We present three versions of the Lax–Milgram theorem in the framework of Hilbert \(C^*\)-modules, two for self-dual ones over \(W^*\)-algebras and one for those over \(C^*\)-algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert \(C^*\)-modules over \(C^*\)-algebras of compact operators, however, the modular Lax–Milgram theorem turns out to be valid for all of them. We also give several examples to illustrate our results, in particular, we show that the main theorem is not true for Hilbert modules over arbitrary \(C^*\)-algebras.

中文翻译：

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Lax–Milgram定理的扩展到Hilbert $$ C ^ * $$ C ∗ -modules

我们在希尔伯特\（C ^ * \）-模块的框架中展示了Lax–Milgram定理的三个版本，两个是\（W ^ * \）-代数上的自对偶，另一个是\（C ^ * \） -紧凑算子的代数。值得注意的是，尽管Riesz定理对于紧凑型算子的\（C ^ * \）-代数的某些Hilbert \（C ^ * \）-模块无效，但是，模块化Lax-Milgram定理证明是有效的对于他们所有人。我们还提供了一些示例来说明我们的结果，特别是，我们证明了在任意\（C ^ * \）-代数上Hilbert模块的主定理都不成立。