| Version 3 |
Version 2 |
| \PMlinkescapephrase{decomposition} |
\PMlinkescapephrase{decomposition} |
|
|
| {\bf [Not finished]} |
{\bf [Not finished]} |
|
|
|
Every real valued function $f$ admits a well-known decomposition into its \PMlinkescapetext{positive} and \PMlinkescapetext{negative} parts: $f = f_+ - f_-$. (These functions are defined within the entry Lebesgue integral.) There is an analogous result for self-adjoint elements in a \PMlinkname{$C^*$-algebra}{CAlgebra} that we will now describe.
|
Every real valued function $f$ admits a well-known decomposition in its \PMlinkescapetext{positive} and \PMlinkescapetext{negative} parts: $f = f_+ - f_-$. (These functions are defined within the entry Lebesgue integral.) There is an analogous result for self-adjoint elements in a \PMlinkname{$C^*$-algebra}{CAlgebra} that we will now describe.
|
|
|
| $\,$ |
$\,$ |
|
|
| {\bf Theorem - } Let $\mathcal{A}$ be a $C^*$-algebra and $a \in \mathcal{A}$ a self-adjoint element. Then there are unique positive elements $a_+$ and $a_-$ in $\mathcal{A}$ such that: |
{\bf Theorem - } Let $\mathcal{A}$ be a $C^*$-algebra and $a \in \mathcal{A}$ a self-adjoint element. Then there are unique positive elements $a_+$ and $a_-$ in $\mathcal{A}$ such that: |
| \begin{itemize} |
\begin{itemize} |
| \item $a= a_+ - a_-$ |
\item $a= a_+ - a_-$ |
| \item $a_+a_- = a_-a_+ = 0$ |
\item $a_+a_- = a_-a_+ = 0$ |
| \item Both $a_+$ and $a_-$ belong to $C^*$-subalgebra generated by $a$. |
\item Both $a_+$ and $a_-$ belong to $C^*$-subalgebra generated by $a$. |
| \item $\|a\| = \max\{\|a_+\|, \|a_-\|\}$ |
\item $\|a\| = \max\{\|a_+\|, \|a_-\|\}$ |
| \end{itemize} |
\end{itemize} |
|
|
| $\,$ |
$\,$ |
|
|
| {\bf Remark - } As a particular case, the result provides a decomposition of each self-adjoint operator $T$ on a Hilbert space as a difference of two positive operators $T=T_+ - T_-$ such that $\mathrm{Ran}\; T_- \subseteq \mathrm{Ker}\; T_+$ and $\mathrm{Ran}\; T_+ \subseteq \mathrm{Ker}\; T_-$, where $\mathrm{Ran}\;$ and $\mathrm{Ker}\;$ denote, respectively, the range and kernel of an operator. |
{\bf Remark - } As a particular case, the result provides a decomposition of each self-adjoint operator $T$ on a Hilbert space as a difference of two positive operators $T=T_+ - T_-$ such that $\mathrm{Ran}\; T_- \subseteq \mathrm{Ker}\; T_+$ and $\mathrm{Ran}\; T_+ \subseteq \mathrm{Ker}\; T_-$, where $\mathrm{Ran}\;$ and $\mathrm{Ker}\;$ denote, respectively, the range and kernel of an operator. |
|
|
| $\,$ |
$\,$ |
|
|
| {\bf \emph{Proof:}} (...) |
{\bf \emph{Proof:}} (...) |
