Year: 2025
Type: preprint
Abstract: In their seminal work, Lau and Mah (1986) study $w^*$-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 ($\ddagger$), for $C^\ast$-algebras, uniform algebras, and $L^1$-predual spaces. In the case of a $C^\ast$-algebra, we prove that property $(\ddagger)$ is equivalent to Lim's condition and consequently, we obtain a geometric characterization of $C^*$-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 $(\ddagger)$ 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).
Source: arXiv (Cornell University)
Authors: Gothwal, Deepak, Rao, T. S. S. R. K.
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4415157442
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