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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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