In this paper a proper class of barrelled spaces which strictly contains the suprabarrelled spaces is considered. A closed graph theorem and some permanence properties are given. This allows us to prove the necessity of a condition of a theorem of S. A. Saxon and P. P. Narayanaswami by constructing an example of a non-suprabarrelled Baire-like space which is a dense subspace of a Fréchet space and is not an (LF)-space under any strong locally convex topology.

A hyperfinite von Neumann algebra satisfies the condition that every o.d. homomorphism is a normal operator if and only if it is a factor of type In.

Let B be an ordered Banach space with ordered Banach dual space. Let N denote the canonical half-norm. We give an alternative proof of the following theorem of Robinson and Yamamuro: the norm on B is α-monotone (α ≥ 1) if and only if for each f in B* there exists g ∈ B* with g ≥ 0, f and ∥g∥ ≤ α N(f). We also establish a dual result characterizing α-monotonicity of B*.

We present two more characterizations of maps which preserve orthogonal decompositions defined on Hilbert spaces ordered by natural cones.

Let f be a continuous function, and u a continuous linear function, from a Banach space into an ordered Banach space, such that f − u satisfies a Lipschitz condition and u satisfies an inequality implicit-function condition. Then f also satisfles an inequality implicit-function condition. This extends some results of Flett, Craven and S. M. Robinson.

In the case when 0 < p < 1 it is proved, using a method of Macphail that the identity map i: lp → lp is not (r, s)-absolutely summing for any r, s.

In this paper we prove an analogue of the separable version of Nachbin's characterization of injective Banach spaces in the setting of Banach lattices. The mappings involved are continuous Riesz homomorphisms defined on ideals of separable Banach lattices which can be extended to Riesz homomorphisms on the whole Banach lattice. We discuss applications to simultaneous extension operators and to extension of continuous mappings between certain topological spaces.

It is shown that for arbitrary ε > 0 there is a function x(t, x) defined on the square [0,1] × [0,1] such that x(t, s) represents an extremal point of the unit ball in the space of Lipschitz continuous functions, and the gradient of x(t, s) is equal to 0 except on a set of measure at most ε.

We continue the study of operators from an Archimedean vector lattice E into a cofinal sublattice H which have the property that there is λ > 0 such that if x ∈ E, h ∈ H and |x|≤|h|, then |Tx| ≤ λ|h|. The collection Z(E|H) of all of those operators forms an algebra under composition. We investigate the relationship between the properties of having an identity, being Abelin and being semi-simple for such-algebras, culminating in a proof that they are equivalent if H is Dedekind complete. We also study various for such an operator T, showing that, apart from 0, its spectrum relative to Z(E|H) is the same as that of T|H relative to Z(H) and that of T relative to ℒ(E) (Provided E is a Banach lattice and H is closed).

This paper studies topological upper and lower semicontinuity of the minimal value multifunction and the solution multifunction for optimization problems, which are defined in terms of cones, subject to perturbations in constraints. It extends the results of Tanino and Sawaragi to finite dimensions and one of Berge to multiple objective optimization problems.

It is shown that a weakly compact convex set in a locally convex space is a zonoform if and only if it is the order continuous image of an order interval in a Dedekind complete Riesz space. While this result implies the Kluv´nek characterization of the range of a vector measure, the techniques of the present paper are purely order theoretic.

In this paper we characterize boundedly laterally complete Riesz spaces, boundedly laterally complete Riesz spaces with the lateral boundedness property and Riesz spaces in which every principal ideal is finite dimensional. The characterizations are given in terms of extension properties of certain Riesz homomorphisms.

If M is a commutative W*-algebra of operators and if ReM is the Dedekind complete Riesz space of self-adjoint elements of M, then it is shown that the set of densely defined self-adjoint transformations affiliated with ReM is a Dedekind complete, laterally complete Riesz algebra containing ReM as an order dense ideal. The Riesz algebra of densely defined orthomorphisms on ReM is shown to coincide with , and via the vector lattice Randon-Nikodym theorem of Luxemburg and Schep, it is shown that the lateral completion of ReM may be identified with the extended order dual of ReM.

Let E be an Archimedean Riesz space and let Orth∞(E) be the f-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T:D→E, with D an order dense ideal in E, such that T(B∩D) ⊆ B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T ∈ Orth∞(F) can be extended to some ∈ Orth∞(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of σ-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.

Let A be an Archimedean, uniformly complete, semiprime f-algebra and F(X1,…Xn) ∈ R+ [X1,…Xn] a homogeneous polynomial of degree p (p∈ N). It is shown that (F(u 1…unn))1/p exists in A+ for all u1…un ∈ A+.

We extend various characterizations of scalar-valued lower semicontinuity and determine their relationship to the continuity of vector-valued convex functions. Order completeness of the range space is not assumed.

Let X be a weakly complete proper cone, contained in an Hausdorff locally convex space E, with continuous dual E′. A positive linear form on the Risz space of functions on X generated by E′ is called a conical measure on X. Let M+ (X) be the set of all conical measures on X. G. Choquet asked the question: when is every conical measure on X given by a Radon measure on (X\0)? Let L be the class of such X. In this paper we show that the fact that X ∈ L only depends, in some sense, on the cofinal subsets of the space E′|x ordered by the order of functions on X. We derive that X ∈ L is equivalent to M+ (M) ∈ L. We show that is closed under denumerable products.

A natural and welcomed decomposition theorem for elements in the positive cone of the tensor product of Archimedean vector lattices leads to substantial simplifications in the theory of tensor products of Archimedean vector lattices.