I have recently started reading the subject of $C^{*} $ - Algebra, But I have several questions in this field.
let $A$ be a $C^{*} $ - algebra and $a , b \in A $ are hermitian , $ ab = ba $.
I would like to know if:
1: $ f : \mathbb{R} \longrightarrow \mathbb{R}$ is a continuous function, and $ a \leq b $ , then $ f ( a ) \leq f ( b )$.
2: $ f : \mathbb{R} \longrightarrow \mathbb{R}$ is a real function , then $ a f( b) = b f( a ) $
thank you.
These aren't even true in the case $A = \mathbb{C}$.
For 1.) just consider $f(x) = -x$.
For 2.) let $a, b\in\mathbb{R}\subset\mathbb{C}$ with $f(x)$ such that $f(a)=0$ and $f(b)\neq 0$.