Definition 34.8.7 (Positive and Negative Parts).label Let $A$ be a unital $C^{*}$-algebra and $x \in A$ be self-adjoint, then there exists unique positive elements $x^{+}, x^{-} \in A$ such that
- (1)
$x = x^{+} - x^{-}$.
- (2)
$x^{+}x^{-} = x^{-}x^{+} = 0$.
The pair $(x^{+}, x^{-})$ are the positive and negative parts of $x$.
Proof. Since $x$ is self-adjoint, $\sigma_{A}(x) \subset \real$ by Proposition 34.4.6. Using the continuous functional calculus, existence is given by the functions $f^{+}(\lambda) = \lambda \vee 0$ and $f^{-}(\lambda) = \lambda \wedge 0$ and Proposition 34.8.3.
On the other hand, for each $p \in \real[z]$ with $p(0) = 0$, (2) implies that $p(x) = p(x^{+}) + p(-x^{-})$. By the Stone-Weierstrass Theorem, $f(x) = f(x^{+}) + f(-x^{-})$ for all $f \in C(\real; \real)$ with $f(0) = 0$. In particular, (1) then implies that $f^{+}(x) = f+(x^{+}) + f^{+}(-x^{-}) = f^{+}(x^{+}) = x^{+}$, and likewise $f^{-}(x) = x^{-}$. Therefore the decomposition is given uniquely by the continuous functional calculus.$\square$
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