The cosine addition and subtraction formulas on non-abelian groups

Abstract

AbstractLet G be a topological group, and let C(G) denote the algebra of continuous, complex valued functions on G. We determine the solutions $$f,g,h \in C(G)$$ f , g , h ∈ C ( G ) of the Levi-Civita equation $$\begin{aligned} g(xy) = g(x)g(y) + f(x)h(y), \ x,y \in G, \end{aligned}$$ g ( x y ) = g ( x ) g ( y ) + f ( x ) h ( y ) , x , y ∈ G , that extends the cosine addition law. As a corollary we obtain the solutions $$f,g \in C(G)$$ f , g ∈ C ( G ) of the cosine subtraction law $$g(xy^*) = g(x)g(y) + f(x)f(y)$$ g ( x y ∗ ) = g ( x ) g ( y ) + f ( x ) f ( y ) , $$x,y \in G$$ x , y ∈ G where $$x \mapsto x^*$$ x ↦ x ∗ is a continuous involution of G. That $$x \mapsto x^*$$ x ↦ x ∗ is an involution, means that $$(xy)^* = y^*x^*$$ ( x y ) ∗ = y ∗ x ∗ and $$x^{**} = x$$ x ∗ ∗ = x for all $$x,y \in G$$ x , y ∈ G .