σ - Quasi Centralizers and Inner Derivations in a Closed Ideal of a Complex Banach Algebra.

Abstract

In this paper we show that, for a closed ideal J of a unital complex Banach algebra A and for a σ -quasi centralizer element a of J in A we have (i) under certain conditions if b is an element in the center of J and π : J → BL( X ) is an irreducible representation of J on the Banach space X , then π(ba) is a scalar operator. (ii) If σA(a) has empty interior and DaJ is the restriction of the inner derivation of a to J then ( DaJ )3 = 0.