Suppose that $A$ is an infinite-dimensional $C^*$-algebra. Is it true that there must be a normal element with non-discrete spectrum? If that is not true must there at least be a normal element with infinite spectrum?
Edit: To clarify, by discrete spectrum I mean that the spectrum consists only of isolated points. Since the spectrum is closed this is the same as saying that it has no accumulation points.
I have also found, via MathOverflow, the following article which claims that there if $A$ is semi-simple there exists a self-adjoint element in $A$ with infinite spectrum.