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

TL;DR

This paper characterizes continuous solutions of the Golab-Schinzel and Goldie functional equations within unital commutative real Banach algebras, providing structure theorems and explicit forms, and explores their solutions in complex Banach algebras.

Contribution

It offers new structure theorems and explicit solutions for these functional equations in Banach algebras, including complex cases, using multi-Popa group theory.

Findings

Explicit forms of solutions in $C[0,1]$ and $R^d$

Distinction between analytic and real-analytic solutions in $C$

Clarification of the equations' structure 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.