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

TL;DR

The paper constructs a large, semisimple, commutative operator algebra within c0 that has a contractive approximate identity but is not generated by its idempotents, revealing new structural properties.

Contribution

It introduces a novel example of a large operator algebra with specific spectral and algebraic properties not previously documented.

Findings

Algebra has a null sequence spectrum including zero

The algebra is not generated by its idempotents

Multiplication is neither compact nor weakly compact

Abstract

We exhibit a singly generated, semisimple commutative operator algebra with a contractive approximate identity, such that the spectrum of the generator is a null sequence and zero, but the algebra is not the closed linear span of the idempotents associated with the null sequence and obtained from the analytic functional calculus. Moreover the multiplication on the algebra is neither compact nor weakly compact. Thus we construct a `large' operator algebra of orthogonal idempotents, which may be viewed as a dense subalgebra of c0.