Frobenius-Schur theorem for $C^{\ast }$-categories

We generalize the Frobenius-Schur theorem to $C^{\ast }$-categories. From this category-theoretical point of view, we introduce the notions of real, complex and quaternionic representations of Hopf $C^{\ast }$-algebras. Based on these definitions, we give another type of the Hopf-algebraic analogue of the Frobenius-Schur theorem, originally due to Linchenko and Montgomery. Also given are similar results for weak Hopf $C^{\ast }$-algebras, table algebras and compact quantum groups.