Let $A$ be a $C^*$-algebra that $A^*$ its dual space (spectrum ) of $A$, and $A^*$ is a non-compact connected Hausdorff space. Why $A$ cannot contain a nonzero projection?

Let $A$ be a $C^*$-algebra and $\hat{A}$ its spectrum of $A$,the set of classes of non-zero irreducible representation of $A$ endowed with hull-kernel topology. suppose $\hat{A}$ is a non-compact connected Hausdorff space.

Why $A$ cannot contain a nonzero projection?

Let $A$ be a $C^*$-algebra that $A^*$ its dual space, and $A^*$ is a non-compact connected Hausdorff space. Why $A$ cannot contain a nonzero projection?

Let $A$ be a $C^*$-algebra that $A^*$ its dual space (spectrum ) of $A$, and $A^*$ is a non-compact connected Hausdorff space. Why $A$ cannot contain a nonzero projection?