**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
**Available:** Pr-A-98-05.ps

**Contact**: Oja, Eve; Faculty of Mathematics, Tartu University, Vanemuise 46, EE2400 Tartu, Estonia (eveoja@math.ut.ee)

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