Noncommutative Geometry - Noncommutative Differentiable Manifolds

Noncommutative Differentiable Manifolds

A smooth Riemannian manifold M is a topological space with a lot of extra structure. From its algebra of continuous functions C(M) we only recover M topologically. The algebraic invariant that recovers the Riemannian structure is a spectral triple. It is constructed from a smooth vector bundle E over M, e.g. the exterior algebra bundle. The Hilbert space L2(M,E) of square integrable sections of E carries a representation of C(M) by multiplication operators, and we consider an unbounded operator D in L2(M,E) with compact resolvent (e.g. the signature operator), such that the commutators are bounded whenever f is smooth. A recent deep theorem states that M as a Riemannian manifold can be recovered from this data.

This suggests that one might define a noncommutative Riemannian manifold as a spectral triple (A,H,D), consisting of a representation of a C*-algebra A on a Hilbert space H, together with an unbounded operator D on H, with compact resolvent, such that is bounded for all a in some dense subalgebra of A. Research in spectral triples is very active, and many examples of noncommutative manifolds have been constructed.