Free Banach $f$-algebras

Abstract

We construct and analyze the free Banach $f$-algebra $\mathrm{FBfA}[E]$ generated by a Banach space $E$, extending recent developments on free Banach lattices to the setting of Banach $f$-algebras, where multiplication interacts with the lattice structure. Starting from the explicit realization of the free Archimedean $f$-algebra as a sublattice-algebra of ${\mathbb{R}}^{E^{\ast }}$, we develop a new structure theorem for normed $f$-algebras that allows us to identify the kernel of the maximal submultiplicative lattice seminorm as precisely those functions vanishing on the unit ball $B_{E^{\ast }}$. This yields a representation of the free normed $f$-algebra inside $C(B_{E^{\ast }})$. We prove that this representation extends to an injective map on the completion $\mathrm{FBfA}[E]$ if and only if $\mathrm{FBfA}[E]$ is semiprime, and we establish that $\mathrm{FBfA}[E]$ is indeed semiprime whenever $E$ is finite-dimensional or $E=L_1(\mu )$. This is closely related to approximating operators into a Banach $f$-algebra by operators into finite-dimensional Banach $f$-algebras. We also use the newly constructed free objects to provide an example of a semiprime normed $f$-algebra whose norm completion is not semiprime. Using the tools developed for the study of free objects, we show the following extension property: if $A$ is a closed sublattice-algebra of a Banach $f$-algebra $B$, then every real-valued lattice-algebra homomorphism on $A$ extends to a real-valued lattice-algebra homomorphism on $B$.