On latticially nonatomic subdifferentials
In [Interuniv. thematic Work Collect., No. 2, Yaroslavl 1978, 132-147 (1978; Zbl 0455.46025)] \textit{G. Ya. Lozanovskij} characterized Banach \(K_\sigma\)-spaces \((E,|\cdot|)\) whose conjugate spaces \(E^*\) are nonatomic. In [Indag. Math., New Ser. 1, No. 3, 391-395 (1990; Zbl 0731.46008)] \textit{B. de Pagter} and \textit{W. Wnuk} extended the Lozanovskij result to the case of Banach lattices. In the present article we study operators that act from a vector lattice into an arbitrary \(K\)-space and establish criteria for latticial nonatomicity of a subdifferential or, in other words, the absence of nonzero lattice homomorphisms in the subdifferential under study. With the help of the criteria, we characterize a Banach-Kantorovich lattice whose conjugate space is latticially nonatomic. The obtained results generalize the criteria of Lozanovskij and de Pagter and Wnuk for nonatomicity of the conjugate of a Banach lattice.
operators that act from a vector lattice into an arbitrary \(K\)-space
latticial nonatomicity of a subdifferential
absence of nonzero lattice homomorphisms
scientific article; zbMATH DE number 917097