Zero Sums of Idempotents and Banach Algebras Failing to be Spectrally Regular

A large class of Banach algebras is identified allowing for non-trivial zero sums of idempotents, hence failing to be spectrally regular. Belonging to it are the C*-algebras known under the name Cuntz algebras. Other Banach algebras lying in the class are those of the form L(X) with X a (non-trivial) Banach space isomorphic to a (finite) direct sum of at least two copies of X. There do exist (somewhat exotic) Banach spaces for which L(X) is spectrally regular.