Dual fixed point property on spaces of continuous functions under equivalent norms (Q2320180)

A Banach space \(X\) has the fixed point property if, for every nonempty closed bounded convex subset \(C\) of \(X\), every nonexpansive mapping \(T:C\to C\) has a fixed point. A Banach space \(X\) is said to have the dual fixed point property if its dual space \(X^*\) has the fixed point property. If \(K\) is a locally compact Hausdorff space and \(C_0(K)\) is the Banach space of continuous functions on \(K\) vanishing at infinity endowed with the supremum norm \(\|\cdot\|_\infty\), \(C_0(K)\) fails to have the dual fixed point property. In the article under review, the author characterizes the locally compact Hausdorff spaces \(K\) for which \(C_0(K)\) can be renormed to have the dual fixed point property. In particular, the author proves that there exists a norm \(|\cdot|_K\) equivalent to \(\|\cdot\|_\infty\) on \(C_0(K)\) such that \((C_0(K), |\cdot|_K)\) has the dual fixed point property if and only if \(K\) is countable. In this case, for every \(\varepsilon>0\), such a norm can be found so that the Banach-Mazur distance \(d( (C_0(K), |\cdot|_K), (C_0(K), \|\cdot\|_\infty)) <1+ \varepsilon\). The author also considers the stability of the dual fixed point property in the following sense: If \(X\) is a Banach space with the dual fixed point property, \(X\) has stability of the dual fixed point property if there exists a constant \(d>1\) such that \(Y\) has the dual fixed point property whenever \(d(X, Y) < d\). The author proves that Banach spaces containing isomorphic copies of \(c_0\) or \(\ell_1\) and Banach spaces with the Dunford-Pettis property fail to have the stability of the dual fixed point property.