| Version 2 |
Version 1 |
| Let $\mathcal{A}$ be a Banach algebra. |
Let $\mathcal{A}$ be a Banach algebra. |
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| A {\bf left approximate identity} for $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which \PMlinkescapetext{satisfies}: |
A {\bf left approximate identity} for $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which \PMlinkescapetext{satisfies}: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$, for some constant $C$. |
\item $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$, for some constant $C$. |
| \item $e_{\lambda}a \longrightarrow a\;$, for every $a \in \mathcal{A}$. |
\item $e_{\lambda}a \longrightarrow a\;$, for every $a \in \mathcal{A}$. |
| \end{enumerate} |
\end{enumerate} |
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| Similarly, a {\bf right approximate identity} for $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which \PMlinkescapetext{satisfies}: |
Similarly, a {\bf right approximate identity} for $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which \PMlinkescapetext{satisfies}: |
| \begin{enumerate} |
\begin{enumerate} |
| \item $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$, for some constant $C$. |
\item $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$, for some constant $C$. |
| \item $ae_{\lambda} \longrightarrow a\;$, for every $a \in \mathcal{A}$. |
\item $ae_{\lambda} \longrightarrow a\;$, for every $a \in \mathcal{A}$. |
| \end{enumerate} |
\end{enumerate} |
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| An {\bf approximate identity} for a $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which is both a left and right approximate identity. |
An {\bf approximate identity} for a $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which is both a left and right approximate identity. |
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| \subsubsection{Remarks:} |
\subsubsection{Remarks:} |
| \begin{itemize} |
\begin{itemize} |
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\item There are examples of Banach algebras that do not have approximate \PMlinkescapetext{identities}.
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\item There are examples of Banach algebras that don't have approximate \PMlinkescapetext{identities}.
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| \item If $A$ has an identity element $e$, then clearly $e$ itself is an approximate identity for $\mathcal{A}$. |
\item If $A$ has an identity element $e$, then clearly $e$ itself is an approximate identity for $\mathcal{A}$. |
| \end{itemize} |
\end{itemize} |
