Some Geometric Aspects Related to Lim's Condition

Abstract

In their seminal work, Lau and Mah (1986) study $w^{\ast }$-normal structure in the space of operators $\mathcal{L}(H)$, on a Hilbert space $H$, using a geometric property of the dual unit ball called Lim's condition. In this paper, we study a weaker form of Lim's condition, which we call property ($‡$), for $C^{\ast }$-algebras, uniform algebras, and $L^1$-predual spaces. In the case of a $C^{\ast }$-algebra, we prove that property $(‡)$ is equivalent to Lim's condition and consequently, we obtain a geometric characterization of $C^{\ast }$-algebras which are $c_0$-direct sum of finite-dimensional operator spaces. For a uniform algebra, we extend a result of Lau and Mah to show that property $(‡)$ 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 Lin and Rao (2007).

Previous version was named as Lim's condition and differentiability. The new file consists of the discussion from a different viewpoint and some new results like the equivalence of property $(‡)$ and Lim's condition in $C^{\ast }$-algebras have been obtained