On a conjectured noncommutative Beals-Cordes-type characterization

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...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2002-07, Vol.130 (7), p.1997-2000
Main Authors: Melo, Severino T., Merklen, Marcela I.
Format: Article
Language:English
Subjects:
Algebra
Exact sciences and technology
Mathematical analysis
Mathematical integrals
Mathematical theorems
Mathematics
Operator theory
Sciences and techniques of general use
Von Neumann algebra
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Summary: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 {\em Heisenberg-smooth} operators as pseudodifferential operators.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-01-06270-0