Operator algebras with contractive approximate identities: A large operator algebra in c0

摘要

We exhibit a singly generated, semisimple commutative operator algebra with acontractive approximate identity, such that the spectrum of the generator is anull sequence and zero, but the algebra is not the closed linear span of theidempotents associated with the null sequence and obtained from the analyticfunctional calculus. Moreover the multiplication on the algebra is neithercompact nor weakly compact. Thus we construct a `large' operator algebra oforthogonal idempotents, which may be viewed as a dense subalgebra of c0.