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

TL;DR

This paper develops a $C^{ ext{*}}$-algebraic framework for principal symbol mapping on compact manifolds, enabling an extension of Connes Trace Theorem without relying on pseudodifferential calculus.

Contribution

It introduces a novel $C^{ ext{*}}$-algebraic approach to principal symbols, broadening the theoretical foundation and extending key theorems in noncommutative geometry.

Findings

Extended Connes Trace Theorem to new settings

Provided a $C^{ ext{*}}$-algebraic construction of principal symbols

Eliminated dependence on pseudodifferential calculus

Abstract

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.