Double-dual types over the Banach space \(C(K)\) (Q2368370)

The author studies the following concept of type over a Banach space \(E\) introduced by \textit{J.--L.\ Krivine} and \textit{B.~Maurey} [Isr.\ J.\ Math.\ 39, 273--295 (1981; Zbl 0504.46013)]. For any fixed \(x \in E\), a function \(\tau_x: E \to \mathbb R\) is defined by \(\tau_x(y) = \| x+y\|\), \(y \in E\). A function \(\tau: E \to \mathbb R\) is said to be a \textit{type over} \(E\) if it is in the closure of the set \(\{\tau_x: x \in E\}\) in the topology of pointwise convergence. In [J.\ Aust.\ Math.\ Soc.\ 77, No.~1, 17--28 (2004; Zbl 1067.46022)], the author obtained a characterization of types over \(C(K)\) in terms of pairs \((\ell,u)\) of functions on \(K\). The main result of the paper under review (Theorem 1.5) establishes a characterization of types over \(C(K)\) that are double-dual types over \(C(K)\). The characterizing condition is formulated in terms of the pairs \((\ell,u)\) as well.