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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On a conjectured noncommutative Beals-Cordes-type characterization


Authors: Severino T. Melo and Marcela I. Merklen
Journal: Proc. Amer. Math. Soc. 130 (2002), 1997-2000
MSC (2000): Primary 47G30
Published electronically: November 15, 2001
MathSciNet review: 1896033
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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.


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Additional Information

Severino T. Melo
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05315-970, São Paulo, Brazil
Email: toscano@ime.usp.br

Marcela I. Merklen
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05315-970, São Paulo, Brazil
Email: marcela@ime.usp.br

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06270-0
PII: S 0002-9939(01)06270-0
Keywords: Pseudodifferential operators, Heisenberg group
Received by editor(s): November 29, 2000
Received by editor(s) in revised form: January 21, 2001
Published electronically: November 15, 2001
Additional Notes: The first author was supported in part by CNPq (Brazil), Processo 300330/88-0.
The second author was also supported by CNPq, Processo 142280/97-6.
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2001 American Mathematical Society