Decomposition of self adjoint elements by positive elements

Question

Let $a \in A$ be a self adjoint element of a $C^*$ algebra.

There exists positive elements $a_+, a_-$, such that

$$a=a_+ - a_{-} $$ $$a_+a_-=a_-a_+=0$$

Is the statement true?

This is used in page 28, proposition 41.2 where the author decomposes a selfadjoint operator.

Answer 1

Yes, of course. It's just functional calculus via the functions $$ f_+(t)=\begin{cases} t,&\ t\geq0 \\ 0,&\ t<0\end{cases} $$ and $$ f_-(t)=\begin{cases} -t,&\ t<0 \\ 0,&\ t\geq0\end{cases} $$