# Extensions of the Lax-Milgram theorem to Hilbert C*-modules

## TL;DR

This paper extends the Lax-Milgram theorem to Hilbert C*-modules over W*-algebras and C*-algebras of compact operators, highlighting differences from classical Hilbert space results and providing illustrative examples.

## Contribution

It introduces three versions of the Lax-Milgram theorem in the context of Hilbert C*-modules, including cases where the Riesz theorem does not hold.

## Key findings

- Lax-Milgram theorem valid over all Hilbert C*-modules of compact operators
- Riesz theorem not valid for certain Hilbert C*-modules of compact operators
- Main theorem not applicable to all Hilbert modules over arbitrary C*-algebras

## 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.