Making use of its smooth structure only, out of a connected oriented smooth
4-manifold a von Neumann algebra is constructed. As a special four dimensional
phenomenon this von Neumann algebra is approximated by algebraic (i.e., formal)
curvature tensors of the underlying 4-manifold and 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. Nevertheless over a fixed
4-manifold this von Neumann algebra admits a 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.
Some consequences of this construction for quantum gravity are also
discussed. Namely reversing the construction by starting not with a particular
smooth 4-manifold but with the unique hyperfinite ${\rm II}_1$ factor, a
conceptually simple but manifestly four dimensional, covariant,
non-perturbative and genuinely quantum theory is introduced whose classical
limit is general relativity in an appropriate sense. Therefore it is reasonable
to consider it as a sort of quantum theory of gravity. In this model, among
other interesting things, the observed positive but small value of the
cosmological constant acquires a natural explanation.