Strictly Cyclic Functionals, Reflexivity, and Hereditary Reflexivity of Operator Algebras

Abstract

This paper is concerned with strictly cyclic functionals of operator algebras on Banach spaces. It is shown that if X is a reflexive Banach space and $\mathcal{A}$ is a norm-closed semisimple abelian subalgebra of B(X) with a strictly cyclic functional $f\in X^{\ast }$, then $\mathcal{A}$ is reflexive and hereditarily reflexive. Moreover, we construct a semisimple abelian operator algebra having a strictly cyclic functional but having no strictly cyclic vectors. The hereditary reflexivity of an algbra of this type can follow from theorems in this paper, but does not follow directly from the known theorems that, if a strictly cyclic operator algebra on Banach spaces is semisimple and abelian, then it is a hereditarily reflexive algebra.