Banach spaces of operators that are complemented in their biduals |
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Authors: | J M Delgado C Piñeiro |
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Institution: | (1) Department of Mathematics, Experimental Sciences Faculty, Campus de El Carmen, 21071 Huelva, Spain |
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Abstract: | Let A, a] be a normed operator ideal. We say that A, a] is boundedly weak*-closed if the following property holds: for all Banach spaces X and Y, if T: X → Y** is an operator such that there exists a bounded net (T
i
)
i∈I
in A(X, Y) satisfying lim
i
〈y*, T
i
x
y*〉 for every x ∈ X and y* ∈ Y*, then T belongs to A(X, Y**). Our main result proves that, when A, a] is a normed operator ideal with that property, A(X, Y) is complemented in its bidual if and only if there exists a continuous projection from Y** onto Y, regardless of the Banach space X. We also have proved that maximal normed operator ideals are boundedly weak*-closed but, in general, both concepts are different.
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Keywords: | normed operator ideal maximal ideal |
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