multiplicative linear functionalMultiplicativeLinearFunctional2007-07-04 21:46:172008-01-19 12:59:23Definitioncharactermaximal ideal spacecharacter space% this is the default PlanetMath preamble. as your knowledge
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\section{Definition}
Let $\mathcal{A}$ be an algebra over $\mathbb{C}$.
A {\bf 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 {\bf characters} of $\mathcal{A}$.
\section{Properties}
\begin{itemize}
\item 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 $\|\phi\| = 1$.
\end{itemize}
\begin{itemize}
\item 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.
\end{itemize}
\begin{itemize}
\item 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
\begin{displaymath}
\phi \longmapsto Ker\; \phi
\end{displaymath}
\end{itemize}
\begin{itemize}
\item Suppose $\mathcal{A}$ is a commutative \PMlinkname{$C^*$-algebra}{CAlgebra}. Multiplicative linear functionals in $\mathcal{A}$ are exactly the \PMlinkname{irreducible representations}{BanachAlgebraRepresentation} of $\mathcal{A}$.
\end{itemize}
\section{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 {\bf \PMlinkescapetext{spectrum}} of $\mathcal{A}$, the {\bf maximal ideal space}, the {\bf character space}.
\section{Examples}
\begin{itemize}
\item Let $X$ be a topological space and $C(X)$ the algebra of continuous functions $X \longrightarrow \mathbb{C}$. Every point evaluation is a multiplicative linear functional of $C(X)$. In other words, for every point $x \in X$, the function
\begin{align*}
ev_x : C(X) \longrightarrow \mathbb{C}\\
ev_x ( f) = f(x)
\end{align*}
that gives the evaluation in $x$, is a multiplicative linear functional of $C(X)$.
\end{itemize}