The Bopp representation of a density matrix is not a density matrix on all of phase space, but its restriction to the image of the wavepacket transform is one.
Dito, Deformation Quantization on a Hilbert Space, Wo rld Scientific, pp
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Phase Space Representation of the Density Operator: Bopp Pseudodifferential Calculus and Moyal Product
The Bopp representation of a density matrix is not a density matrix on all of phase space, but its restriction to the image of the wavepacket transform is one.