On amorphic $C$-algebras

An amorphic association scheme has the property that any of its fusion is also an association scheme. In this paper we generalize the property to be amorphic to an arbitrary $C$-algebra, and prove that any amorphic $C$-algebra is determined up to isomorphism by the multiset of its diagonal structure constants and an additional integer equal $\pm 1$. We show that any amorphic $C$-algebra with rational structure constants is the fusion of an amorphic homogeneous $C$-algebra. As a special case of our results we obtain the well-known Ivanov's characterization of intersection numbers of amorphic association schemes.