Simple proof that $x^*x$ is positive in a commutative $C^*$-Algebra

Question

Is there any simple proof (one that does not use continuous functional calculus) for the statement that $\sigma(x^*x) \subseteq [0,\infty)$ for any $x \in \mathcal{A}$ where $\mathcal{A}$ is a commutative $C^*$-Algebra?

Answer 1

Alright, I should have spend more time before asking. Yes there is such a proof: We have $\sigma(y) = \{\phi(y) \mid \phi \in \mathcal{A}'\}$ where $\mathcal{A}'$ is the set of all multiplicative linear functions $\mathcal{A} \to \mathbb{C}$. Such functions are automatically continuous and *-preserving. But then $\phi(x^*x) = |\phi(x)|^2 \geq 0$.