The Chern-Connes Character for pseudodifferential operators on the sphere

TL;DR

This paper calculates the Chern-Connes character for the algebra of classical zero-order pseudodifferential operators on the sphere, linking $K$-theory to Lie algebra cohomology under SO(3) action.

Contribution

It explicitly computes the Chern-Connes character for these operators, revealing its image as the real numbers when using the principal symbol trace.

Findings

The Chern-Connes character maps to  in general.

When the trace is the integral of the principal symbol, the image is .

The work connects operator algebra $K$-theory with Lie algebra cohomology.

Abstract

We compute the Chern-Connes character (a map from the $K$-theory of a C$^*$-algebra under the action of a Lie group to the cohomology of its Lie algebra) for the $L^2$-norm closure of the algebra of all classical zero-order pseudodifferential operators on the sphere under the canonical action of ${\rm SO}(3)$. We show that its image is $\mathbb{R}$ if the trace is the integral of the principal symbol.