The authors consider finite-dimensional *-algebras over \(\mathbb{C}\). Such an algebra is called a \(C\)-algebra, if it has a vector space basis \(R\) satisfying several given conditions (expressed in terms of so-called structure constants). The main result states that the category of positive \(C\)-algebras is equivalent to a (suitably defined) category of pairs of algebras in Plancherel duality. The authors also indicate some relations between the concepts considered and some generalizations of Hopf algebras which they are going consider in a subsequent publication.