The free Banach $f$-algebra generated by a Banach space

TL;DR

The paper constructs the free Banach $f$-algebras generated by a Banach space $E$ by first developing free Archimedean $f$-algebras, then implementing a structure theorem that embeds Banach $f$-algebras into $X_0oxplus_ abla C(K)$. It provides an explicit kernel description $ ext{ker } ho=iglbox{$f|_{B_{E^{*}}}=0$}igrbox{$}$, enabling a concrete representation of the free normed algebra inside $C(B_{E^{*}})$ and, in the finite-dimensional case, an explicit isomorphism $ ext{FBFA}[E]\cong C([0,1] imes S_{E^{*}})$ with a weighted product. The work proves semiprimeness for $ ext{FNFA}[E]$ in general and for $ ext{FBFA}[E]$ in the finite-dimensional or $E=L_1( u)$ cases, and shows representability in $C(B_{E^{*}})$ is equivalent to semiprimeness. It also analyzes norm control via the $ au$-norm, representation by continuous function spaces, and isometric rigidity results when duals are smooth, linking free objects to concrete function spaces and their operator-theoretic behavior. Overall, the paper lays foundational steps for free Banach $f$-algebras and clarifies how lattice and algebraic structures interact in this setting, with explicit finite-dimensional descriptions and pathways to representability and isometry results.

Abstract

We construct and analyze the free Banach $f$-algebra $\mathrm{FB{\it f}A}[E]$ generated by a Banach space $E$, extending recent developments in 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 into $C(B_{E^*})$. We prove that this representation extends to an injective map on the completion $\mathrm{FB{\it f}A}[E]$ if and only if $\mathrm{FB{\it f}A}[E]$ is semiprime, and establish that $\mathrm{FB{\it f}A}[E]$ 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. For a finite-dimensional Banach space $E$, a complete description of $\mathrm{FB{\it f}A}[E]$ is provided: $\mathrm{FB{\it f}A}[E]$ is lattice-algebra isomorphic to $C([0,1]\times S_{E^{*}})$ equipped with pointwise order and the product given by \[ (f\star g)(r,u)=rf(r,u)g(r,u). \] As a consequence, Banach spaces of the same dimension generate isomorphic free Banach $f$-algebras. The interplay between the lattice and algebraic structures also leads to unexpected behavior: for instance, the free normed $f$-algebra is always order dense in $\mathrm{FB{\it f}A}[E]$, whereas for free Banach lattices this only holds when $E$ is finite-dimensional.

Figures (1)

• Figure 1: Representation of three functions in $C([0,1]\times S_{\ell_{2}^2})$. In the plots, $[0,1]$ is identified with $0\le z\le 1$ and $S_{\ell_{2}^2}$ is identified with the unit circle in the plane $z=0$. The elements $e_1,e_2$ denote the canonical basis of $\ell_{2}^2$.

Theorems & Definitions (126)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.6
• proof