arXiv subjects
Publications and source records attributed to A. G. Kusraev.
The aim of the present article is to extend the Stone--Weierstrass theorem to functions ranging in a lattice normed space and order rather than topological approximation. We proceed with the machinery of Boolean valued transfer from lattice normed space to normed space.
The paper deals with the study of Banach spaces whose duals are injective Banach lattices. Davies in 1967 proved that an ordered Banach space is an $L^1$-predual space if and only if it is a simplex space. In 2007 Duan and Lin proved that a real Banach space is an $L^1$-predual space if and only if its every four-point subset is centerable. We prove the counterparts of these remarkable results for injectives by the new machinery of Boolean valued transfer from $L^1$-spaces to injective Banach lattices.
The paper contains two main results that are obtained by Boolean valued analysis. The first asserts that a universally complete vector lattice without locally one-dimensional bands can be decomposed into a direct sum of two vector sublattices that are laterally complete and invariant under all band projections and there exists a band preserving linear isomorphism of each of these sublattices to the original lattice. The second result establishes a counterpart of the Ando Theorem on the joint characterization of $A\!L^p$ and $c_0$ for the class of cyclic Banach lattices, using the Boolean valued transfer for injective Banach lattices.
This is a survey of some recent applications of Boolean valued models of set theory to order bounded operators in vector lattices.
This is an overview of the quasidifferential calculus for the mappings that arrive at Kantorovich spaces. The necessary optimality conditions are also derived for multiple criteria optimization problems with quasidifferentiable data.
This is a survey of some recent applications of Boolean valued analysis to operator theory and harmonic analysis. Under consideration are pseudoembedding operators, the noncommutative Wickstead problem, the Radon-Nikodym Theorem for JB-algebras, and the Bochner Theorem for lattice-valued positive definite mappings on locally compact groups.
In 1977 Anthony Wickstead raised the question of the conditions for all band preserving linear operators to be order bounded in a vector lattice. This article overviews the main ideas and results on the Wickstead problem and its variations, focusing primarily on the case of band preserving operators in a universally complete vector lattice.
This is an overview of the recent results of interaction of Boolean valued analysis and vector lattice theory.