Operator Algebras Associated with Quantized Canonical Transformations

TL;DR

This paper reviews an approach to index theory for operator algebras linked to Lie groups of quantized canonical transformations, unifying various classical and shift operator index problems.

Contribution

It introduces an ellipticity condition, defines localized algebraic and analytic indices, and proves their equality within a comprehensive framework.

Findings

Ellipticity condition ensures Fredholm property.

Defined localized algebraic and analytic indices.

Unified classical and shift operator index theories.

Abstract

We review an approach to the index theory of operator algebras associated with Lie groups of quantized canonical transformations. Main points are an ellipticity condition ensuring the Fredholm property, the definition of localized algebraic and analytic indices and the proof of their equality. This framework encompasses many well-known index problems, such as the classical theory on closed manifold, the Atiyah-Weinstein problem and the index theory for operators with shifts.