Algebras of distributions suitable for phase-space quantum mechanics. I

TL;DR

This paper extends the twisted product to a $*$-algebra of tempered distributions, exploring its properties and invariance under Fourier transform, and constructs Banach algebras using a matrix representation.

Contribution

It introduces a new $*$-algebra of distributions compatible with phase-space quantum mechanics, including a matrix-based Banach algebra construction.

Findings

The extended twisted product is invariant under Fourier transformation.

The algebra includes smooth functions, distributions, and polynomials.

A matrix representation facilitates the construction of Banach algebras.

Abstract

The twisted product of functions on $R^{2N}$ is extended to a $*$-algebra of tempered distributions which contains the rapidly decreasing smooth functions, the distributions of compact support, and all polynomials, and moreover is invariant under the Fourier transformation. The regularity properties of the twisted product are investigated. A matrix presentation of the twisted product is given, with respect to an appropriate orthonormal basis, which is used to construct a family of Banach algebras under this product.