$C$-algebras and algebras in Plancherel duality

For an arbitrary $C$-algebra (possibly non-commutative) a positivity condition generalizing the Krein condition for a commutative case is defined. We show that the class of positive $C$-algebras includes those arising in algebraic combinatorics from association schemes (possibly non-commutative). It is proved that the category of positive $C$-algebras is equivalent to the category of pairs of algebras in Plancherel duality one of which being commutative.