For Banach spaces [Formula: see text] and [Formula: see text], we establish a natural bijection between preduals of [Formula: see text] and preduals of [Formula: see text] that respect the right [Formula: see text]-module structure. If [Formula: see text] is reflexive, it follows that there is a unique predual making [Formula: see text] into a dual Banach algebra. This removes the condition that [Formula: see text] have the approximation property in a result of Daws. We further establish a natural bijection between projections that complement [Formula: see text] in its bidual and [Formula: see text]-linear projections that complement [Formula: see text] in its bidual. It follows that [Formula: see text] is complemented in its bidual if and only if [Formula: see text] is (either as a module or as a Banach space). Our results are new even in the well-studied case of isometric preduals.

中文翻译：

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有界线性算子空间的预对与补

对于 Banach 空间 [Formula: see text] 和 [Formula: see text]，我们在 [Formula: see text] 的谓词和 [Formula: see text] 的谓词之间建立了一个自然双射，尊重右 [Formula: see text] -模块结构。如果 [Formula: see text] 是自反的，则有一个独特的前对偶使 [Formula: see text] 成为对偶 Banach 代数。这消除了 [Formula: see text] 在 Daws 结果中具有近似属性的条件。我们进一步在补充 [公式：见文本] 的双射和补充 [公式：见文本] 的 [公式：见文本] 的线性投影之间建立了自然双射。因此，当且仅当 [Formula: see text] 是（作为模块或作为 Banach 空间）时，[Formula: see text] 在其对偶中被补充。