Sine Subtraction Laws on Semigroups

We consider two variants of the sine subtraction law on a semi-group S. The main objective is to solve f(xy^∗ ) = f(x)g(y) − g(x)f(y) for unknown functions f, g : S → ℂ, where x ↦ x^* is an anti-homomorphic involution. Until now this equation was not solved even when S is a non-Abelian group and x* = x⁻¹. We find the solutions assuming that f is central. A secondary objective is to solve f(xσ(y)) = f(x)g(y) − g(x)f(y), where σ : S → S is a homomorphic involution. Until now this variant was solved assuming that S has an identity element. We also find the continuous solutions of these equations on topological semigroups.