$f$-algebra products on AL and AM-spaces
David Muñoz-Lahoz
TL;DR
The paper develops a complete description of $f$-algebra products on AM- and AL-spaces by introducing a canonical associated space $W_X$ for each AM-space $X$, providing a bijection between $f$-algebra products on $X$ and $(W_X)_+$. It generalizes the standard $C(K)$-space characterization and identifies when an AM-space can be embedded as a closed subalgebra of a $C(K)$-space, yielding the AM-algebra construct when such a product exists and is unique. It also gives an explicit atom-based description of $f$-algebra products on AL-spaces and shows that AL-spaces admit only the zero product precisely when they have no atoms. Collectively, these results offer concrete criteria and representations for $f$-algebra products on AM-/AL-spaces and clarify when algebraic embeddings into function spaces are possible, including characterizations equivalent to being lattice isometric to closed ideals of $C(K)$ or to $C_0(\, ext{Ω}\,)$.
Abstract
We characterize all $f$-algebra products on AM-spaces by constructing a canonical AM-space $W_X$ associated to each AM-space $X$, such that the $f$-algebra products on $X$ correspond bijectively to the positive cone $(W_X)_+$. This generalizes the classical description of $f$-algebra products on $C(K)$ spaces. We also identify the unique product (when it exists) that embeds $X$ as a closed subalgebra of $C(K)$, and study AM-spaces for which this product exists -- the so-called AM-algebras. Finally, we investigate AM-spaces that admit only the zero product, providing a characterization in the AL-space case and examples showing that no simple characterization exists in general.