Banach spaces with many boundedly complete basic sequences failing PCP (Q992799)

It is shown that, if \(X\) is a Banach space not containing \(c_0\) with a semi-normalized supershrinking basis, then every nontrivial weak Cauchy sequence has a boundedly complete basic subsequence. The author then applies this to the natural predual \(B_\infty\) of the James tree space \(JT_\infty\) constructed by \textit{N.\,Ghoussoub} and \textit{B.\,Maurey} [J.~Funct.\ Anal.\ 61, 72--97 (1985; Zbl 0565.46011)]. In particular, \(B_\infty\) fails the point of continuity property (PCP), does not contain \(\ell_1\), and every semi-normalized basic sequence in \(B_\infty\) has a boundedly complete subsequence spanning a reflexive or an order one quasireflexive subspace. This last result answers a question of \textit{H.\,Rosenthal} [J.~Funct.\ Anal.\ 253, No.\,2, 772--781 (2007; Zbl 1139.46006)].