A closer look at the $B$-spline interpolation problem in the setting of Hilbert $C^*$-modules

(A joint work with R. Eskandari, M. Frank, and V. M. Manuilov)
In this talk, we introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module, investigate its fundamental properties and explore the uniqueness of solution. We 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.
Moving to the set-up of Hilbert $W^*$-modules, we characterize the case when the spline interpolation problem for the extended $C^*$-valued sesquilinear form has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathcal{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathcal{X}$.