O carater de Chern-Connes para C$^*$-sistemas dinamicos calculado em algumas algebras de operadores pseudodiferenciais

TL;DR

This paper explicitly computes the Chern-Connes character for certain C$^*$-dynamical systems involving pseudodifferential operators on spheres, linking K-theory with de Rham cohomology.

Contribution

It provides explicit calculations of the Chern-Connes character for specific pseudodifferential operator algebras on spheres, illustrating the theory with concrete examples.

Findings

Computed the Chern-Connes character for $(ar{ ext{Ψ}}_{cl}^0(S^1), S^1, ext{α})$

Computed the Chern-Connes character for $(ar{ ext{Ψ}}_{cl}^0(S^2), SO(3), ext{α})$

Linked K-theory elements to de Rham cohomology in these examples.

Abstract

Given a C$^*$-dynamical system $(A, G, \alpha)$ one defines a homomorphism, called the Chern-Connes character, that take an element in $K_0(A) \oplus K_1(A)$, the K-theory groups of the C$^*$-algebra $A$, and maps it into $H_{\mathbb{R}}^*(G)$, the real deRham cohomology ring of $G$. We explictly compute this homomorphism for the examples $(\overline{\Psi_{cl}^0(S^1)}, S^1, \alpha)$ and $(\overline{\Psi_{cl}^0(S^2)}, SO(3), \alpha)$, where $\overline{\Psi_{cl}^0(M)}$ denotes the C$^*$-algebra generated by the classical pseudodifferential operators of zero order in the manifold $M$ and $\alpha$ the action of conjugation by the regular representation (translations).