The dual of the James tree space is asymptotically uniformly convex (Q2759169)

Let \(X\) denotes an infinite-dimensional real Banach space, \(X^*\) its dual space, \(B(X)\) its closed unit ball, \(S(X)\) its unit sphere, \([Y]\) the closed linear space of the subspace \(Y\) of \(X\) and \(n(X)= \{[x^*_i]^\perp\), \(1\leq i\leq n\), \(x^*_i\in X^*\) and \(n\in\mathbb{N}\}\) the collection of norm-closed finite-codimensional subspaces of \(X\). The modulues of asymptotic conversity \(\overline\delta_X:[0,1]\to [0,1]\) of \(X\) is defined as \(\overline\delta_X(\varepsilon)= \inf_{x\in S(X)} \sup_{y\in n(X)} \inf_{y\in S(Y)} [\|x+\varepsilon y\|- 1]\) and \(X\) is called asymptotically uniformly convex (AUC) iff \(\overline\delta_X(\varepsilon)> 0\) for each \(\varepsilon\in (0,1]\). The space \(X\) has the Radon-Nikodým property (RNP) if each bounded subset of \(X\) has non-empty slices of arbitrary small diameter. The space \(X\) has the point of continuity property (PCP) if each bounded subset of \(X\) has non-empty relatively weakly open subsets of arbitrary small diameter.NEWLINENEWLINENEWLINEThe main result of the paper (Theorem 5) shows that the dual \(JT^*\) of the James tree space \(JT\) is asymptotically uniformly convex \textit{W. B. Johnson}, \textit{J. Lindenstrauss}, \textit{D. Preiss} and \textit{G. Schechtman} [``Almost Fréchet differentiability of Lipschitz mapping between infinite-dimensional Banach spaces'' (Preprint)] showed that AUC space has PCP and asked wether AUC space has RNP. It is well known that the predual \(JT^*\) and \(JT^*\) both have the PCP yet fails the RNP. It follows from Theorem 5 of this paper that \(JT^*\) is an AUC space without the RNP. Thus \(JT_*\) is a separable AUC space without the RNP. Perhaps, these are the first known examples of AUC spaces without the RNP.