The Go{\l}\k{a}b-Schinzel and Goldie functional equations in Banach algebras

Abstract

We are concerned below with the characterization in a unital commutative real Banach algebra $\mathbb{A}$ of continuous solutions of the Go{\l}\k{a}b-Schinzel functional equation (below), the general Popa groups they generate and the associated Goldie functional equation. This yields general structure theorems involving both linear and exponential homogeneity in $\mathbb{A}$ for both these functional equations and also explict forms, in terms of the recently developed theory of multi-Popa groups [BinO3,4], both for the ring $C[0,1]$ and for the case of $\mathbb{R}^{d}$ with componentwise product, clarifying the context of recent developments in [RooSW]. The case $\mathbb{A}=\mathbb{C}$ provides a new viewpoint on continuous complex-valued solutions of the primary equation by distinguishing analytic from real-analytic ones.