Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set? Is there an example of this situation such that $A$ has trivial center? Is there an example of this situation such that $A$ is a simple $C^*$ algebra?
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$\begingroup$ Compact in what topology -- do you mean the norm topology inherited from A? $\endgroup$– Yemon ChoiCommented Dec 8, 2020 at 17:48
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$\begingroup$ @YemonChoi Yes I mean the norm topology. $\endgroup$– Ali TaghaviCommented Dec 8, 2020 at 20:11
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1$\begingroup$ I don't know much about functional analysis, but if I'm reading things right, then the trace on a type II factor is a surective map from the set of projections to $[0,\infty]$ which I think has connected fibers. If this map is continuous, that would tell us the set of projections in a type II factor is connected. I suspect that the set of projections is rarely compact in the norm topology though, especially if one is deleting the trivial projections from an apparently-connected set. $\endgroup$– Tim CampionCommented Mar 9, 2021 at 1:01
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