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Jul 3, 2017 at 22:18 comment added Bert Lindenhovius You are right. I edited my post, but as it is now it is actually quite irrelevant, since his algebras must be the commutative scattered C*-algebras.
Jul 3, 2017 at 21:45 history edited Bert Lindenhovius CC BY-SA 3.0
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Jul 3, 2017 at 17:48 comment added Christian Remling $K(H)$ doesn't work as an example: $\left( \begin{smallmatrix} 0 & 1 \\ 0 & 0\end{smallmatrix} \right)$ (for $H=\mathbb C^2$) is an element with connected spectrum. (I think Ali points out a general version of this in his answer.)
Jul 3, 2017 at 15:08 history answered Bert Lindenhovius CC BY-SA 3.0