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A note on order topologies on ordered tensor products
Published online by Cambridge University Press: 17 April 2009
Abstract
If E, F are regularly ordered vector spaces the tensor product E ⊗ F can be ordered by the conic hull Kπ of tensors, x ⊗ y with x ≥ 0 in E and y ≥ 0 in F, or by the cone K⊗ of tensors Φ ∈ E ⊗ F such that Φ(x′, y′) ≥ 0 for positive linear functionals x′, y′ on E, F.
If E, F are locally convex spaces the tensor product can te given the π-topology which is defined by seminorms pα ⊕ qβ where {pα}, {qβ} are classes of seminorms defining the topologies on E, F. The tensor product can also be given the ε-topology which is the topology of uniform convergence on equicontinuous subsets J x H of E′ x F′. The main result of this note is that if the regularly ordered vector spaces E, F carry their order topologies then the order topology on E ⊕ F is the π-topology when E ⊕ F is ordered by kπ, and the ε-topology when E ⊕ F is ordered by K⊕.
- Type
- Research Article
- Information
- Bulletin of the Australian Mathematical Society , Volume 3 , Issue 3 , December 1970 , pp. 385 - 390
- Copyright
- Copyright © Australian Mathematical Society 1970