Positive part of an element of $C_0(X)$

Question

I'm currently working on the Isem24 $C^*$-Algebras notes and there's a question I cannot answer.

Let $A$ be a commutative $C^*$-Algebra, let be $x$ and $h$ two self-adjoint, positive elements of $A$ such that $h \geq x$.

Let's denote by $x_+$ and $x_-$ the positive and negative part of $x$.

Show that $h \geq x_+$.

Here's my sketch of proof :

I need to answer this question in $C_0(X)$ where $X$ is a locally compact space, then I'll be able to use the Gelfand isomorphism that preserves order to get the result for A.

My issue is that I'm struggling with solving this inequation in $C_0(X)$. May you give me some hints to find my way to go ?

Thank you.

Answer 1

In $C_0(X)$ is self-adjoint (i.e. real-valued), we have $$f= f^+-f^-$$ with $f^+ = \max\{f,0\}$ and $f^-= \max\{-f,0\}$. Then clearly $f^+ \le h$:

If $x\in X$ with $f(x)\ge 0$, then $f^+(x) = f(x)\le h(x)$. If on the other hand $f(x) \le 0$, then $f^+(x) = 0 \le h(x)$. In any case, $f^+(x)\le h(x)$ for all $x\in X$, i.e. $f^+ \le h$.