Gelfand Representation - The Gelfand Representation of A Commutative Banach Algebra

The Gelfand Representation of A Commutative Banach Algebra

Let A be a commutative Banach algebra, defined over the field C of complex numbers. A non-zero algebra homomorphism φ: A → C is called a character of A; the set of all characters of A is denoted by Φ_A.

It can be shown that every character on A is automatically continuous, and hence Φ_A is a subset of the space A* of continuous linear functionals on A; moreover, when equipped with the relative weak-* topology, Φ_A turns out to be locally compact and Hausdorff. (This follows from the Banach-Alaoglu theorem.) The space Φ_A is compact (in the topology just defined) if and only if the algebra A has an identity element.

Given a ∈ A, one defines the function by . The definition of Φ_A and the topology on it ensure that is continuous and vanishes at infinity, and that the map defines a norm-decreasing, unit-preserving algebra homomorphism from A to C₀(Φ_A). This homomorphism is the Gelfand representation of A, and is the Gelfand transform of the element a. In general the representation is neither injective nor surjective.

In the case where A has an identity element, there is a bijection between Φ_A and the set of maximal proper ideals in A (this relies on the Gelfand-Mazur theorem). As a consequence, the kernel of the Gelfand representation A → C₀(Φ_A) may be identified with the Jacobson radical of A. Thus the Gelfand representation is injective if and only if A is (Jacobson) semisimple.