$f$-algebra products on AL and AM-spaces

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.