The fixed point property and the Opial condition on tree-like Banach spaces (Q888119)

Let \({\mathcal D}=\bigcup^\infty_{n=0} \{0,1\}^n\) be the standard dyadic tree. The space \(X\) is the completion of \(c_{00}({\mathcal D})\) with respect to the norm \[ \| x\|=\sup\Biggl[\sum^r_{i=1} |{\mathcal I}^*_i(x)|^2\Biggr]^{{1\over 2}}, \] where the supremum is taken over all finite admissible families \(\{{\mathcal I}_i\}^r_{i=1}\), the pairwise disjoint segments. The Banach space \(X\) is separable with nonseparable dual and it does not contain a subspace isomorphic to \(l_1\). The Banach space \(Y\) is the completion of \(c_{00}({\mathcal D})\) with respect to the norm \[ \| x\|=\sup \Biggl(\sum_{{\mathcal I}\in{\mathcal S}}|{\mathcal I}^*(x)|^2\Biggr)^{{1\over 2}}, \] where the supremum is taken over all finite admissible (in a different sense) families \({\mathcal S}\) of pairwise disjoint segments. The space \(Y\) is separable with nonseparable dual and it does not contain an isomorphic copy of \(l_1\). In the present paper, the author proves that each of the Banach spaces \(X\) and \(Y\) satisfies the following: (1) the space has the fixed point property (Theorem 4.1, 6.2), (2) the space does not satisfy the Opial condition (Theorem 5.6, 6.5). In addition, the space \(X\) contains subspaces (having the fixed point property) isomorphic to \(c_0\), whose Banach-Mazur distance from \(c_0\) becomes arbitrarily large (Corollary 5.2).