Sine Subtraction Laws on Semigroups

Abstract

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.

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

“…The sine subtraction law which is named after the formula sin(x − y) = sin x cos y − cos x sin y where x, y ∈ R, from elementary trigonometry, has a long history, but new facets of it are still being discovered. Thus Aserrar and Elqorachi [3,4] and Ebanks [8,9,11] have recently studied various extensions of it from R to semigroups. For the classic case of an abelian group see Aczél and Dhombres [1, pp.…”

“…Ebanks [11,Example 3.3] presents on the (ax + b)-group for x * = x −1 a solution of (8.1) such that f is not central. Thus his assumption about f being central is a real restriction.…”

“…Another justification for our study is that we from our results for (1.3) can derive the set of solution (f, g) of the sine subtraction law (8.1) on groups (Theorem 8.3). Ebanks [11,Theorem 3.2] solved (8.1) on semigroups, but only for f central.…”