APPROXIMATE IDENTITIES IN BANACH-ALGEBRAS OF COMPACT-OPERATORS

Let X be a Banach space and let A be a uniformly closed algebra of compact operators on X, containing the finite rank operators. We set up a general framework to discuss the equivalence between Banach space approximation properties and the existence of right approximate identities in A. The appropriate properties require approximation in the dual X* by operators which are adjoints of operators on X. We show that the existence of a bounded right approximate identity implies that of a bounded left approximate identity. We give examples to show that these properties are not equivalent, however. Finally, we discuss the well known result that, if X* has a basis, then X has a shrinking basis. We make some attempts to generalize this to various bounded approximation properties.