we have a beautiful picture of RS (M). There is a flow on RS(M) taking any Riemannian metric to a metric of constant curvature. But the space of constant curvature metrics is the Teichmuller space described in [section 6.6.2]...
Thus one sees a picture of RS(M) as a fiber bundle whose base (Teichmuller space) is finite dimensional, and whose fibers are infinite dimensional and contractible.
We will briefly discuss dimensions three and four, but first let us see how terrible RS(M) is for all manifolds of dimension five or more. ...
... We want to survey the topography of RS(M), where the notion of height of a mountain or depth of a valley is measured using one of the functionals F: RS (M) -> R ... In particular, we are interested in the critical points of that functional, concerning ourselves with where these points appear on our relief map of RS(M). ...
... it is a very natural notion to exploit Morse theory to describe the topology of RS(M) by using the fibration
Diff(M) -> RM(M) -> RS (M) = RM(M) / Diff(M) ... we find that the total space RM(M) is topologically trivial, since it is convex, and then we can use a spectral sequence to uncover a rich topology on RS(M) from the rich topology of Diff(M). Sadly, unlike the theory of compact manifolds from [section 10.3.3]... here we are at sea because the topology of Diff(M), one of the most natural objects of study in differential topology, is almost completely a mystery ...
... One of Nabutovsky's theorems will show ... that there are infinitely many connected components, and that there are infinitely many connected components of every level set of many functionals ...
... We ... enter the realm of algorithmic computability theory, and the proof is a deep Riemannian refinement of Novikov's proof of the algorithmic unsolvability of the problem of recognizing a sphere Sn of dimension n > 5. ...
... For the moment, Nabutovsky has only achieved partial results for the Hilbert functional (the integral of the scalar curvature), which ones imagines will yield Einstein structures.
It seems that the common moral of these stories is that (at least for standard functionals) looking for critical points, even for local minima, will not yield natural "best" metrics when one has too many minima (uncomputably growing numbers) looking for the absolute minimum is not a tractable job.
This also explains why the problem of distributing points spaced apart on a sphere, which is problem 7 in Smale's 1998 list ... is today apparently a hopeless task ...
... On three dimensional manifolds, the above theorems as stated are certainly false, since Thurston has proven that there is an algorithm to recognize the three dimensional sphere. But this does not solve his geometrization programme ...
... On four dimensional manifolds, Nabutovsky thinks that most of his results may be valid and approachable by the same avenues. ...
... Finally special attention should be given to Connes ...[whose Noncommutative Geometry is]... the beginning of a complete program to put most (perhaps all) geometries into a very general frame. The frame is that of algebras of operators ... In this frame, according to various extra suitable axioms one can recover almost any kind of geometry, including of course Riemannian ones, from families of operators. The next step in the program will be to generalize every concept one could wish, e.g. curvature.
For the metric itself this is done ...
... in Connes ...Noncommutative Geometry]... one finds a formula giving the distance between any two points using the Dirac operator ... on spinors ... which has the advantage that it carries over to noncommutative geometry ...".
Note that in General Relativity the space of Riemannian structures on spacetime RS(spacetime) corresponds to the space of possible configurations of matter/energy in spacetime. Compare similar formulations of possible configurations for Bohm-many-worlds string quantum theory, for timelike brane M-theory, and for spacelike brane F-theory., which may be useful in sum-over-histories interpretations, particularly with respect to prime numbers and zeta functions.
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