  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.
