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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Delgado, J.M., Piñeiro, C. Banach spaces of operators that are complemented in their biduals. Acta Math Hung 115, 49–58 (2007). https://doi.org/10.1007/s10474-006-0534-6
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DOI: https://doi.org/10.1007/s10474-006-0534-6