Definition

A multiplicative linear functional is an nontrivial algebra homomorphism $\phi :\mathcal{A} \longrightarrow \mathbb{C}$ , i.e. $\phi$ is a non-zero linear functional such that $\;\phi(x\cdot y) = \phi(x)\cdot\phi(y), \;\;\;\forall x,y \in \mathcal{A}$ .

Multiplicative linear functionals are also called characters of $\mathcal{A}$ .

Properties

• If $\phi$ is a multiplicative linear functional in a Banach algebra $\mathcal{A}$ over $\mathbb{C}$ then $\phi$ is continuous. Moreover, if $\mathcal{A}$ has an identity element then $\|\phi\| = 1$ .

• Suppose $\mathcal{A}$ is a Banach algebra over $\mathbb{C}$ . The set of multiplicative linear functionals in $\mathcal{A}$ is a locally compact Hausdorff space in the weak-* topology. Moreover, this set is compact if $\mathcal{A}$ has an identity element.

• Suppose $\mathcal{A}$ is a commutative Banach algebra over $\mathbb{C}$ with an identity element. There is a bijective correspondence between the set of maximal ideals in $\mathcal{A}$ and the set of multiplicative linear functionals in $\mathcal{A}$ . This correspondence is given by
  

• Suppose $\mathcal{A}$ is a commutative $C^*$ -algebra. Multiplicative linear functionals in $\mathcal{A}$ are exactly the irreducible representations of $\mathcal{A}$ .

Character space of a Banach algebra

There are several designations for this space, such as: the spectrum of $\mathcal{A}$ , the maximal ideal space, the character space.

Examples

• Let $X$ be a topological space and $C(X)$ the algebra of continuous functions $X \longrightarrow \mathbb{C}$ . Every point evaluation is a multiplicative linear functional of $C(X)$ . In other words, for every point $x \in X$ , the function
  
  that gives the evaluation in $x$ , is a multiplicative linear functional of $C(X)$ .