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

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

  • Suppose 𝒜 is a commutativePlanetmathPlanetmathPlanetmath C*-algebra ( Multiplicative linear functionals in 𝒜 are exactly the irreducible representations ( 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


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

Title multiplicative linear functional
Canonical name MultiplicativeLinearFunctional
Date of creation 2013-03-22 17:22:25
Last modified on 2013-03-22 17:22:25
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 29
Author asteroid (17536)
Entry type Definition
Classification msc 46H05
Synonym character (of an algebra)
Related topic LinearFunctional
Related topic GelfandTransform
Related topic BanachAlgebra
Defines character
Defines maximal ideal space
Defines character space