On Inessential Elements in Different Banach Algebras

Abstract

If B is a subalgebra of a Banach algebra A we consider the problem of determining conditions that ensure that an element of B that exhibits a property in B also exhibits this property when viewed as an element of A. Further, we consider the converse problem whereby an element of B exhibiting a property in A also exhibits this property in B. We extend the analysis of this problem in the setting of inessential and quasi inessential elements. Secondly, we characterise positive quasi inessential elements in the setting of ordered Banach algebras. The quasi inessential elements are the analogues of quasi compact operators.