60 lines
2.0 KiB
TeX
60 lines
2.0 KiB
TeX
\section{Ideals}
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\label{section:banach-algebra-ideals}
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\begin{definition}[Ideal]
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\label{definition:banach-algebra-ideal}
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Let $A$ be a Banach algebra and $I \subset A$, then $I$ is a \textbf{left ideal} if:
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\begin{enumerate}
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\item $I$ is a subspace of $A$.
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\item For each $x \in A$, $aI \subset I$.
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\end{enumerate}
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and $I$ is a \textbf{two-sided ideal} if in addition to the above,
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\begin{enumerate}[start=2]
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\item For each $x \in A$, $Ia \subset I$.
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\end{enumerate}
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\textit{In this document, the "sidedness" of ideals are omitted. Within each block, the statement should hold as long as the interpretation is consistent. }
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\end{definition}
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\begin{definition}[Proper Ideal]
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\label{definition:banach-algebra-proper-ideal}
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Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is \textbf{proper} if $I \subsetneq A$.
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\end{definition}
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\begin{definition}[Maximal Ideal]
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\label{definition:maximal-ideal}
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Let $A$ be a Banach algebra and $I \subset A$ be an ideal, then $I$ is \textbf{maximal} if:
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\begin{enumerate}
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\item $I$ is proper.
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\item For any proper ideal $J \subsetneq A$ with $J \supset I$, $I = J$.
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\end{enumerate}
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The set $\cm(A)$ of maximal two-sided ideals of $A$ is the \textbf{maximal ideal space} of $A$.
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\end{definition}
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\begin{proposition}
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\label{proposition:ideal-gymnastics}
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Let $A$ be a unital Banach algebra and $I \subset A$ be a proper ideal, then:
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\begin{enumerate}
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\item $I$ contains no invertible elements.
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\item $\ol I$ is also a proper ideal.
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\item $I$ is contained in a maximal ideal.
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\item If $I$ is maximal, then $I$ is closed.
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\end{enumerate}
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\end{proposition}
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\begin{proof}[Proof, {{\cite{FollandHarmonic}}}. ]
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(1): If $I$ contains an invertible element, then $1 \in I$ and thus $A = I$.
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(2): By \autoref{proposition:banach-algebra-inverse}, $G(A)$ is open. Since $G(A) \cap I = \emptyset$, $G(A) \cap \ol I = \emptyset$ as well.
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(3): By Zorn's lemma.
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(4): By (2).
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\end{proof}
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