Lattice-ordered algebras admitting a polynomial growth continuous function calculus

Abstract

We characterize the Archimedean lattice-ordered algebras with identity that admit a polynomial growth continuous function calculus. More precisely, for an $n$-tuple $\mathbf{x}=(x_1,\dots ,x_n)$ in an Archimedean lattice-ordered algebra $X$ with identity $1_X$, we prove that the existence of a lattice-algebra homomorphism from the algebra $P{G}_n$ of continuous functions on ${\mathbb{R}}^n$ of polynomial growth, sending the coordinate projections to $x_1,\dots ,x_n$ and the constant function to $1_X$, is equivalent to the existence of $f\ge 1_X\vee |x_1|\vee \cdots \vee |x_n|$ and an $f$-subalgebra $Y$ of $X$ such that $1_X,x_1,\dots ,x_n\in Y$ and, for every $m\in \mathbb{N}$, the norm $\Vert \cdot {\Vert }_{f^m}$ is complete on $Y\cap I_{f^m}$. This result may be viewed as an analogue, for lattice-ordered algebras, of the characterization of positively homogeneous continuous function calculus for Archimedean vector lattices due to Laustsen and Troitsky. As a by-product, we describe the finitely generated free objects in the category of uniformly complete Archimedean $f$-algebras and also show that the existence of a nontrivial polynomial growth continuous function calculus on a vector space forces it to be a commutative $f$-algebra.