multiplicative linear functional
1 Definition
Let $\mathcal{A}$ be an algebra over $\u2102$.
A multiplicative linear functional is an nontrivial algebra homomorphism $\varphi :\mathcal{A}\u27f6\u2102$, i.e. $\varphi $ is a nonzero linear functional^{} such that $\varphi (x\cdot y)=\varphi (x)\cdot \varphi (y),\forall x,y\in \mathcal{A}$.
Multiplicative linear functionals are also called characters^{} of $\mathcal{A}$.
2 Properties

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If $\varphi $ is a multiplicative linear functional in a Banach algebra^{} $\mathcal{A}$ over $\u2102$ then $\varphi $ is continuous. Moreover, if $\mathcal{A}$ has an identity element^{} then $\parallel \varphi \parallel =1$.

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Suppose $\mathcal{A}$ is a Banach algebra over $\u2102$. 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.

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Suppose $\mathcal{A}$ is a commutative Banach algebra over $\u2102$ 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
$$\varphi \u27fcKer\varphi $$

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Suppose $\mathcal{A}$ is a commutative^{} ${C}^{*}$algebra (http://planetmath.org/CAlgebra). Multiplicative linear functionals in $\mathcal{A}$ are exactly the irreducible representations (http://planetmath.org/BanachAlgebraRepresentation) of $\mathcal{A}$.
3 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 of $\mathcal{A}$, 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 $X\u27f6\u2102$. Every point evaluation is a multiplicative linear functional of $C(X)$. In other words, for every point $x\in X$, the function
$e{v}_{x}:C(X)\u27f6\u2102$ $e{v}_{x}(f)=f(x)$ that gives the evaluation in $x$, is a multiplicative linear functional of $C(X)$.
Title  multiplicative linear functional 
Canonical name  MultiplicativeLinearFunctional 
Date of creation  20130322 17:22:25 
Last modified on  20130322 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 