multiplicative linear functional
1 Definition
Let 𝒜 be an algebra over ℂ.
A multiplicative linear functional is an nontrivial algebra homomorphism ϕ:𝒜⟶ℂ, i.e. ϕ is a non-zero linear functional such that ϕ(x⋅y)=ϕ(x)⋅ϕ(y),∀x,y∈𝒜.
Multiplicative linear functionals are also called characters of 𝒜.
2 Properties
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If ϕ is a multiplicative linear functional in a Banach algebra
𝒜 over ℂ then ϕ is continuous. Moreover, if 𝒜 has an identity element
then ∥ϕ∥=1.
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Suppose 𝒜 is a Banach algebra over ℂ. The set of multiplicative linear functionals in 𝒜 is a locally compact Hausdorff space
in the weak-* topology
. Moreover, this set is compact
if 𝒜 has an identity element.
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Suppose 𝒜 is a commutative Banach algebra over ℂ with an identity element. There is a bijective correspondence between the set of maximal ideals
in 𝒜 and the set of multiplicative linear functionals in 𝒜. This correspondence is given by
ϕ⟼Kerϕ
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Suppose 𝒜 is a commutative
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
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Let X be a topological space and C(X) the algebra of continuous functions X⟶ℂ. Every point evaluation is a multiplicative linear functional of C(X). In other words, for every point x∈X, the function
evx:C(X)⟶ℂ evx(f)=f(x) 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 |