A characterisation of the Fourier transform on the Heisenberg group (Q455458)

An operator that takes the convolution product into the pointwise product is essentially the Fourier transform. Taking this characterization of the Fourier transform as a model, analogous results are obtained in this paper for the Weyl transform and the group Fourier transform on the Heisenberg group \(\mathbb H^n\): 1. It is proved that a continuous homomorphism of the algebra \(L^1(\mathbb C^n)\) with twisted convolution multiplication into the operator algebra \(\mathcal B(L^2(\mathbb R^n))\) is `essentially' the Weyl transform. 2. A characterization, in the same spirit, of the group Fourier transform on the Heisenberg group, as a map \(L^1(\mathbb H^n) \rightarrow L^{\infty}(\mathbb R^*, \mathcal S_2, \mu)\) is obtained. Here \(\mathcal S_2\) denotes the algebra of Hilbert-Schmidt operators on \(L^2(\mathbb R^n)\) and the measure \(\mu\) on \(\mathbb R^*\) is defined by \(d\mu(\lambda) = (2\pi)^{-n-1}|\lambda|^n d\lambda\).