Definition 6.4.11 (Jordan Decomposition).label Let $h\in A_{h}$ then the absolute value, positive part and negative part of $h$ are
\begin{align*}\abs{h}\define (h^{2})^{1/2}&&h_{+}\define\frac{1}{2}(\abs{h}+h)&&h_{-}\define\frac{1}{2}\parens{\abs{h}-h}\end{align*}
and the decomposition $h=h_{+}-h_{-}$ is called the Jordan decomposition of $h$. We have $\qspec{A}{\abs{h}},\qspec{A}{h_+},\qspec{A}{h_-}\suf\R_{\geq0}$ and $h_{+},h_{-}$ are characterized as elements in $A_{h}$ such that
\begin{align*}h=h_{+}-h_{-}&&h_{+}h_{-}=0&&\qspec{A}{h_+}\cup\qspec{A}{h_-}\suf \R_{\geq 0}\end{align*}