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单词 MultiplicativeLinearFunctional
释义

multiplicative linear functional


1 Definition

Let 𝒜 be an algebra over .

A multiplicative linear functional is an nontrivial algebra homomorphism ϕ:𝒜, i.e. ϕ is a non-zero linear functionalMathworldPlanetmath such that ϕ(xy)=ϕ(x)ϕ(y),x,y𝒜.

Multiplicative linear functionals are also called charactersMathworldPlanetmath of 𝒜.

2 Properties

  • If ϕ is a multiplicative linear functional in a Banach algebraMathworldPlanetmath 𝒜 over then ϕ is continuous. Moreover, if 𝒜 has an identity elementMathworldPlanetmath then ϕ=1.

  • Suppose 𝒜 is a Banach algebra over . The set of multiplicative linear functionals in 𝒜 is a locally compact Hausdorff spacePlanetmathPlanetmath in the weak-* topologyMathworldPlanetmath. Moreover, this set is compactPlanetmathPlanetmath if 𝒜has an identity element.

  • Suppose 𝒜 is a commutative Banach algebra over with an identity element. There is a bijective correspondencebetween the set of maximal idealsMathworldPlanetmath in 𝒜 and the set of multiplicative linear functionalsin 𝒜. This correspondence is given by

    ϕKerϕ
  • Suppose 𝒜 is a commutativePlanetmathPlanetmathPlanetmath C*-algebra (http://planetmath.org/CAlgebra). Multiplicative linear functionals in 𝒜 are exactly the irreducible representations (http://planetmath.org/BanachAlgebraRepresentation) of 𝒜.

3 Character space of a Banach algebra

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

There are several designations for this space, such as:the of 𝒜, the maximal ideal space, the character space.

4 Examples

  • Let X be a topological space and C(X) the algebra of continuous functions X. Every point evaluation is a multiplicative linear functional of C(X). In other words, for every point xX, the function

    evx:C(X)
    evx(f)=f(x)

    that gives the evaluation in x, is a multiplicative linear functional of C(X).

Titlemultiplicative linear functional
Canonical nameMultiplicativeLinearFunctional
Date of creation2013-03-22 17:22:25
Last modified on2013-03-22 17:22:25
Ownerasteroid (17536)
Last modified byasteroid (17536)
Numerical id29
Authorasteroid (17536)
Entry typeDefinition
Classificationmsc 46H05
Synonymcharacter (of an algebra)
Related topicLinearFunctional
Related topicGelfandTransform
Related topicBanachAlgebra
Definescharacter
Definesmaximal ideal space
Definescharacter space
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更新时间:2025/5/4 21:29:46