The universal von Neumann algebra of smooth four-manifolds revisited

Abstract

Making use of its smooth structure only, out of a connected oriented smooth $4$-manifold a von Neumann algebra is constructed. It is geometric in the sense that is generated by local operators and as a special four dimensional phenomenon it contains all algebraic (i.e., formal or coming from a metric) curvature tensors of the underlying $4$-manifold. The von Neumann algebra itself is a hyperfinite factor of ${\rm II}_1$-type hence is unique up to abstract isomorphisms of von Neumann algebras. Over a fixed $4$-manifold this universal von Neumann algebra admits a particular representation on a Hilbert space such that its unitary equivalence class is preserved by orientation-preserving diffeomorphisms consequently the Murray--von Neumann coupling constant of this representation is well-defined and gives rise to a new and computable real-valued smooth $4$-manifold invariant. Its link with Jones' subfactor theory is noticed as well as computations in the simply connected closed case are carried out. Application to the cosmological constant problem is also discussed. Namely, the aforementioned mathematical construction allows to reformulate the classical vacuum Einstein equation with cosmological constant over a $4$-manifold as an operator equation over its tracial universal von Neumann algebra such that the trace of a solution is naturally identified with the cosmological constant. This framework permits to use the observed magnitude of the cosmological constant to estimate by topological means the number of primordial black holes about the Planck era. This number turns out to be negligable which is in agreement with known density estimates based on the Press--Schechter mechanism.