Corollary 34.6.3.label Let $A$ be a commutative unital $C^{*}$-algebra and $x \in A$ be normal, then:

1. (1)

   $x$ is self-adjoint if and only if $\sigma_{A}(x) \subset \real$.

2. (2)

   $x$ is unitary if and only if $\sigma_{A}(x) \subset \partial B_{\complex}(0, 1)$.

3. (3)

   $x$ is positive if and only if $\sigma_{A}(x) \subset [0, \infty)$.

4. (4)

   $x$ is a projection if and only if $\sigma_{A}(x) \subset \bracs{0,1}$.