Lim's condition and differentiability

TL;DR

The paper studies a weaker form of Lim's condition $(\ddag)$ for $C^*$-algebras and $L^1$-predual spaces, linking these geometric properties to differentiability and fixed-point phenomena. It proves that $(\ddag)$ forces a separable $C^*$-algebra to decompose as a $c_0$-direct sum of finite-dimensional operator algebras $\bigoplus_{c_0} \mathcal{L}(H_{\alpha})$, yielding a strongly subdifferentiable norm, and shows $(\ddag)$ is separably commutatively determined. For $L^1$-predual spaces, $(\ddag)$ implies that every unit vector is $k$-smooth, with the corresponding state space $S_x$ being a finite-dimensional simplex; the work also extends these ideas to commutative function spaces $C(\Omega)$, $A(K)$, and related dual structures, highlighting the interplay between geometric properties and differentiability. The results have implications for weak$^*$-normal structure and fixed-point properties for nonexpansive maps, and suggest a structured, $c_0$-type decomposition as a unifying theme across these settings.

Abstract

In their seminal work on quasi-normal structures, Lau and Mah studied weak$^\ast$-normal structure in spaces of operators on a Hilbert space 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 for $C^\ast$-algebras and $L^1$-predual spaces, i.e., Banach spaces whose dual is isometric to $L^1(μ)$ for a positive measure $μ$. In the case of a separable $C^\ast$-algebra ${\mathcal A}$, we show that this condition implies weak$^*$-normal structure and in the general case, the norm on ${\mathcal A}$ is strongly subdifferentiable. In the case of $L^1$-predual spaces, we show that the condition implies $k$-smoothness of the norm.

Theorems & Definitions (39)

• Definition 1
• Definition 2
• Theorem 3
• Definition 4
• Proposition 5
• proof
• Corollary 6
• Example 7
• Theorem 8
• proof