Banach Algebra - Ideals and Characters

Ideals and Characters

Let A  be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A. Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ(A) of all nonzero homomorphisms from A  to C. The set Δ(A) is called the "structure space" or "character space" of A, and its members "characters."

A character χ is a linear functional on A which is at the same time multiplicative, χ(ab) = χ(a) χ(b), and satisfies χ(1) = 1. Every character is automatically continuous from A  to C, since the kernel of a character is a maximal ideal, which is closed. Moreover, the norm (i.e., operator norm) of a character is one. Equipped with the topology of pointwise convergence on A (i.e., the topology induced by the weak-* topology of A∗), the character space, Δ(A), is a Hausdorff compact space.

For any x ∈ A,

where is the Gelfand representation of x defined as follows: is the continuous function from Δ(A) to C given by   The spectrum of in the formula above, is the spectrum as element of the algebra C(Δ(A)) of complex continuous functions on the compact space Δ(A). Explicitly,

.

As an algebra, a unital commutative Banach algebra is semisimple (i.e., its Jacobson radical is zero) if and only if its Gelfand representation has trivial kernel. An important example of such an algebra is a commutative C*-algebra. In fact, when A is a commutative unital C*-algebra, the Gelfand representation is then an isometric *-isomorphism between A and C(Δ(A)) .