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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.

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up vote 5 down vote accepted

Yes it is true. As mentioned in the article you linked to in the Remark on page 4, every infinite dimensional C*-algebra contains a self-adjoint element with infinite spectrum. (In context it helps to know that every C*-algebra is semi-simple.) A reference is given to Ogasawara's "Finite dimensionality of certain Banach algebras," which doesn't appear to be readily available online. (I do not have a more easily accessible reference or proof off hand.) Note that the only compact discrete subsets of $\mathbb C$ are the finite subsets, so your two questions are the same.

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Without normality this seems to be due to Irving Kaplansky, Ring isomorphisms of Banach algebras, Canad. J. Math. 6 (1954), 374-381, Lemma 7 on page 376. See also Alan W. Tullo, Conditions on Banach algebras which imply finite dimensionality, Proceedings of the Edinburgh Mathematical Society (Series 2) (1976), 20, 1-5, lemma on page 4. The remark after the lemma states that Ogasawara's result follows from that. – t.b. Nov 19 '11 at 8:39
Ah, thank you. $ $ – Johan Nov 20 '11 at 10:47

The operators on an infinite-dimensional Hilbert space of the form $\alpha I + K$ where $K$ is a compact operator form an infinite-dimensional $C^*$-algebra in which every element has discrete spectrum.

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Maybe I misspoke but by discrete spectrum I meant that all points in the spectrum are isolated. That is not true for a general compact operator. – Johan Nov 18 '11 at 11:54

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