D’Alembert’s functional equation on topological groups

Summary.D’Alembert’s equation f(xy) + f(xy−1) = 2f(x)f(y) is solved for all compact groups: all solutions come from continuous homomorphisms from the group to $$SU_2 (\mathbb{C})$$. We introduce the notion of a basic d’Alembert function: a continuous solution for which f(xy) = f(y) for all y implies that x = e. It is shown that every d’Alembert function factors through a basic d’Alembert function. We show that the only compact groups that support a basic d’Alembert function are topologically isomorphic to compact subgroups of $$SU_2 (\mathbb{C})$$. Each subgroup (compact or not) of $$SU_2 (\mathbb{C})$$ supports a basic d’Alembert function.