Let G be a locally compact group and H an open subgroup of G. The embeddings of group C*-algebras associated with H into the group C*-algebras associated with G are studied. Three conditions for the embeddings given in terms of C*-norms of the group algebras, group representations and positive definite functions are shown to be equivalent. As corollary, we prove that the full C*-algebra of H can be embedded into the full C*-algebra of G in a natural way as well as the case for the reduced group C*-algebras. We also show that the embeddings hold for their duals and double duals.
