Symbol calculus of square-integrable operator-valued maps

TL;DR

This paper introduces an abstract framework for quantization and dequantization that generalizes existing methods, encompassing various examples like magnetic Weyl calculus and representations of Lie groups, and demonstrates its compatibility with infinite tensor products.

Contribution

It develops a novel abstract approach to operator calculus that does not rely solely on group representations, broadening the scope of quantization techniques.

Findings

Framework unifies multiple quantization methods

Shows compatibility with infinite tensor products

Includes examples from magnetic Weyl calculus and Lie group representations

Abstract

We develop an abstract framework for the investigation of quantization and dequantization procedures based on orthogonality relations that do not necessarily involve group representations. To illustrate the usefulness of our abstract method we show that it behaves well with respect to the infinite tensor products. This construction subsumes examples coming from the study of magnetic Weyl calculus, the magnetic pseudo-differential Weyl calculus, the metaplectic representation on locally compact abelian groups, irreducible representations associated with finite-dimensional coadjoint orbits of some special infinite-dimensional Lie groups, and the square-integrability properties shared by arbitrary irreducible representations of nilpotent Lie groups.