f-algebra products on AL and AM-spaces

Abstract We characterize all $$f\!$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> -algebra products on AM-spaces by constructing a canonical AM-space $$W_X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>X</mml:mi> </mml:msub> </mml:math> associated to each AM-space X , such that the $$f\!$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> -algebra products on X correspond bijectively to the positive cone $$(W_X)_+$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>W</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> </mml:math> . This generalizes the classical description of $$f\!$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mspace/> </mml:mrow> </mml:math> -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.