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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?

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    $\begingroup$ That does not mean anythings: $A^*$ is a Banach space space so it is always connected, Hausdorff and non compact... so there is no reason that $A$ cannot have a non zero projection. $\endgroup$
    – Simon Henry
    Commented Mar 24, 2016 at 14:31
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    $\begingroup$ $A^*$ or $\hat{A}$ is the spectrum of $A$ $\endgroup$
    – M.fouladi
    Commented Mar 24, 2016 at 16:53
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    $\begingroup$ Dual space and spectrum are not the same things ! but ok now that you have edited the question makes sense. $\endgroup$
    – Simon Henry
    Commented Mar 24, 2016 at 17:26

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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.

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