TITLE: When does a $C^*$-algebra have no nonzero projection?
QUESTION [7 upvotes]: 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?

REPLY [6 votes]: Given $x\in A$ and $\alpha>0$ the set of $\pi\in \hat A$
such that $\|\pi(x)\|\geq \alpha$ is compact (Proposition 3.3.7 in Dixmier's ``C*-algebras"). Applied to a non-zero projection $p$, the set
of $\pi\in \hat A$ such that $\pi(p)\neq 0$ is compact (since then $\|\pi(p)\|=1$)
and non-empty. This set is also open (because $\pi\mapsto \|\pi(x)\| $ is l.s.c.; Proposition 3.3.2 in Dixmier's). In the question the spectrum is Hausdorff and connected. So we get a non-empty clopen set which must be the whole space. But the spectrum has also been assumed noncompact.