$C^{\ast }$-algebraic approach to the principal symbol. III

We treat the notion of principal symbol mapping on a compact smooth manifold as a $\ast$-homomorphism of $C^{\ast }$-algebras. Principal symbol mapping is built from the ground, without referring to the pseudodifferential calculus on the manifold. Our concrete approach allows us to extend Connes Trace Theorem for compact Riemannian manifolds.