Connes trace theorem for contact manifolds

Connes trace theorem relates the asymptotic behavior of the eigenvalues of a negative order pseudodifferential operator on a compact manifold to its principal symbol. Its corollary is Connes integration theorem, which allows us to compute the integral of a continuous function on a compact Riemannian manifold with respect to the Riemannian volume form in terms of the Laplace-Beltrami operator. These results are of fundamental importance in noncommutative geometry, since they allow us to introduce the general concepts of noncommutative integral and noncommutative Yang-Mills action. We discuss analogs of Connes theorems for contact sub-Riemannian manifolds and the contact sub-Laplacian. The proofs of these theorems rely on new constructions of the pseudodifferential calculus and the principal symbol map on compact contact manifolds based on the theory of $C^*$-algebras. The results were obtained jointly with F. Sukochev and D. Zanin.