Free Banach $f$-algebras

We construct and analyze the free Banach $f\!$-algebra $\operatorname{FB{\it f}A}[E]$ 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 $\mathbb{R}^{E^*}$, 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 $B_{E^*}$. This yields a representation of the free normed $f\!$-algebra inside $C(B_{E^*})$. We prove that this representation extends to an injective map on the completion $\operatorname{FB{\it f}A}[E]$ if and only if $\operatorname{FB{\it f}A}[E]$ is semiprime, and we establish that $\operatorname{FB{\it f}A}[E]$ is indeed semiprime whenever $E$ is finite-dimensional or $E = L_1(μ)$. 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 $B$.