2021/05/17 by Ν. H. Bingham, Bingham, N. H., A. J. Ostaszewski +1
Mathematics · Computer Science · #Functional Equations Stability Results #Advanced Algebra and Logic #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2105.07794
We are concerned below with the characterization in a unital commutative real Banach algebra \mathbbA of continuous solutions of the Gołąb-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 \mathbbA 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 ℝd with componentwise product, clarifying the context of recent developments in [RooSW]. The case \mathbbA=ℂ provides a new viewpoint on continuous complex-valued solutions of the primary equation by distinguishing analytic from real-analytic ones.