Preprint No. A-98-05

Eve Oja

On the geometry of Banach spaces having shrinking approximations of the identity

Abstract: Let $a,c\geq0$ and let $B$ be a compact set of scalars. We introduce property $M^\ast(a,B,c)$ of Banach spaces $X$ by the requirement that $$ \limsup_\nu\| ax_\nu^\ast+bx^\ast+cy^\ast\|\leq\limsup_\nu\| x_\nu^\ast\|\quad\forall b\in B $$ whenever $(x_\nu^\ast)$ is a bounded net converging weak$^\ast$ to $x^\ast$ in $X^\ast$ and $\| y^\ast\|\leq\| x^\ast\|$. Using $M^\ast(a,B,c)$ with $\max|B|+c>1$, we characterize the existence of certain shrinking approximations of the identity (in particular, those related to $M$-, $u$-, and $h$-ideals of compact or approximable operators). We also show that the existence of these approximations of the identity is separably determined.

Mathematics Subject Classification (MSC91): 46B20 Geometry and structure of normed linear spaces

Language: ENG


Contact: Oja, Eve; Faculty of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia (

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