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On the structure of ideals and multipliers: A unified approach


Authors: Mostafa Mbekhta and Matthias Neufang
Journal: Proc. Amer. Math. Soc. 147 (2019), 4757-4769
MSC (2010): Primary 43A10, 43A20, 46H10
DOI: https://doi.org/10.1090/proc/14676
Published electronically: August 7, 2019
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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).


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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
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
Email: mneufang@math.carleton.ca; matthias.neufang@univ-lille.fr

DOI: https://doi.org/10.1090/proc/14676
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
Article copyright: © Copyright 2019 American Mathematical Society