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

We establish the following result.

Theorem. Let $\alpha:G\to {\mathcal L}(X)$ be a $\sigma(X,X_*)$ integrable bounded group representation whose Arveson spectrum $\operatorname{Sp}(\alpha)$ is scattered. Then the subspace generated by all eigenvectors of the dual representation $\alpha^*$ is $w^*$ dense in $X^*.$ Moreover, the $\sigma(X,X_*)$ closed subalgebra $W_\alpha$ generated by the operators $\alpha _t$ ($t\in G$) is semisimple.

If, in addition, $X$ does not contain any copy of $c_0,$ then the subspace spanned by all eigenvectors of $\alpha$ is $\sigma(X,X_*)$ dense in $X.$ Hence, the representation $\alpha$ is almost periodic whenever it is strongly continuous.