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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The behavior of Fourier transforms for nilpotent Lie groups
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by Ronald L. Lipsman and Jonathan Rosenberg PDF
Trans. Amer. Math. Soc. 348 (1996), 1031-1050 Request permission

Abstract:

We study weak analogues of the Paley-Wiener Theorem for both the scalar-valued and the operator-valued Fourier transforms on a nilpotent Lie group $G$. Such theorems should assert that the appropriate Fourier transform of a function or distribution of compact support on $G$ extends to be “holomorphic” on an appropriate complexification of (a part of) $\hat G$. We prove the weak scalar-valued Paley-Wiener Theorem for some nilpotent Lie groups but show that it is false in general. We also prove a weak operator-valued Paley-Wiener Theorem for arbitrary nilpotent Lie groups, which in turn establishes the truth of a conjecture of Moss. Finally, we prove a conjecture about Dixmier-Douady invariants of continuous-trace subquotients of $C^{*}(G)$ when $G$ is two-step nilpotent.
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Additional Information
  • Ronald L. Lipsman
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • Email: rll@math.umd.edu
  • Jonathan Rosenberg
  • Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742
  • MR Author ID: 298722
  • ORCID: 0000-0002-1531-6572
  • Email: jmr@math.umd.edu
  • Received by editor(s): September 4, 1994
  • Additional Notes: Both authors were partially supported by NSF grant DMS-92-25063.
  • © Copyright 1996 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 348 (1996), 1031-1050
  • MSC (1991): Primary 22E27; Secondary 43A30, 44A12, 22D25
  • DOI: https://doi.org/10.1090/S0002-9947-96-01583-8
  • MathSciNet review: 1370646