Banach spaces of operators that are complemented in their biduals (Q2460640)

The authors define the concept of a boundedly weak*-closed operator ideal as follows: A normed operator ideal \([\mathcal A,a]\) is boundedly weak*-closed if all weak*-limits in \(\mathcal A(X,Y^{**})\) of bounded nets \((T_i)_{i\in I}\subseteq \mathcal A(X,Y)\) belong to \(\mathcal A(X,Y^{**})\). The main result of the paper is that a boundedly weak*-closed operator ideal \(\mathcal A(X,Y)\) is complemented in \(\mathcal A(X,Y)^{**}\) if and only if \(Y\) is complemented in \(Y^{**}\). In particular, this is independent of the first space \(X\). Moreover, maximal normed ideals are boundedly weak*-closed, and an example is provided of a boundedly weak*-closed ideal which is not maximal.