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On a conjectured noncommutative Beals-Cordes-type characterization

Given a skew-symmetric matrix J, we prove that a bounded operator A on L 2 (R d ), for which (z,ζ) → T z MζAM -1 ζ T -1 z is smooth, and which commutes with all pseudodifferential operators G(x + JD), G ∈ S(R d ), is of the form F(x - JD), with F possessing bounded derivatives of all orders on R d . Here, T z and M ζ denote the translation and the gauge representations of R d . This was conjectured by Rieffel (1993) and is an application of the well-known Cordes' characterization of the the Heisenberg-smooth operators as pseudodifferential operators.