Embedding uniformly convex spaces into spaces with very few operators (Q665509)

It is shown that every separable reflexive Banach space \(X\) with Szlenk index \(\omega_0\) embeds into a Banach space \(Z\) with the property that every operator \(T\) on \(Z\) has the form \(T = \lambda \text{Id} + K\), where \(K\) is compact. This is the case in particular if \(X\) is separable and uniformly convex. To prove this the authors combine two different modifications of the Bourgain-Delbaen construction, producing \({\mathcal L}_\infty\) spaces whose duals are isomorphic to \(\ell_1\). The first one is that used by \textit{S. Argyros} and \textit{R. Haydon} [``A hereditarily indecomposable \({\mathcal L}_{\infty}\)-space that solves the scalar-plus-compact problem'', Acta Math. 206, No. 1, 1--54 (2011; Zbl 1223.46007)] to solve the ``scalar plus compact problem'' and produces a hereditarily indecomposable Banach space. The second one is that used by \textit{D. Freeman, E. Odell} and \textit{Th. Schlumprecht} [``The universality of \(\ell _{1}\) as a dual space'', Math. Ann. 351, No. 1, 149--186 (2011; Zbl 1236.46010)] to show that every Banach space with separable dual embeds into an \({\mathcal L}_\infty\) space with a shrinking basis and with dual isomorphic to \(\ell_1\). Reflexivity of \(X\) intervenes as follows: the construction shows that for any operator \(T\) on \(Z\), there exists a scalar \(\lambda\) such that \(S = T - \lambda \text{Id}\) factors through \(X\). Since \(X\) is reflexive, \(S\) is weakly compact. It is therefore compact since \(Z^\ast\), being isomorphic to \(\ell_1\), has the Schur property. That \(X\) has Szlenk index \(\omega_0\) is used to dominate block sequences of \(X\) by the unit vector basis of the mixed Tsirelson space used in the construction (see Theorem A of \textit{E. Odell}, \textit{Th. Schlumprecht} and \textit{A. Zsák} [``Banach spaces of bounded Szlenk index'', Stud. Math. 183, No.~1, 63--97 (2007; Zbl 1138.46005)]).