The equation $$\pmb {f(xy) = f(x)h(y) + g(x)f(y)}$$ and representations on $$\pmb {\mathbb {C}^2}$$

Abstract

AbstractLetGbe a topological group, and letC(G) denote the algebra of continuous, complex valued functions onG. We find the solutions$$f,g,h \in C(G)$$f,g,h∈C(G)of the Levi-Civita equation$$\begin{aligned} f(xy) = f(x)h(y) + g(x)f(y), \ x,y \in G, \end{aligned}$$f(xy)=f(x)h(y)+g(x)f(y),x,y∈G,which is an extension of the sine addition law. Representations ofGon$$\mathbb {C}^2$$C2play an important role. As a corollary we get the solutions$$f,g \in C(G)$$f,g∈C(G)of the sine subtraction law$$f(xy^*) = f(x)g(y) - g(x)f(y)$$f(xy∗)=f(x)g(y)-g(x)f(y),$$x,y \in G$$x,y∈G, in which$$x \mapsto x^*$$x↦x∗is a continuous involution, meaning that$$(xy)^* = y^*x^*$$(xy)∗=y∗x∗and$$x^{**} = x$$x∗∗=xfor all$$x,y \in G$$x,y∈G.