A Quantum Harmonic Analysis Approach to Segal Algebras

TL;DR

This paper introduces quantum Segal algebras within a Banach algebra framework on $L^1(R^{2n}) igoplus T^1$, exploring their properties, ideals, and Gelfand theory, extending classical Segal algebra concepts into quantum harmonic analysis.

Contribution

It develops the concept of quantum Segal algebras, extending classical Segal algebra properties to a quantum harmonic analysis setting, and analyzes their structure and examples.

Findings

Quantum Segal algebras share many properties with classical Segal algebras.

Quantum Segal algebras are not ideals of the ambient space.

Examples of constructions of quantum Segal algebras are provided.

Abstract

In this article, we study a commutative Banach algebra structure on the space $L^1(\mathbb{R}^{2n})\oplus \mathcal{T}^1$, where the $\mathcal{T}^1$ denotes the trace class operators on $L^2(\mathbb{R}^{n})$. The product of this space is given by the convolutions in quantum harmonic analysis. Towards this goal, we study the closed ideals of this space, and in particular its Gelfand theory. We additionally develop the concept of quantum Segal algebras as an analogue of Segal algebras. We prove that many of the properties of Segal algebras have transfers to quantum Segal algebras. However, it should be noted that in contrast to Segal algebras, quantum Segal algebras are not ideals of the ambient space. We also give examples of different constructions that yield quantum Segal algebras.