正则多重线性算子的Arens延拓的序连续性
Order continuity of Arens extensions of regular multilinear operators
AI总结：
本文研究了正则多重线性算子Arens延拓的序连续性，通过反例说明现有结果无法推广至整个双对偶乘积，并建立了使其在全双对偶乘积上序连续的条件，同时证明了延拓至少在单变量上序连续。
摘要：
首先，我们给出一个反例，表明关于多重线性算子的Arens延拓的分别序连续性的近期结果无法改进为在整个双对偶的乘积上具有分别序连续性。然后，我们对算子和/或底层Riesz空间/Banach格建立条件，使得这些延拓在整个双对偶的乘积上是序连续的。我们还证明了任何正则多重线性算子的所有Arens延拓至少在一个变量中是序连续的，并且研究了Banach格$E$上正则齐次多项式的Arens延拓何时在$E^{\sim\sim}$上是序连续的。
英文摘要：
First we give a counterexample showing that recent results on separate order continuity of Arens extensions of multilinear operators cannot be improved to get separate order continuity on the product of the whole of the biduals. Then we establish conditions on the operators and/or on the underlying Riesz spaces/Banach lattices so that the extensions are order continuous on the product of the whole biduals. We also prove that all Arens extensions of any regular multilinear operator are order continuous in at least one variable and we study when Arens extensions of regular homogeneous polynomials on a Banach lattice $E$ are order continuous on $E^{\sim\sim}$.
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