Strict deformation quantization of locally convex algebras and modules

TL;DR

This paper extends Rieffel's strict deformation quantization to locally convex algebras and modules using vector-valued oscillatory integrals, broadening the framework and including new examples with compactly supported actions.

Contribution

It introduces symbol spaces for locally convex vector spaces and develops a generalized deformation quantization framework incorporating these spaces.

Findings

Extended deformation quantization to locally convex algebras and modules.

Unified various integral formulas for star products within this framework.

Constructed new examples involving compactly supported actions on n.

Abstract

In this work various symbol spaces with values in a sequentially complete locally convex vector space are introduced and discussed. They are used to define vector-valued oscillatory integrals which allow to extend Rieffel's strict deformation quantization to the framework of sequentially complete locally convex algebras and modules with separately continuous products and module structures, making use of polynomially bounded actions of $\mathbb{R}^n$. Several well-known integral formulas for star products are shown to fit into this general setting, and a new class of examples involving compactly supported $\mathbb{R}^n$-actions on $\mathbb{R}^n$ is constructed.