A net (xα) in a vector lattice X is said to uo-converge to x if |xα−x|∧u o −→ 0 for every u ≥ 0. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several ...

We introduce and study the class of weak almost Dunford-Pettis operators. As an application, we characterize Banach lattices with the weak Dunford-Pettis property. Also, we establish some sufficient conditions for which each weak almost Dunford-Pettis operator is weak Dunford-Pettis. Finally, we derive some interesting results. Keywords—eak almost Dunford-Pettis operator, almost DunfordPettis o...

A Banach space has the weak fixed point property if its dual space has a weak∗ sequentially compact unit ball and the dual space satisfies the weak∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε > 0 such that, for every infinite subset A of the unit sphere of the dual space, A ∪ (−A) fails to be (2 − ε)-separated. In particular, E-convex Banach spaces, a cla...

We will use Garcı́a-Falset and Lloréns Fuster’s paper on the AMC-property to prove that a Banach space X that 1 δ embeds in a subspace Xδ of a Banach space Y with a 1-unconditional basis has the property AMC and thus the weak fixed point property. We will apply this to some results by Cowell and Kalton to prove that every reflexive real Banach space with the property WORTH and its dual have the ...

In this paper, we introduce a new geometric property (UÃ2) and we show that if a separable Banach space has this property, then both X and its dual X∗ have the weak fixed point property. We also prove that a uniformly Gateaux differentiable Banach space has property (UÃ2) and that if X∗ has property (UÃ2), then X has the (UKK)-property. Criteria for Orlicz spaces to have the properties (UA2), (...

A subset B of an algebra subsets a set Ω has property (N) if each B-pointwise bounded sequence the Banach space ba(A) is in ba(A), where real or complex finitely additive measures defined on endowed with variation norm. (G) [(VHS)] for [if sequence] convergence implies its weak convergence. (sN) [(sG) (sVHS)] every increasing covering {Bn:n∈N} contains Bp [(G) (VHS)], and (wN) [(wG) (wVHS)] web...