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Title:On some spaces related to weak L(p) and their duals
Author(s):Chung, Si Kit
Doctoral Committee Chair(s):Lotz, Heinrich P.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:In Cwikel's paper "On the dual of Weak $L\sp{p}$", it is shown that (Weak $L\sp{p})\sp\prime = L(p\sp\prime, 1)\oplus S\sb0\oplus S\sb\infty.$ For non-atomic measure spaces, Cwikel obtains a representation of elements in $S\sb0$ and $S\sb\infty.$ However, we show that this representation is incorrect by proving that if E is a non-reflexive weakly sequentially complete Banach lattice, then the disjoint complement of E in $E\sp{\prime\prime}$ is non-reflexive. So, we would like to obtain more information on (Weak $L\sp{p})\sp\prime$.
For simplicity, we consider the Lebesgue measure space on (0,1) so that $S\sb\infty=\{0\}.$ We introduce three lattice semi-norms $\rho\sb0,\rho\sb1$ and $\rho\sb\omega$ on $L\sp\infty(0,1)$ so that ($L\sp\infty(0,1),\rho\sb\omega)$ can be identified as an ideal of a quotient of Weak $L\sp\rho(0,1).$ The dual of $(L\sp\infty(0,1),\rho\sb{i})$ where i = 0, 1, $\omega$ is studied using the result that if E is a normed vector lattice and ${\cal A}$ is a bounded subset of $E\sb+\sp\prime,$ then the unit ball of $(E,\rho\sb{\cal A})\sp\prime$ is the solid hull of the $\sigma(E\sp\prime, E)$-closed convex hull generated by ${\cal A},$ where $\rho\sb{\cal A}$ is the lattice semi-norm defined by $\rho\sb{\cal A}(x) = \sup\sb{x\sp\prime\in {\cal A}}\langle \vert x\vert, x\sp\prime\rangle.$ We prove that the maximal elements in the unit ball of $(L\sp\infty(0,1),\rho\sb0)\sp\prime$ are the non-increasing means concentrated at 0 and that these elements are weak*-limits of nets of non-increasing, non-negative functions with $L\sb1$-norms equal to one and supports shrinking to 0. We introduce the idea of dual admissibility of an ordered pair $(\Vert\cdot\Vert\sb1,\Vert\cdot\Vert\sb0)$ of lattice norms defined on a vector lattice. Characterizations of and sufficient conditions for dual admissibility are obtained. From this, we show that the unit ball of ($L\sp\infty(0,1),\rho\sb{\omega})\sp\prime$ can be obtained by taking the weak*-closure in $L\sp\infty(0,1)\sp\prime$ of a certain subset of $(L\sp\infty(0,1),\rho\sb0)\sp\prime.$ Then we show that every element in $(L\sp\infty(0,1),\rho\sb{\omega})\sp\prime$ has a unique norm preserving "extension" in $S\sb0$ and that $S\sb0$ can be "generated" by these norm preserving "extensions" together with a family of operators.
Finally, we consider questions related to dual admissibility. Results on the equivalence of order continuous norm topologies on order intervals as well as that on the $\sigma(E\sp\prime,E)$-density in $E\sbsp{+}{\prime}$ of the positive part of a sublattice of the dual of a normed vector lattice E are obtained.
Issue Date:1993
Rights Information:Copyright 1993 Chung, Si Kit
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9411594
OCLC Identifier:(UMI)AAI9411594

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