Symmetric coalgebras

We construct a structure of a ring with local units on a co-Frobenius coalgebra. We study a special class of co-Frobenius coalgebras whose objects we call symmetric coalgebras. We prove that any semiperfect coalgebra can be embedded in a symmetric coalgebra. We use a dual version of Brauerʹs equivalence theorem to characterize symmetric coalgebras by comparing certain functors. We define an automorphism of the ring with local units constructed from a co-Frobenius coalgebra, which we call the Nakayama automorphism. This is used to give a new characterization to symmetric coalgebras and to describe Hopf algebras that are symmetric as coalgebras. As a corollary we obtain as a consequence the known characterization of Hopf algebras that are symmetric as algebras.