Unbounded approximate identities in algebras of compact operators on Banach spaces

Abstract

A left (right) approximate identity $(e_\alpha)$ for a normed algebra ${\cal A}$ is said to satisfy condition (U) if $e_{\alpha}\circ a({\rm resp.} a\circ e_{\alpha})$ converges to a uniformly for a in compact sets of ${\cal A}$. Let ${\cal A}$ be an operator algebra on a Banach space $X$, containing finite rank operators and being contained in the algebra of compact operators. It is shown that ${\cal A}$ has a left approximate identity satisfying condition (U) if and only if the identity operator on the space $X$ is approximable uniformly on compact sets of $X$ by operators in ${\cal A}$. The algebra ${\cal A}$ has a right approximate identity satisfying condition (U) if and only if the identity operator on the dual space $X^*$ is approximable uniformly on compact sets of $X^*$ by operators which are adjoints of the operators in ${\cal A}$. Moreover, ${\cal A}$ has a two-sided approximate identity satisfying condition (U) if it has a right approximate identity satisfying condition (U).