f-algebra products on AL and AM-spaces

We characterize all $$f\!$$ f -algebra products on AM-spaces by constructing a canonical AM-space WX W X associated to each AM-space X , such that the $$f\!$$ f -algebra products on X correspond bijectively to the positive cone (WX)₊ ( W X ) + . This generalizes the classical description of $$f\!$$ 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.