Vector-Valued Reproducing Kernel Hilbert $$C^*$$-Modules

The aim of this paper is to present a unified framework in the setting Hilbert $$C^*$$ -modules for scalar- and vector-valued reproducing kernel spaces -valued spaces. We investigate conditionally negative definite kernels with values -algebra adjointable operators acting on -module. In addition, we show that there exists two-sided connection between positive -modules. Furthermore, explore some conditions under which function module an interpolation theorem. Moreover, study basic properties so-called relative give characterization dual modules. Among other things, prove every gives us -module certain map. Several examples illustrate our investigation.