On ideals and generalized centers of finite sets in Banach spaces (Q695146)

Let \(X\) be a Banach space and \(Y\) be a subspace. Then \(Y\) is an ideal in \(X\) if the annihilator of \(Y\), \(Y^\perp\), is the kernel of a norm one projection on the dual \(X^*\). This is a generalization of being norm one complemented. It is known that \(Y\) is an ideal in \(X\) if and only if \(Y\) is an ideal in every closed subspace \(Z\) of \(X\), with \(Y \subset Z \subset X\) and \(\dim Z/Y < \infty\). The question studied in this paper is: When is it enough to consider only those \(Z\) with \(\dim Z/Y = 1\)? The author shows that if the dual of \(X\) is isometric to an \(L_1(\mu)\) space, then a subspace \(Y\) of \(X\) is an ideal in \(X\) if \(Y\) is an ideal in every closed subspace \(Z\) with \(Y \subset Z \subset X\) and \(\dim Z/Y = 1\). Furthermore, \(Y\) has generalized centers of finite sets in this case. Another result is that if a subspace \(Y\) of \(\ell_1\) has the finite intersection property and \(Y\) is an ideal in every closed subspace \(Z\) with \(Y \subset Z \subset \ell_1\) and \(\dim Z/Y = 1\), then \(Y\) is not only an ideal but is also norm one complemented in \(\ell_1\).