Geometric conditions which imply $w^{\ast }$-normal structure

Abstract

Let $X$ be a complex Banach space. In their seminal work, Lau and Mah (\cite{LM}) study a geometric property called Lim's condition which, for separable $X$, implies that for any non-empty $w^{\ast }$-compact convex set $K\subset X^{\ast }$, any nonexpansive mapping $T:K\to K$ has a fixed point. In this paper, we study a weaker form of Lim's condition, which we call weak-Lim's condition. We prove that if ${\ell }_1/\hookrightarrow X$, then weak-Lim's condition on $X^{\ast }$ implies the fixed point property for non-expansive maps on weakly compact subsets of $X$. In the case of a commutative $C^{\ast }$-algebra, we prove that weak-Lim's condition is equivalent to Lim's condition. We show that Lim's condition characterizes $C^{\ast }$-algebras which are a $c_0$-direct sum of finite-dimensional operator spaces, while weak-Lim's condition characterizes $C^{\ast }$-algebras which are a $c_0$-direct sum of the spaces of compact operators on Hilbert spaces. For a uniform algebra, we extend a result of Lau and Mah to show that weak-Lim's condition implies that the space is finite-dimensional. In the case of an $L^1$-predual space, we show that this condition implies $k$-smoothness of the norm in the sense considered in \cite{LR}.

arXiv commentPrevious version was named as Geometric conditions related to Lim's condition. The new file consists of the discussion from the fixed point property point of view