# Monotonic norms in ordered banach spaces

01 Oct 1988-Journal of The Australian Mathematical Society (Cambridge University Press (CUP))-Vol. 45, Iss: 2, pp 217-219

Abstract: 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 a-monotone (a > 1) if and only if for each / in B* there exists g 6 B* with g > 0, / and ||g|| < aN(f). We also establish a dual result characterizing a-monotonicity of B'.

Topics: Eberlein–Šmulian theorem (61%), Banach manifold (60%), Lp space (59%), Banach space (59%), Dual norm (58%)

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Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces \(X\) and \(Y\) with closed cones we investigate normality of \(B(X,Y)\) in terms of normality and conormality of the underlying spaces \(X\) and \(Y\). Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples \(X\) and \(Y\) that are not Banach lattices, but for which \(B(X,Y)\) is normal. In particular, we show that a Hilbert space \(\mathcal {H}\) endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if \(\dim \mathcal {H}\ge 3\)), and satisfies an identity analogous to the elementary Banach lattice identity \(\Vert |x|\Vert =\Vert x\Vert \) which holds for all elements \(x\) of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

15 citations

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Abstract: We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$.
Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that are not Banach lattices, but for which $B(X,Y)$ is normal. In particular, we show that a Hilbert space $\mathcal{H}$ endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and satisfies an identity analogous to the elementary Banach lattice identity $\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.

2 citations

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01 Aug 2019Abstract: Consider the following still-open problem: for any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅+ : X → C and ⋅− : X → C satisfying x = x+ − x− for every x ∈ X?

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