  In this paper we introduce the notion of a Hopf C*-algebra and construct the
counit and antipode. A Hopf C*-algebra is a C*-algebra with comultiplication
satisfying some extra condition which makes possible the construction of the
counit and antipode. The leading example is of course the C*-algebra of
continuous, vanishing at infinity functions on a locally compact group. Also
locally compact quantum groups will be examples. We include several formulas
for the counit and antipode which are familiar from Hopf algebra theory.
