Modulation Spaces and Representations for Rieffel's Quantization

TL;DR

This paper introduces modulation spaces tailored for Rieffel's deformation quantization, enabling the construction of Hilbert space representations of the associated $C^*$-algebras, with special results for Abelian cases.

Contribution

It develops localized modulation maps and spaces specific to Rieffel's quantization, facilitating representation theory for the deformed algebras.

Findings

Construction of Hilbert space representations from covariant systems.

Orthogonal relations in the Abelian case.

Enhanced understanding of representations for deformed $C^*$-algebras.

Abstract

We define localized modulation maps and modulation spaces of symbols suited to the study of Rieffel's deformation quantization pseudodifferential calculus. They are used to generate Hilbert space representations for the quantized $C^*$-algebras, starting from covariant representations of the corresponding twisted $C^*$-dynamical system. In the case of an Abelian undeformed algebra, orthogonal relations and extra information about the representations are obtained.