Properties of d’Alembert functions

Summary.

We study properties of solutions f of d’Alembert’s functional equations on a topological group G. For nilpotent groups and for connected, solvable Lie groups G, we prove that f has the form $f(x) = (\gamma(x)+\gamma(x^{-1}))/2, x \in G$, where γ is a continuous homomorphism of G into the multiplicative group ${\mathbb{C}} \setminus \{0\}$. We give conditions on G and/or f for equality in the inclusion $\{u\, \in\, G | f(xu) = f(x)\, {\rm for\, all}\, x \in G \}\, \subseteq \{u\, \in\, G | f(u) = 1\}$.