B-spline interpolation problem in Hilbert C*-modules
Rasoul Eskandari, Michael Frank, Vladimir Manuilov, Mohammad Sal Moslehian
TL;DR
The paper addresses the $B$-spline interpolation problem for bounded $C^*$-valued sesquilinear forms on Hilbert $C^*$-modules, extending the analysis to the module’s second dual and to Hilbert $W^*$-modules. It establishes representation theorems in the self-dual setting, derives necessary and sufficient solvability conditions, and investigates uniqueness through radical submodules, while generalizing results to $C^*$-ideals of $W^*$-algebras via Paschke duality and multiplier algebras. The main contributions include concrete solvability criteria, a detailed study of dual and double-dual contexts, and a range of examples illustrating when solutions exist or fail to exist, along with implications for orthogonal complementation under alternative inner products. These results provide a robust framework for operator-valued interpolation in noncommutative settings, with potential applications in approximation theory and noncommutative geometry.
Abstract
We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.