Amenable Banach algebras with discrete spectrum

The spectrum of an amenable commutative Banach algebra A is known to be uniformly discrete under the norm topology and discrete under the weak topology. Here we are interested in amenable algebras whose spectrum is discrete under the weak* topology; such will be the case, for example, whenever A is an ideal in A**. We consider various results for such algebras, including their relation to the commutative amenable radical algebra problem and indicate some of the open questions in the area.