Toeplitz operators and the Roe-Higson type index theorem in Riemannian surfaces

TL;DR

This paper extends the Roe-Higson index theorem to two-dimensional Riemannian surfaces, linking Toeplitz operators with cyclic cohomology and providing explicit examples of non-trivial pairings.

Contribution

It establishes a two-dimensional analogue of the Roe-Higson index theorem for partitioned manifolds, connecting Toeplitz operators with Connes' pairing and cyclic cohomology.

Findings

Proves the equivalence of Connes' pairing and Fredholm index of Toeplitz operators in 2D surfaces.

Uses properties of circles and Higson's argument in the proof.

Provides an example of a non-cylindrical partitioned manifold with non-trivial pairing.

Abstract

We study a two dimensional analogue of the Roe-Higson index theorem for a partitioned manifold. We prove that Connes' pairing of some invertible element with Roe's cyclic one-cocycle coincides to the Fredholm index of a Toeplitz operator. In the proof of this paper, we use some properties of a circle and use Higson's argument. In the last section, there is a example of partitioned manifold, which is not a cylinder, with non-trivial pairing.