The index of Toeplitz operators on compact Lie groups and simply connected closed 3-manifolds

TL;DR

This paper develops a method to compute the index of Toeplitz operators on compact Lie groups and applies it to simply connected closed 3-manifolds like the 3-sphere, combining advanced index theorems and symbolic calculus.

Contribution

It introduces a novel approach using operator-valued symbols and combines the Connes index theorem with symbolic calculus to analyze Toeplitz operators on Lie groups and 3-manifolds.

Findings

Computed the index of Toeplitz operators on compact Lie groups.

Extended the index computation to simply connected closed 3-manifolds.

Connected index theory with topological results like Poincaré's theorem.

Abstract

In this paper we use the notion of operator-valued symbol in order to compute the index of Toeplitz operators on compact Lie groups. Our approach combines the Connes index theorem and the infinite-dimensional operator-valued symbolic calculus of Ruzhansky-Turunen. We also give applications to the index of Toeplitz operators on simply connected closed $3$-manifolds $\mathbb{M}\simeq \mathbb{S}^3\simeq \textnormal{SU}(2) ,$ by using, as a fundamental tool, the Poincar\'e theorem (see Perelman [31,32,33,34])