MR2595826 (2011c:46026) Domínguez Benavides, T. The Szlenk index and the fixed point property under renorming. Fixed Point Theory Appl. 2010, Art. ID 268270, 9 pp. (Reviewer: Diana Caponetti)

Abstract

It is known that not every Banach space can be renormed so that the resultant space satisfies the weak Fixed Point Property (w-FPP). In the paper under review the author gives a further contribution to identify classes of Banach spaces which can be renormed to satisfy the w-FPP. Let $X$ be a Banach space and $X^*$ its dual. The dual norm is $UKK^*$ if for every $\varepsilon >0$ there is $\theta(\varepsilon)>0$ such that every $u$ in the closed unit ball $B_{X^*}$ of $X^*$ with $\|u\| > 1 - \theta(\varepsilon)$ has a weak$^*$ open neighborhood $\mathcal{U}$ with diam$(B_{X^*}\cap\mathcal{U})< \epsilon$. In [Bull. Lond. Math. Soc. 42 (2010), no. 2, 221--228; MR2601548] M. Raya showed that if $X$ is an Asplund space and the Szlenk index $S_z(X) \le \omega$, where $\omega$ denotes the first ordinal number, then there is an equivalent norm on $X$ such that the dual norm on $X^*$ is $UKK^*$. In the paper under review it is proved that whenever $X$ is endowed with this norm, then $R(X) <2$, where $R(X)= \sup \{ \lim \inf \|x_n +x\| : x_n \ \mbox{is weakly null with } \ \|x_n\| \le 1, \|x\|=1 \}$ is the Garc\'ia-Falset“'s coefficient. Since the author and S. Phothi in [Nonlinear Anal. 72 (2010), no. 3-4, 1409-1416; MR2577541] proved that when $X$ is a Banach space which can be continuously embedded in a Banach space $Y$ with $R(Y) <2$, then $X$ can be renormed to satisfy the w-FPP, the results about the Szlenk index lead to the main result of the paper: Let $Y$ be a Banach space with $S_z(Y) \le \omega$, then any Banach space $X$ which can be continuously embedded in $Y$ can be renormed to satisfy the w-FPP. The result applies to Banach spaces which can be continuously embedded in $C(K)$, where $K$ is a scattered compact topological space such that the $\omega$th-derived set $K^{(\omega)}= \emptyset$. In the paper the author also proves that if $(X, \| \cdot\|)$ is a Banach space and $\mathcal{D}$ is the space of all norms in $X$ equivalent to the given one endowed with the metric $ \rho(p,q)= \sup \{ |p(x)-q(x)| \}$, where the supremum is taken over all $x$ in the closed unit ball of $X$ and $S_z(X) \le \omega$, then for almost all norms (in the sense of porosity) in $\mathcal{D}$, $X$ satisfies the w-FPP.