approximate identity
ApproximateIdentity
2007-08-25 20:54:57
2008-12-31 16:36:55
Definition
left approximate identity
right approximate identity
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Let $\mathcal{A}$ be a Banach algebra.
A {\bf left approximate identity} for $\mathcal{A}$ is a net $(e_{\lambda})_{\lambda \in \Lambda}$ in $\mathcal{A}$ which \PMlinkescapetext{satisfies}:
\begin{enumerate}
\item $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$, for some constant $C$.
\item $e_{\lambda}a \longrightarrow a\;$, for every $a \in \mathcal{A}$.
\end{enumerate}
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}
\item $\|e_{\lambda}\| < C \;\;\;\; \forall_{\lambda \in \Lambda} \;$, for some constant $C$.
\item $ae_{\lambda} \longrightarrow a\;$, for every $a \in \mathcal{A}$.
\end{enumerate}
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.
\subsubsection{Remarks:}
\begin{itemize}
\item There are examples of Banach algebras that do not have approximate \PMlinkescapetext{identities}.
\item If $\mathcal{A}$ has an identity element $e$, then clearly $e$ itself is an approximate identity for $\mathcal{A}$.
\end{itemize}