$C^*$-algebras have approximate identities
CAlgebrasHaveApproximateIdentities
2007-08-28 21:25:07
2007-08-28 21:25:07
Theorem
approximate identity
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In this entry $\leq$ has three different meanings:
\begin{enumerate}
\item - The \PMlinkname{ordering of self-adjoint elements}{OrderingOfSelfAdjoints} of a given \PMlinkname{$C^*$-algebra}{CAlgebra}.
\item - The usual \PMlinkname{order}{PartialOrder} in $\mathbb{R}$.
\item - The \PMlinkescapetext{order} of a directed set taken as the domain of a given net.
\end{enumerate}
It will be clear from the context which one is being used.
{\bf Theorem -} Every $C^*$-algebra has an approximate identity $(e_{\lambda})_{\lambda \in \Lambda}$. Moreover, the approximate identity $(e_{\lambda})_{\lambda \in \Lambda}$ can be chosen to \PMlinkescapetext{satisfy} the following \PMlinkescapetext{properties}:
\begin{itemize}
\item $0\leq e_{\lambda}\;\;\;\;\forall_{\lambda \in \Lambda}$
\item $\|e_{\lambda}\| \leq 1\;\;\;\;\forall_{\lambda \in \Lambda}$
\item $\lambda \leq \mu\; \Rightarrow \;e_{\lambda}\leq e_{\mu}$, i.e. $(e_{\lambda})_{\lambda \in \Lambda}$ is increasing.
\end{itemize}
For \PMlinkname{separable}{Separable} $C^*$-algebras the approximate identity can be chosen as an increasing sequence $0\leq e_1 \leq e_2 \leq \dots$ of norm-one elements.