Simultaneous power factorization in modules over Banach algebras

TL;DR

This paper extends power factorization theorems in Banach algebra modules, providing constructive methods for simultaneous factorizations, including positive factorizations in ordered contexts, with applications to bounded maps and specific examples.

Contribution

It generalizes previous factorization theorems by constructing explicit factorizations in modules over Banach algebras, including positive and pointwise factorizations, with detailed properties and examples.

Findings

Existence of simultaneous power factorizations under certain conditions

Positive factorizations are possible in some ordered Banach modules

Constructive methods yield explicit factorizations and examples

Abstract

Let $A$ be a Banach algebra with a bounded left approximate identity $\{e_\lambda\}_{\lambda\in\Lambda}$, let $\pi$ be a continuous representation of $A$ on a Banach space $X$, and let $S$ be a non-empty subset of $X$ such that $\lim_{\lambda}\pi(e_\lambda)s=s$ uniformly on $S$. If $S$ is bounded, or if $\{e_\lambda\}_{\lambda\in\Lambda}$ is commutative, then we show that there exist $a\in A$ and maps $x_n: S\to X$ for $n\geq 1$ such that $s=\pi(a^n)x_n(s)$ for all $n\geq 1$ and $s\in S$. The properties of $a\in A$ and the maps $x_n$, as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair's power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of $\mathrm{C}_0({\mathbb R})$ and unbounded $S$ is included.