The fixed point property in \(c_0\) with an equivalent norm
The authors study fixed point properties of the Banach space \(c_0\) endowed with the norm \( \| (x_n)\|_D = \sup_{i,j\in \mathbb{N}} |x_i-x_j|\,. \) The norm \(\|\cdot\|_D\) is equivalent to the usual norm \(\|\cdot\|_\infty\) and, like \((c_0, \|\cdot\|_\infty)\), the space \((c_0, \|\cdot\|_D)\) has the weak fixed point property for nonexpansive mappings. Also, analogous to the situation in \((c_0, \|\cdot\|_\infty)\), every infinite-dimensional subspace of \((c_0, \|\cdot\|_D)\) contains an asymptotically isometric copy of \(c_0\), ensuring that no infinite-dimensional subspace of \((c_0, \|\cdot\|_D)\) has the fixed point property for nonexpansive mappings. However, unlike the situation in \((c_0, \|\cdot\|_\infty)\), there exist closed, bounded, convex subsets of \((c_0, \|\cdot\|_D)\) that are not weakly compact and do not contain asymptotically isometric \(c_0\)-summing basic sequences. The authors also prove that the dual space of \((c_0, \|\cdot\|_D)\) is Bynum's space \(\ell_{1\infty}\) and that every infinite-dimensional subspace of \(\ell_{1\infty}\) contains an asymptotically isometric copy of \(\ell_1\). Consequently, no infinite-dimensional subspace of \(\ell_{1\infty}\) has the fixed point property for nonexpansive mappings.
scientific article; zbMATH DE number 5997809