Convergence structures and Hausdorff uo-Lebesgue topologies on vector lattice algebras of operators
Yang Deng, Marcel de Jeu
备注：
33 pages. Some typos corrected. The use of a result on f-algebras has led to stronger versions of Proposition 2.6, Corollary 2.7, and Theorem 6.2. Final version, to appear in Positivity
A vector sublattice of the order bounded operators on a Dedekind complete vector lattice can be supplied with the convergence structures of order convergence, strong order convergence, unbounded order convergence, strong unbounded order convergence, and, when applicable, convergence with respect to a Hausdorff uo-Lebesgue topology and strong convergence with respect to such a topology. We determine the general validity of the implications between these six convergences on the order bounded operator and on the orthomorphisms. Furthermore, the continuity of left and right multiplications with respect to these convergence structures on the order bounded operators, on the order continuous operators, and on the orthomorphisms is investigated, as is their simultaneous continuity. A number of results are included on the equality of adherences of vector sublattices of the order bounded operators and of the orthomorphisms with respect to these convergence structures. These are consequences of more general results for vector sublattices of arbitrary Dedekind complete vector lattices. The special attention that is paid to vector sublattices of the orthomorphisms is motivated by explaining their relevance for representation theory on vector lattices.
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Aliprantis's questions on locally solid topologies
Universidad Autónoma de Madrid（马德里自治大学）；ETH Zürich（苏黎世联邦理工学院）；Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas（西班牙高等科学研究理事会数学科学研究所）
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