Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Banach lattice AM-algebras
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by David Muñoz-Lahoz and Pedro Tradacete;
Proc. Amer. Math. Soc. 153 (2025), 2565-2577
DOI: https://doi.org/10.1090/proc/17173
Published electronically: March 24, 2025

Abstract:

An analogue of Kakutani’s representation theorem for Banach lattice algebras is provided. We characterize Banach lattice algebras that embed as a closed sublattice-algebra of $C(K)$ precisely as those with a positive approximate identity $(e_\gamma )$ such that $x^{*}(e_\gamma )\to \|x^{*}\|$ for every positive functional $x^{*}$. We also show that every Banach lattice algebra with identity other than $C(K)$ admits different product operations which are compatible with the order and the algebraic identity. This complements the classical result, due to Martignon, that on $C(K)$ spaces pointwise multiplication is the unique compatible product.
References
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Bibliographic Information
  • David Muñoz-Lahoz
  • Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13–15, Campus de Cantoblanco UAM, 28049 Madrid, Spain
  • Email: david.munnozl@uam.es
  • Pedro Tradacete
  • Affiliation: Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13–15, Campus de Cantoblanco UAM, 28049 Madrid, Spain
  • MR Author ID: 840453
  • ORCID: 0000-0001-7759-3068
  • Email: pedro.tradacete@icmat.es
  • Received by editor(s): September 26, 2024
  • Received by editor(s) in revised form: December 5, 2024, and December 10, 2024
  • Published electronically: March 24, 2025
  • Additional Notes: Research was supported by grants PID2020-116398GB-I00, CEX2023-001347-S, and CEX2019-000904-S funded by MICIU/AEI/10.13039/501100011033. Research of the first author was supported by an FPI–UAM 2023 contract funded by Universidad Autónoma de Madrid. Research of the second author was also supported by a 2022 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation.
  • Communicated by: Stephen Dilworth
  • © Copyright 2025 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 153 (2025), 2565-2577
  • MSC (2020): Primary 46B42, 46J10, 46J30, 06F25
  • DOI: https://doi.org/10.1090/proc/17173
  • MathSciNet review: 4892628