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Canonical Schwartz spaces and generalized operators

For a C∞ multiplier ω, on Rn we define the concepts of differentiability and codifferentiability in the Von Neumann algebra generated by the regular ω representation of Rn. Analogs of the classical Schwartz space and its dual are formulated and the case where ω is fully antisymmetric is studied. Connections with the canonical Fourier transform of an earlier paper are investigated.