Abstract
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.
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Eskandari, R., Frank, M., Manuilov, V.M. et al. Extensions of the Lax–Milgram theorem to Hilbert \(C^*\)-modules. Positivity 24, 1169–1180 (2020). https://doi.org/10.1007/s11117-019-00726-9
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DOI: https://doi.org/10.1007/s11117-019-00726-9