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Spaces with the Namioka-Phelps property have trivial \(L\)-structure. (English) Zbl 0818.46021

We show that an infinite-dimensional Banach space \(X\) with the Namioka- Phelps property does not have any non-trivial \(L\)-summands and we give some applications of this to \(M\)-structure theory. In particular we show that for a reflexive Banach space \(X\), if the space of compact operators forms an \(M\)-ideal in the space of bounded operators then it is the minimal proper \(M\)-ideal.

MSC:

46B20 Geometry and structure of normed linear spaces
47L05 Linear spaces of operators
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