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multiplicative linear functional

(Definition)

Definition

Let $\mathcal{A}$ be an algebra over $\mathbb{C}$ .

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 $\Vert\phi\Vert = 1$ .

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

$\displaystyle \phi \longmapsto Ker\; \phi$

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

Character space of a Banach algebra

As stated above, the set of all multiplicative linear functionals in a Banach algebra $\mathcal{A}$ is a locally compact Hausdorff space with the weak-* topology. It becomes a compact set if $\mathcal{A}$ has an identity element.

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

Examples

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

$\displaystyle ev_x : C(X) \longrightarrow \mathbb{C}$

$\displaystyle ev_x ( f) = f(x)$

that gives the evaluation in , is a multiplicative linear functional of .

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Other names:

character (of an algebra)

Also defines:

character, maximal ideal space, character space

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Cross-references: function, point, topological space, compact set, maximal ideals, bijective, commutative, compact, weak-* topology, locally compact Hausdorff space, identity element, continuous, Banach algebra, linear functional, homomorphism, algebra
There are 9 references to this entry.

This is version 26 of multiplicative linear functional, born on 2007-07-04, modified 2008-01-19.
Object id is 9737, canonical name is MultiplicativeLinearFunctional.
Accessed 2233 times total.

Classification:

AMS MSC:

46H05 (Functional analysis :: Topological algebras, normed rings and algebras, Banach algebras :: General theory of topological algebras)

Pending Errata and Addenda

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