We establish a multilinear generalisation of Synnatzschke’s Theorem for regular operators on Banach lattices, with the Arens adjoint taking the place of the transpose. Using this result, we show that the Aron–Berner extension of a regular homogeneous polynomial to the order continuous bidual preserves the absolute value. As a consequence, it follows that the regular norm is preserved by the Aron–Berner extension.

中文翻译：

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Synnatzschke多项式定理

我们为Banach格上的正则算子建立了Synnatzschke定理的多线性推广，其中Arens伴随代替了转置。使用该结果，我们证明了将常规齐次多项式的Aron–Berner扩展扩展为连续双阶的阶可以保留绝对值。因此，可以得出结论，Aron-Berner扩展保留了常规规范。