Generalized small-dimension lemma and d’Alembert type functional equation on compact groups

Abstract

Let ${\mathbb {C}}$ be the set of complex numbers and $\sigma$ be a continuous automorphism and $\tau$ be a continuous anti-automorphism such that $\sigma ^{2}=\tau ^{2}=id.$ The purpose of this paper is to generalize the small-dimension lemma [20, Small Dimension Lemma] and by help of it we find on any compact group G the non-zero continuous solutions $f:G\rightarrow {\mathbb {C}}$ of the functional equation

$$\begin{aligned} f(x\sigma (y))+f(\tau (y)x)=2f(x)f(y), \ \ \ x,y \in G, \end{aligned}$$

in terms of continuous characters of G.