Already von Neumann demonstrated that group algebras of groups G satisfying certain additional constraints give rise to von Neumann algebras. For finite groups they correspond to factors of type I in finite-D Hilbert spaces.
The group G must have an infinite number of elements and satisfy some additional conditions to give a HFF. First of all, all its conjugacy classes must have an infinite number of elements. Secondly, G must be amenable. This condition is not anymore algebraic. Braid groups define HFFs.
To see what is involved, let us start from the group algebra of a finite group G. It gives a finite-D Hilbert space, factor of type I.
- Consider next the braid groups Bn, which are coverings of Sn. One can check from Wikipedia that the relations for the braid group Bn are obtained as a covering group of Sn by giving up the condition that the permutations sigmai of nearby elements ei,ei+1 are idempotent. Could the corresponding braid group algebra define HFF?
It is. The number of conjugacy classes gn σign-1, gn == σ n+1 is infinite. If one poses the additional condition ei2= U× 1, U a root of unity, the number is finite. Amenability is too technical a property for me but from Wikipedia one learns that all group algebras, also those of the braid group, are hyperfinite factors of type II1 (HFFs).
- Any finite group is a subgroup G of some Sn. Could one obtain the braid group of G and corresponding group algebra as a sub-algebra of group algebra of Bn, which is HFF. This looks plausible.
- Could the inclusion for HFFs correspond to an inclusion for braid variants of corresponding finite group algebras? Or should some additional conditions be satisfied? What the conditions could be?
- In the TGD framework, I am primarily interested in Galois groups, which are finite groups. The vision/conjecture is that the inclusion hierarchies of extensions of rationals correspond to the inclusion hierarchies for hyperfinite factors. The hierarchies of extensions of rationals defined by the hierarchies of composite polynomials Pn ˆ ...ˆ P1 have Galois groups which define a hierarchy of relative Galois groups such that the Galois group Gk is a normal subgroup of Gk+1. One can say that the Galois group G is a semidirect product of the relative Galois groups.
- One can decompose any finite subgroup to a maximal number of normal subgroups, which are simple and therefore do not have a further decomposition. They are primes in the category of groups.
- Could the prime HFFs correspond to the braid group algebras of simple finite groups acting as Galois groups? Therefore prime groups would map to prime HFFs and the inclusion hierarchies of Galois groups induced by composite polynomials would define inclusion hierarchies of HFFs just as speculated.
One would have a deep connection between number theory and HFFs. This would also give a rather precise mathematical formulation of the number theoretic vision.
For a summary of earlier postings see Latest progress in TGD.