p-adic representative functions on abelian groups

The main result is the following. Let G be an abelian group, let K be an algebraically closed field of characteristic zero. Let A be any shift-invariant K-algebra with unit element of representative functions G→K, invariant under the antipode. Then the additive homomorphisms G→K in A together with the multiplicative homomorphisms $\text{G→K}{}^{\mathrm{\ast }}$ in A generate A (Theorem 1.1). In § 2 a few consequences for p-adic representative functions are discussed.