# The principal-symbol index map for an algebra of pseudodifferential operators

## TL;DR

This paper describes the K-theory index map for a C*-algebra generated by zero-order pseudodifferential operators on a cylinder, linking algebraic K-theory classes to Fredholm indices of elliptic operators.

## Contribution

It provides a detailed description of the principal-symbol index map for a specific algebra of pseudodifferential operators on a cylinder, including its target in K_0 of the commutator ideal.

## Key findings

- The index map takes values in Z^2, isomorphic to K_0 of the commutator ideal.
- The index map relates K_1-classes of invertible operators to Fredholm indices of elliptic operators.
- The algebra includes operators with periodic symbols on a cylindrical manifold.

## Abstract

A C*algebra A generated by a class of zero-order classical pseudodifferential operator on a cylinder RxB, where B is a compact riemannian manifold, containing operators with periodic symbols, is considered. A description of the K-theory index map associated to the continuous extension to A of the principal-symbol map is given. That index map takes values in K_0 of the commutator ideal E of the algebra, which is isomorphic to Z^2. It maps the K_1-class of an operator invertible modulo E to the Fredholm indices of a pair of elliptic pseudodifferentail operators on SxB, where S denotes the circle.