Superstability of Generalized Multiplicative Functionals

Abstract

Let X be a set with a binary operation ∘ such that, for each x,y,z∈X, either (x∘y)∘z=(x∘z)∘y, or z∘(x∘y)=x∘(z∘y). We show the superstability of the functional equation g(x∘y)=g(x)g(y). More explicitly, if ε≥0 and f:X→ℂ satisfies |f(x∘y)−f(x)f(y)|≤ε for each x,y∈X, then f(x∘y)=f(x)f(y) for all x,y∈X, or |f(x)|≤(1+1+4ε)/2 for all x∈X. In the latter case, the constant (1+1+4ε)/2 is the best possible.