Central and almost constrained subspaces of Banach spaces (Q2912206)

This paper deals with the notions of central, almost constrained and norm-one complemented subspaces of a real Banach space. The paper consists of 4 sections. In the introduction, the author motivates the notion of almost constrained subspaces, gives a thorough and clear account of the main definitions and reviews important results that triggered the theorems in the remaining sections. In Section 2, the author gives necessary and sufficient conditions for:NEWLINENEWLINE 1. A Banach space to be almost constrained in every super space;NEWLINENEWLINE 2. a subspace of an \(L^1\)-predual space to be a central subspace; andNEWLINENEWLINE3. a complex Banach space of dimension greater or equal to 3 to have the property that every finite dimensional subspace is a central subspace. It is shown that this is only possible in a Hilbert space setting. Hence this gives a new characterization of Hilbert space. In Section 3, the author investigates these three notions for \(c_0\), \(\ell^1\), and \(\ell^{\infty}\) direct sums of Banach spaces and also for certain subspaces of those direct sums. One of the main results of this section reads as follows: ``Theorem 8. Let \(\{X_i\}_{\i \in I}\) be an infinite family of reflexive Banach spaces and \(X= \bigoplus_{c_0} X_i\). Let \(Y\) be a factor reflexive and proximinal subspace. \(Y\) is a central subspace if and only if it is almost constrained.''NEWLINENEWLINESection 4 starts with the notion of an ideal introduced in [\textit{G. Godefroy, N. J. Kalton} and \textit{P. D. Saphar}, Stud. Math. 104, No. 1, 13--59 (1993; Zbl 0814.46012)] and developed by the author in [Rocky Mt. J. Math. 31, No. 2, 595--609 (2001; Zbl 0988.46012)]. Most of the results in this section explore the geometric structure of \(M\)-ideals and how it relates to the concepts investigated in this paper. The author also gives an example showing that \(M\)-ideals may not be central subspaces.NEWLINENEWLINEWe now quote the last two results.NEWLINENEWLINEProposition 27. Let \(Y \subset \ell^1\) be a finite co-dimensional subspace that is an intersection of hyperplanes that are ranges of projections of norm one. Then \(Y\) is the range of a projection of norm one.NEWLINENEWLINETheorem 28. Let \(\Omega\) be an extremely disconnected compact Hausdorff space. Let \(Y \subset X\) be a finite dimensional central subspace. Then \(C(\Omega, Y)\) is an almost constrained subspace of \(C(\Omega, X).\)NEWLINENEWLINEThe proofs are clear and very interesting.