Abstract. We introduce and study the class of almost limited sets in Banach lattices, that is, sets on which every disjoint weak * null sequence of functionals converges uniformly to zero. It is established that a Banach lattice has order continuous norm if and only if almost limited sets and L -weakly compact sets coincide. In particular, in terms of almost Dunford-Pettis operators into c0, we give an operator characterization of those σ-Dedekind complete Banach lattices whose relatively weakly compact sets are almost limited, that is, for a σ-Dedekind Banach lattice E, every relatively weakly compact set in E is almost limited if and only if every continuous linear operator T : E → c 0 is an almost Dunford-Pettis operator.

Let E and F be vector lattices and L r (E, F ) the ordered space of all regular operators, which turns out to be a (Dedekind complete) vector lattice if F is Dedekind complete. We show that every lattice isomorphism from E onto F is a finite element in L r (E, F ), and that if E is an AL-space and F is a Dedekind complete AM -space with an order unit, then each regular operator is a finite element in L r (E, F ). We also investigate the finiteness of finite rank operators in Banach lattices. In particular, we give necessary and sufficient conditions for rank one operators to be finite elements in the vector lattice L r (E, F ). Mathematics Subject Classification (2000). 46B42, 47B07, 47B65.

LetAandBbe ordered algebras overℝ, whereAhas a generating positive cone andBsatisfies the property thatb2>0if0≠b∈B. We give some conditions for a mapT:A→Bwhich is supra-additive and supra-multiplicative for all positive and negative elements to be linear and multiplicative; that is,Tis a homomorphism of algebras. Our results generalize some known results on supra-additive and supra-multiplicative maps between spaces of real functions.

Abstract. Let X and Y be compact Hausdorff spaces, and E, F be Banach lattices. Let C(X, E) denote the Banach lattice of all continuous E-valued functions on X equipped with the pointwise ordering and the sup norm. We prove that if there exists a Riesz isomorphism Φ :that Φf is non-vanishing on Y if and only if f is non-vanishing on X, then X is homeomorphic to Y , and E is Riesz isomorphic to F . In this case, Φ can be written as a weighted composition operator: Φf (y) = Π(y)(f (ϕ(y))), where ϕ is a homeomorphism from Y onto X, and Π(y) is a Riesz isomorphism from E onto F for every y in Y . This generalizes some known results obtained recently.

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