  Supervised learning in reproducing kernel Hilbert space (RKHS) and
vector-valued RKHS (vvRKHS) has been investigated for more than 30 years. In
this paper, we provide a new twist to this rich literature by generalizing
supervised learning in RKHS and vvRKHS to reproducing kernel Hilbert
$C^*$-module (RKHM), and show how to construct effective positive-definite
kernels by considering the perspective of $C^*$-algebra. Unlike the cases of
RKHS and vvRKHS, we can use $C^*$-algebras to enlarge representation spaces.
This enables us to construct RKHMs whose representation power goes beyond
RKHSs, vvRKHSs, and existing methods such as convolutional neural networks. Our
framework is suitable, for example, for effectively analyzing image data by
allowing the interaction of Fourier components.
