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Property $$(M)$$, $$M$$-ideals, and almost isometric structure of Banach spaces. (English) Zbl 0823.46018
In a paper from 1993 the first-named author settled the problem how to characterize spaces such that the compact operators are an M-ideal [Ill. J. Math. 37, No. 1, 147-169 (1993)], a problem which was open for many years. An essential part in this description played the following definitions: A Banach space $$X$$ is said to satisfy property (M) (resp. (M$$^*$$)) if $$\limsup\| u+ x_ n\|= \limsup\| v+ x_ n\|$$ (resp. $$\limsup\| u^*+ x^*_ n\|= \limsup\| v^*+ x^*_ n\|$$) whenever $$\| u\|= \| v\|$$ and $$(x_ n)$$ tends to zero weakly (resp. $$\| u^*\|= \| v^*\|$$ and $$(x^*_ n)$$ tends to zero with respect to the weak$$^*$$-topology).
Here these properties, some variants and a number of important consequences are thoroughly discussed. Let’s mention a few of these results:
– (M) and (M$$^*$$) are equivalent if $$X$$ does not contain $$\ell^ 1$$.
– Kalton’s result from 1993 can be refined: The compact operators on a separable space $$X$$ are an M-ideal in all bounded operators iff $$X$$ has (M), $$X$$ contains no copy of $$\ell^ 1$$ and $$X$$ has the metric approximation property.
– For a closed infinite-dimensional subspace $$X$$ of $$L_ p$$ (where $$1\leq p\leq \infty$$, $$p\neq 2$$) the unit ball is compact with respect to the $$L^ 1$$-norm iff $$X$$ is isomorphic with arbitrarily small constants to subspaces of $$\ell^ p$$.
– Similar results hold for subspaces of the Schatten classes.
In the end an operator version of (M) is also given, and the paper closes with some open problems.

MSC:
 46B20 Geometry and structure of normed linear spaces 46B04 Isometric theory of Banach spaces
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