is finite, where λ denotes Lebesgue measure. Let ρ be the gauge functional of the convex hull of the unit ball {f : q(f) ≤ 1} of the quasi-norm q, and let N be the null space of ρ. The normed envelope of WeakL, which we denote by W , is the space (WeakL/N, ρ). The Banach envelope of WeakL, W , is the completion of W . We show that W is isometrically lattice isomorphic to a sublattice of W . It ...

Let Z be a Banach lattice endowed with positive cone C and an order-continuous norm j.j . Let G be a left-amenable semigroup of positive linear endomorphisms of Z . Then the positive fixed points Co of Z under G form a lattice cone, and their linear span Z0 is a Banach lattice under an order-continuous norm ||.||0 which agrees with |.| on Co. A counterexample shows that under the given conditio...

S. Sherman has shown [4] that if the self adjoint elements of a C* algebra form a lattice under their natural ordering the algebra is necessarily commutative. In this note we extend this result to real Banach algebras with an identity and arbitrary Banach * algebras with an identity. The central fact for a real Banach algebra A is that if the positive cone is defined to be the uniform closure o...

Let E be a Dedekind σ -complete Banach lattice, let Ea denote the order continuous part of E, and suppose that Ea is order dense in E. If E contains a lattice copy of ∞ (equivalently, E = Ea) then E/Ea contains a lattice copy of ∞/c0, and hence a lattice-isometric copy of ∞. This is one of the consequences of more general results presented in this paper.