An isomorphic characterization of L-spaces |
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Authors: | Vlad Timofte |
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Institution: | aInstitute of Mathematics “Simion Stoilow” of the Romanian Academy, P.O. Box 1-764, RO-014700, Bucharest, Romania |
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Abstract: | We show that a sequentially (τ)-complete topological vector lattice Xτ is isomorphic to some L1(μ), if and only if the positive cone can be written as X+ = +B for some convex, (τ)-bounded, and (τ)-closed set B X+ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y+ = + (for u >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials. |
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Keywords: | Primary 46A40 Secondary 46E30 46B03 |
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