Completeness of eigenvectors of group representations of operators whose Arveson spectrum is scattered

We establish the following result. Theorem. Let alpha : G --> L(X) be a sigma(X, X-*) integrable bounded group representation whose Arveson spectrum Sp(alpha) is scattered. Then the sub space generated by all eigenvectors of the dual representation alpha* is w* dense in X*. Moreover, the sigma(X, X-*) closed subalgebra W alpha generat ed by the operators alpha(t) (t is an element of G) is semisimple. If, in addition, X does not contain any copy of c(0,) then the subspace spa nned by all eigenvectors of alpha is sigma(X, X (*)) dense in X. Hence, the representation alpha is almost periodic whenever it is strongly continuous .