Retracted Journal Nonlinear Analysis and Application 2014 (2014) 1-6 Available online at www.ispacs.com/jnaa Volume 2014, Year 2014 Article ID jnaa-00257, 6 Pages doi:10.5899/2014/jnaa-00257 Research Article Weak* almost Dunford-Pettis operators in Banach lattices H. Ardakani 1 ∗ , M. Salimi 1 , S. M. S. Modarres Mosadegh 1 (1) Department of Mathematics, University of Yazd, Yazd, IRAN. Copyright 2014 c ⃝ H. Ardakani, M. Salimi and S. M. S. Modarres Mosadegh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We introduce and study the class of weak* almost Dunford-Pettis operators. As an application, we characterize Ba- nach lattices with the weak DP* property. Also, we establish some sufficient conditions under which the class of order bounded weak* almost Dunford-Pettis operator coincide with that of almost limited. Finally, we derive some interesting results. Keywords: Weak* Dunford- Pettis operator, Weak* almost Dunford- Pettis operator, Almost limited operator, Almost limited set, Weak Dp* property. 1 Introduction Throughout this paper E and F will denote real Banach lattices. B E is the closed unit ball of E and sol (A) denotes the solid hull of a subset A of a Banach lattice. We will use the term operator T : E → F between two Banach spaces to mean a bounded linear mapping. It is positive if T (x) ≥ 0 in F whenever x ≥ 0 in E. The objective of this paper is to study the class of weak* almost Dunford-Pettis operators. Also, we derive the following interesting consequences: 1. some characterizations of the class of weak* almost Dunford-Pettis operators; 2. some characterizations of the weak DP* property; 3. the coincidence of this class of operators with that of almost limited operators; 4. the domination property of the class of weak* almost Dunford-Pettis operators. To state our results, we need to fix some notation and recall some defnitions. A Riesz space (or a vector lattice) is an ordered vector space E with the additional property that for each pair of vectors x, y ∈ E the supremum and the infimum of the set {x, y} both exist in E. Following the classical notation, we shall write x ∨ y := sup{x, y} and x ∧ y := in f {x, y}. A Banach lattice E is a Banach space (E , ∥.∥) such that E is a vector lattice and its norm satisfies the following property: For each x, y ∈ E such that |x|≤|y|, we have ∥x∥≤∥y∥. If E is a Banach lattice, its topological dual E ′ , endowed with the dual norm, is also a Banach lattice. A norm ∥.∥ of a Banach lattice E is order continuous if for each generalized nets (x α ) such that (x α ) ↓ 0 in E , (x α ) converges to 0 for the norm ∥.∥ where the notation (x α ) ↓ 0 means that the (x α ) is decreasing, its infimum exists and in f (x α )= 0. A Riesz space is said to be σ -Dedekind complete if every countable subset that is bounded above has a supremum (or, equivalently, whenever 0 ≤ x n ↑≤ x implies the existence of sup{x n }).The lattice operations in a Banach lattice E are weakly sequentially cotinuous if for every weakly null sequence (x n ) in E , |x n |→ 0 for σ (E , E ′ ).The lattice operations in a Banach lattice E ′ are weak* sequentially cotinuous if for every weak* null sequence ( f n ) in E ′ , | f n |→ 0 ∗ Corresponding author. Email address: halimeh ardakani@yahoo.com