Hilbert generated Banach spaces need not have a norming Markushevich basis
Weakly compactly generated (for short, WCG) Banach spaces (i.e., spaces with a weakly compact linearly dense subset, e.g., separable spaces, reflexive spaces, \(c_0(\Gamma)\) for every \(\Gamma\), \(L_1(\mu)\) for a \(\sigma\)-finite measure \(\mu\), \(\ldots\)) form a fundamental class in the study of general (nonseparable) Banach spaces. The presence of a kind of ``coordinate system'' (Schauder bases and, more generally, Markushevich (M, for short) bases) throws a strong light on the structure of the space and allows, in particular, for renormings. WCG Asplund spaces are characterized as Banach spaces admitting a shrinking M-basis (i.e., the closed linear span \(F\) of the set of all functional coefficients is the whole dual). If the supremum on the closed unit ball of \(F\) gives an equivalent norm on the space, the M-basis is said to be norming (\(1\)-norming if it gives the original norm). It has been an important open problem for fifty years whether every WCG Banach space admits a norming M-basis. The present paper solves this problem in the negative. Amazingly, the counterexample enjoys a bunch of extra properties: it is a \(C(K)\)-space, where \(K\) ia a zero-dimensional uniform Eberlein compact, so the space is Hilbert generated -- i.e., there exists a bounded linear operator from some \(\ell_2(\Gamma)\) into \(C(K)\) with dense range. The last part of the paper shows that the space \(C(\sigma_1(\Gamma)^{\omega})\) has a \(1\)-norming M-basis (here, \(\sigma_1(\Gamma)\) is the one-point compactification of a discrete set \(\Gamma\), and \(\sigma_1(\Gamma)^{\omega}\) is just the Cartesian product of \(\omega\) copies of this space). This provides a test for checking whether a compact space \(K\) is a continuous image of \(\sigma_1(\Gamma)^{\omega}\). Another consequence is that having a norming M-basis for a Banach space \(C(K)\) does not necessarily pass to \(C(L)\), where \(L\subset K\). The paper is an extraordinary piece of delicate technical work.
scientific article; zbMATH DE number 7078821
scientific article