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
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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