Isomorphic structure of Ces\`aro and Tandori spaces

We investigate the isomorphic structure of the Ces\`aro spaces and their duals, the Tandori spaces. The main result states that the Ces\`aro function space $Ces_{\infty}$ and its sequence counterpart $ces_{\infty}$ are isomorphic, which answers to the question posted in \cite{AM09}. This is rather surprising since $Ces_{\infty}$ has no natural lattice predual similarly as the known Talagrand's example \cite{Ta81}. We prove that neither $ces_{\infty}$ is isomorphic to $l_{\infty}$ nor $Ces_{\infty}$ is isomorphic to the Tandori space $\widetilde{L_1}$ with the norm $\|f\|_{\widetilde{L_1}}= \|\widetilde{f}\|_{L_1},$ where $\widetilde{f}(t):= \esssup_{s \geq t} |f(s)|.$ Our investigation involves also an examination of the Schur and Dunford-Pettis properties of Ces\`aro and Tandori spaces. In particular, using Bourgain's results we show that a wide class of Ces{\`a}ro-Marcinkiewicz and Ces{\`a}ro-Lorentz spaces have the latter property.