The equation f(xy) = f(x)h(y) + g(x)f(y) and representations on C2

Let G be a topological group, and let C ( G ) denote the algebra of continuous, complex valued functions on G . We find the solutions f , g ,h ∈ C ( G ) of the Levi-Civita equation f ( xy ) = f ( x ) h ( y ) + g ( x ) f ( y ) for x , y ∈ G , which is an extension of the sine addition law. Representations of G on C 2 play an important role. As a corollary we get the solutions f,g in C ( G ) of the sine subtraction law f ( xy *) = f ( x ) g ( y ) - g ( x ) f ( y ), x , y ∈ G , in which x ↦ x * is a continuous involution, meaning that ( xy )* = y * x * and x ** = x for all x,y ∈ G . 2020 Mathematics Subject Classification: 39B32 and 39B52.