Extensions of the Lax-Milgram theorem to Hilbert C*-modules
R. Eskandari, M. Frank, V. M. Manuilov, M. S. Moslehian
TL;DR
This paper extends the Lax--Milgram theorem to Hilbert $C^*$-modules in two broad settings: self-dual Hilbert $W^*$-modules and modules over $C^*$-algebras of compact operators. By leveraging Paschke's extension to self-dual modules and multiplier modules, the authors establish existence and uniqueness results for representations of bounded $\mathscr{A}$-sesquilinear forms as $B(x,y)=\langle T x, y\rangle$ with $T$ adjointable and invertible. The results hold under carefully formulated normal-state lower bounds, with the range of the associated operator $T$ shown to be closed. A counterexample demonstrates that these Lax--Milgram extensions do not generally transfer to arbitrary $C^*$-algebras, while the multiplier-extended approach provides a robust framework in the compact-operator and $W^*$-contexts. Overall, the work broadens the functional-analytic toolkit for bilinear form representations in noncommutative Hilbert-module settings, highlighting both the reach and the limits of Riesz-type representations in this setting.
Abstract
We present three versions of the Lax-Milgram theorem in the framework of Hilbert C*-modules, two for those 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, our 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.