The Gelfand transform x↦x^^ is defined as follows. If ϕ:B->C is linear and multiplicative in the senses ϕ(a x + b y) = a ϕ(x) + b ϕ(y) and ϕ(x y) = ϕ(x) ϕ(y), where B is a commutative Banach algebra, then write x^^(ϕ) = ϕ(x). The Gelfand transform is automatically bounded. For example, if B = L^1(R) with the usual norm, then B is a Banach algebra under convolution and the Gelfand transform is the Fourier transform.