Operator Algebras Associated with Quantized Canonical Transformations

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.

This is a survey article for the proceedings of the RIMS Workshop 'Symmetry and Singularity of Geometric Structures and Differential Equations', December 2018, based on joint work of the authors and Boris Sternin

References in corpus (2)

• Type III and spectral triples
• Analytic and algebraic indices of elliptic operators associated with discrete groups of quantized canonical transformations