Free Banach $f$-algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915945290661888 |
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| author | Muñoz-Lahoz, David Tradacete, Pedro |
| author_facet | Muñoz-Lahoz, David Tradacete, Pedro |
| contents | We construct and analyze the free Banach $f\!$-algebra $\operatorname{FB{\it f}A}[E]$ generated by a Banach space $E$, extending recent developments on free Banach lattices to the setting of Banach $f\!$-algebras, where multiplication interacts with the lattice structure. Starting from the explicit realization of the free Archimedean $f\!$-algebra as a sublattice-algebra of $\mathbb{R}^{E^*}$, we develop a new structure theorem for normed $f\!$-algebras that allows us to identify the kernel of the maximal submultiplicative lattice seminorm as precisely those functions vanishing on the unit ball $B_{E^*}$. This yields a representation of the free normed $f\!$-algebra inside $C(B_{E^*})$. We prove that this representation extends to an injective map on the completion $\operatorname{FB{\it f}A}[E]$ if and only if $\operatorname{FB{\it f}A}[E]$ is semiprime, and we establish that $\operatorname{FB{\it f}A}[E]$ is indeed semiprime whenever $E$ is finite-dimensional or $E = L_1(μ)$. This is closely related to approximating operators into a Banach $f\!$-algebra by operators into finite-dimensional Banach $f\!$-algebras. We also use the newly constructed free objects to provide an example of a semiprime normed $f\!$-algebra whose norm completion is not semiprime. Using the tools developed for the study of free objects, we show the following extension property: if $A$ is a closed sublattice-algebra of a Banach $f\!$-algebra $B$, then every real-valued lattice-algebra homomorphism on $A$ extends to a real-valued lattice-algebra homomorphism on $B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13299 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Free Banach $f$-algebras Muñoz-Lahoz, David Tradacete, Pedro Functional Analysis 06F25, 46B42, 46A40, 46M15 We construct and analyze the free Banach $f\!$-algebra $\operatorname{FB{\it f}A}[E]$ generated by a Banach space $E$, extending recent developments on free Banach lattices to the setting of Banach $f\!$-algebras, where multiplication interacts with the lattice structure. Starting from the explicit realization of the free Archimedean $f\!$-algebra as a sublattice-algebra of $\mathbb{R}^{E^*}$, we develop a new structure theorem for normed $f\!$-algebras that allows us to identify the kernel of the maximal submultiplicative lattice seminorm as precisely those functions vanishing on the unit ball $B_{E^*}$. This yields a representation of the free normed $f\!$-algebra inside $C(B_{E^*})$. We prove that this representation extends to an injective map on the completion $\operatorname{FB{\it f}A}[E]$ if and only if $\operatorname{FB{\it f}A}[E]$ is semiprime, and we establish that $\operatorname{FB{\it f}A}[E]$ is indeed semiprime whenever $E$ is finite-dimensional or $E = L_1(μ)$. This is closely related to approximating operators into a Banach $f\!$-algebra by operators into finite-dimensional Banach $f\!$-algebras. We also use the newly constructed free objects to provide an example of a semiprime normed $f\!$-algebra whose norm completion is not semiprime. Using the tools developed for the study of free objects, we show the following extension property: if $A$ is a closed sublattice-algebra of a Banach $f\!$-algebra $B$, then every real-valued lattice-algebra homomorphism on $A$ extends to a real-valued lattice-algebra homomorphism on $B$. |
| title | Free Banach $f$-algebras |
| topic | Functional Analysis 06F25, 46B42, 46A40, 46M15 |
| url | https://arxiv.org/abs/2511.13299 |