Hilbert space with non-associative scalars I

Abstract

The theory of a Hilbert space over a finite associative algebra is formulated, and the spectral resolution theorem for bounded Hermitian operators on this space is obtained. The properties of series representations are discussed and are found to be analogous to the usual ones of the complex Hilbert space. It is then shown that the theory of the non-associative Hilbert space developed in our previous paper is contained in the more general theory for the special case in which the finite algebra is chosen to be the Cayley ring.