  We generalize the Frobenius-Schur theorem to $C^*$-categories. From this
category-theoretical point of view, we introduce the notions of real, complex
and quaternionic representations of Hopf $C^*$-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 $C^*$-algebras, table algebras and compact
quantum groups.
