Am I understanding the reduced group C*-algebras correctly?

Question

Let $G$ be a group and $\mathbb{C}G$ be its group algebra. $\mathbb{C}G$ contains finite linear combinations of group elements of $G$. Let $C^{*}_{r}(G)$ be the reduced group C-algebra. Let $a, b\in \mathbb{C}$, $g\neq h\in G$. If I understand correctly, $||ag+bh||^{2}_{r}\leq|a|^{2}+|b|^{2}$ by triangle inequality, and consider $(ag+bh)$ act on the identity $e$, which equals to $ag+bh$, and the norm square of $(ag+bh)(e)$ is less than or equal to the norm square of $ag+bh$. But norm square of $(ag+bh)(e)$ is just $|a|^{2}+|b|^{2}$, so the norm square of $ag+bh$ is just $|a|^{2}+|b|^{2}$. Is that correct? Am I understand the concept of reduced group C norm correctly?