| Version 12 |
Version 11 |
| Let $\mathcal{A}$ be an algebra over $\mathbb{C}$. |
Let $\mathcal{A}$ be an algebra over $\mathbb{C}$. |
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| A {\bf multiplicative linear functional} is an nontrivial algebra homomorphism $\phi :\mathcal{A} \longrightarrow |
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 $\;\phi(x\cdot y) = \phi(x)\cdot\phi(y), \;\;\;\forall x,y \in \mathcal{A}$. |
\mathbb{C}$, i.e. $\phi$ is a non-zero linear functional such that $\;\phi(x\cdot y) = \phi(x)\cdot\phi(y), \;\;\;\forall x,y \in \mathcal{A}$. |
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| Multiplicative linear functionals are also called {\bf characters} of $\mathcal{A}$. |
Multiplicative linear functionals are also called {\bf characters} of $\mathcal{A}$. |
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| Properties: |
Properties: |
| \begin{enumerate} |
\begin{enumerate} |
| \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 $\|\phi\| = 1$. |
\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 $\|\phi\| = 1$. |
| \item Suppose $\mathcal{A}$ is a Banach algebra over $\mathbb{C}$ with an identity element. There is a bijective correspondence |
\item Suppose $\mathcal{A}$ is a 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 |
between the set of maximal ideals in $\mathcal{A}$ and the set of multiplicative linear functionals |
| in $\mathcal{A}$. |
in $\mathcal{A}$. |
| \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 $\mathcal{A}$ |
\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. |
has an identity element. |
| \end{enumerate} |
\end{enumerate} |
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| There are several designations for the set of all multiplicative linear functionals in $\mathcal{A}$, such as: |
There are several designations for the set of all multiplicative linear functionals in $\mathcal{A}$, such as: |
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the {\bf \PMlinkescapetext{spectrum}} of $\mathcal{A}$, the {\bf maximal ideal space}, the {\bf character space}.
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the {\bf spectrum} of $\mathcal{A}$, the {\bf maximal ideal space}, the {\bf character space}.
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