The universal von Neumann algebra of smooth four-manifolds

Making use of its smooth structure only, out a connected oriented $4$-manifold von Neumann algebra is constructed. As special four dimensional phenomenon this approximated by algebraic (i.e., formal) curvature tensors the underlying and itself hyperfinite factor ${\rm II}_1$ type hence unique up to abstract isomorphisms algebras. Nevertheless over fixed admits representation on Hilbert space such that unitary equivalence class preserved orientation-preserving diffeomorphisms. Consequently Murray--von coupling constant well-defined gives rise new computable real-valued invariant. Some consequences construction for quantum gravity are also discussed. Namely reversing starting not with particular but factor, conceptually simple manifestly dimensional, covariant, non-perturbative genuinely theory introduced whose classical limit general relativity in an appropriate sense. Therefore it reasonable consider as sort gravity. In model, among other interesting things, observed positive small value cosmological acquires natural explanation.