  In this article -that has also the intention to survey some known results in
the theory of compact quantum groups using methods different from the standard
and with a strong algebraic flavor- we consider compact o-coalgebras and Hopf
algebras. In the case of a o-Hopf algebra we present a proof of the
characterization of the compactness in terms of the existence of a positive
definite integral, and use our methods to give an elementary proof of the
uniqueness - up to conjugation by an automorphism of Hopf algebras- of the
compact involution appearing in [4]. We study the basic properties of the
positive square root of the antipode square that is a Hopf algebra automorphism
that we call the positive antipode. We use it -as well as the unitary antipode
and the Nakayama automorphism- in order to enhance our understanding of the
antipode itself.
