Gelfand-Naimark Theorem of Commutative Hopf C^*-Algebras

Let A be a commutative C*-algebra. By the Gelfand-Naimark theorem, there exists a locally compact space G such that A is isomorphic to C₀（G）, the C*-algebra of all complex continuous functions on G vanishing at infinity. The result is generalized to the case of Hopf C*-algebra, where G is altered by a locally compact group. Using the isomorphic representation, the counit ε and the antipode S of a commutative Hopf C*-algebra are proposed.