Gaussian elements in CCR algebras

TL;DR

This paper constructs a matrix system in the Weyl algebra using Gaussian elements, revealing the structure of the associated C*-algebra as a compact operator algebra and analyzing Gaussian spectral decompositions.

Contribution

It introduces a Gaussian-based matrix system in the Weyl algebra that explicitly includes minimal projections, clarifying the algebra's structure.

Findings

The constructed matrix system includes von Neumann's minimal projection.

The associated C*-algebra is shown to be a compact operator algebra.

Spectral decomposition of Gaussian elements is explicitly characterized.

Abstract

A system of matrix units in the Weyl algebra of convolution type is constructed with the aid of a Gaussian element so that it includes von Neumann's minimal projection, which explicitly shows that the associated C*-algebra is a compact operator algebra. The spectral decomposition of an arbitrary Gaussian element is then worked out in terms of the diagonal projections in the matrix units.