Let S(t) be an exponentially bounded C-semigroup on a Banach space X and let S~*(t) be its adjoint semigroup on the dual space X~*. This paper gives a closed subspace X~＋ of X~* such that the restriction S~＋(t) of S~*(t) to X~＋ is a C~*｜_(X~＋)-semigroup on X~＋ and gives the generator of S~＋(t). The subspace X~＋ is maximal in a certain sense.