The Fourier transform

Usually we shall just call A an algebra if the field k is clear from the context. The algebra A is associative if multiplication is associative i.e. for all a, b, c ∈ A, (ab)c = a(bc), and unital if there is a multiplicative identity, i.e. an element usually denoted by 1 such that, for all a ∈ A, 1a = a1 = a. Note that, in this case, 1 = 0 ⇐⇒ A = {0}. Otherwise, the map k → A defined by t 7→ t·1 is injective and identifies k with the subset k·1 = {t·1 : t ∈ k} of A. For us, all algebras will be associative and unital (although there are many interesting classes of non-associative algebras). A k-algebra homomorphism f : A → B is a function from A to B which is both a k-linear map and a ring homomorphism; equivalently, f is k-linear and f(ab) = f(a)f(b) for all a, b ∈ A. If A and B are unital, then we will also require that f(1) = 1. The algebra homomorphism f is an isomorphism if it is a bijection. In this case, f−1 is also an algebra homomorphism. A subalgebra A′ of A is defined in the obvious way, as a vector subspace closed under multiplication. If A is unital then we also require that 1 ∈ A′. In this case, k · 1 is a subalgebra of A and is in fact the smallest subalgebra of A.