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

TL;DR

This paper characterizes all $f$-algebra products on AM-spaces by constructing a canonical associated space, generalizing classical results, and explores conditions for the existence of specific algebraic structures.

Contribution

It introduces a canonical AM-space $W_X$ for each AM-space $X$, establishing a bijective correspondence with $f$-algebra products and analyzing special cases like AM-algebras.

Findings

All $f$-algebra products on AM-spaces correspond to the positive cone of a canonical space $W_X$.

The unique embedding product into $C(K)$ spaces is identified when it exists.

Characterizations of AM-spaces with only the zero product are provided, especially in the AL-space case.

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.