Banach lattice AM-algebras
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912047981133824 |
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| author | Muñoz-Lahoz, David Tradacete, Pedro |
| author_facet | Muñoz-Lahoz, David Tradacete, Pedro |
| contents | 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 $(e_γ)$ such that $x^{*}(e_γ)\to \|x^{*}\|$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Banach lattice AM-algebras Muñoz-Lahoz, David Tradacete, Pedro Functional Analysis 46B42, 46J10, 46J30, 06F25 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 $(e_γ)$ such that $x^{*}(e_γ)\to \|x^{*}\|$ 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. |
| title | Banach lattice AM-algebras |
| topic | Functional Analysis 46B42, 46J10, 46J30, 06F25 |
| url | https://arxiv.org/abs/2409.18500 |