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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On the structure of ideals and multipliers: A unified approach
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by Mostafa Mbekhta and Matthias Neufang PDF
Proc. Amer. Math. Soc. 147 (2019), 4757-4769 Request permission

Abstract:

We study the structure of one-sided ideals in a Banach algebra $\mathcal {A}$. We find very general conditions under which any left (right) ideal is of the form $\mathcal {A} q$ ($q \mathcal {A}$) for some idempotent right (left) multiplier on $\mathcal {A}$. We further show that a large class of one-sided multipliers can be realized as a product of an invertible and an idempotent multiplier. Applying our results to algebras over locally compact quantum groups and $C^*$-algebras, we demonstrate that our approach generalizes and unifies various theorems from abstract harmonic analysis and operator algebra theory. In particular, we generalize results of Bekka (and Reiter), Berglund, Forrest, and Lau–Losert. We also deduce the Choquet–Deny theorem for compact groups as an application of our approach. Moreover, we answer, for a certain class of measures on a compact group, a question of Ülger which, in the abelian case, goes back to Beurling (1938).
References
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Additional Information
  • Mostafa Mbekhta
  • Affiliation: Laboratoire de Mathématiques Paul Painlevé (UMR CNRS 8524), Université de Lille, Département de Mathématiques, 59655 Villeneuve d’Ascq Cedex, France
  • MR Author ID: 121980
  • Email: mostafa.mbekhta@univ-lille.fr
  • Matthias Neufang
  • Affiliation: School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6, Canada; and Laboratoire de Mathématiques Paul Painlevé (UMR CNRS 8524), Université de Lille, Département de Mathématiques, 59655 Villeneuve d’Ascq Cedex, France
  • MR Author ID: 718390
  • Email: mneufang@math.carleton.ca; matthias.neufang@univ-lille.fr
  • Received by editor(s): January 2, 2019
  • Published electronically: August 7, 2019
  • Additional Notes: This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). The second author was partially supported by NSERC Discovery Grant RGPIN-2014-06356. This support is gratefully acknowledged.
  • Communicated by: Stephen Dilworth
  • © Copyright 2019 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 147 (2019), 4757-4769
  • MSC (2010): Primary 43A10, 43A20, 46H10
  • DOI: https://doi.org/10.1090/proc/14676
  • MathSciNet review: 4011510