Direct sums of renormings of \(\ell _{1}\) and the fixed point property (Q984030)

We say that \((X,\|\cdot\|)\) has the fixed point property if for all nonempty closed bounded convex subsets \(C\) of \(X\) and for all nonexpansive maps \(T: C\to C\), \(T\) has a fixed point in \(C\). Let \(\{\gamma_n\}_{n\geq 1}\) be a strictly increasing sequence in \((0, 1)\) that converges to \(1\), and let \(|||\cdot|||\) be the equivalent norm on \(l_1\) defined by \[ |||(a_i)|||:= \sup\Biggl\{\gamma_n \sum^\infty_{i=n} |a_i|: n\geq 1\Biggr\}. \] Fix \(m\in\mathbb{N}\) and let \(X= (\sum^m_{i=1}\oplus(l_1,|||\cdot|||))_1\) be the finite \(l_1\)-sum. In the present paper the authors prove the following theorems. Theorem 3. \(X\) is not isometrically isomorphic to any subspace of \((l_1,|||\cdot|||)\). Theorem 4. The space \(X\) has the fixed point property.