A characterisation of the Fourier transform on the Heisenberg group

TL;DR

This paper characterizes the Fourier transform on the Heisenberg group by showing that continuous *-homomorphisms correspond to Weyl transforms, linking algebraic structure with harmonic analysis.

Contribution

It provides a novel characterization of the Fourier transform on the Heisenberg group via *-homomorphisms and convolution, connecting algebraic and analytical frameworks.

Findings

Any continuous *-homomorphism of L1(Cn) is a Weyl transform

Characterization of the group Fourier transform on the Heisenberg group

Establishes a link between algebraic homomorphisms and harmonic analysis

Abstract

The aim of this paper is to show that any continuous *-homomorphism of L1(Cn)(with twisted convolution as multipli- cation) into B(L2(Rn)) is essentially a Weyl transform. From this we deduce a similar characterisation for the group Fourier transform on the Heisenberg group, in terms of convolution.