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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.

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