On a conjectured noncommutative Beals-Cordes-type characterization

Abstract:

Given a skew-symmetric matrix $J,$ we prove that a bounded operator $A$ on $L^2({{\mathbb R}^{d}}),$ for which $(z,\zeta )\mapsto T_zM_\zeta AM_\zeta ^{-1}T_z^{-1}$ is smooth, and which commutes with all pseudodifferential operators $G(x+JD),$ $G\in {{\mathcal S}({{\mathbb R}^{d}})},$ is of the form $F(x-JD),$ with $F$ possessing bounded derivatives of all orders on ${{\mathbb R}^{d}}.$ Here, $T_z$ and $M_\zeta$ denote the translation and the gauge representations of ${{\mathbb R}^{d}}.$ 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.