Completeness of Eigenvectors of Group Representations of Operators Whose Arveson Spectrum Is Scattered

We establish the following result. Theorem. Let α : G → L(X) be a σ(X, X∗) integrable bounded group representation whose Arveson spectrum Sp(α) is scattered. Then the subspace generated by all eigenvectors of the dual representation α∗ is w∗ dense in X∗. Moreover, the σ(X, X∗) closed subalgebra Wα generated by the operators αt (t ∈ G) is semisimple. If, in addition, X does not contain any copy of c0, then the subspace spanned by all eigenvectors of α is σ(X, X∗) dense in X. Hence, the representation α is almost periodic whenever it is strongly continuous. 1. Spectral theory for integrable bounded group representations Throughout this paper G will denote a locally compact abelian (LCA) group with identity e and Ĝ will denote the dual group of G. The multiplication on LCA groups will be written by addition. Let L(G) (resp. M(G)) be the usual group algebra (resp. measure algebra) with convolution as product operation. We refer to [11] or [21] for basic knowledge of Harmonic Analysis on LCA groups. Given a complex Banach space X, let L(X) be the Banach algebra of all bounded linear operators on X. Take a LCA group G. A bounded group representation α of G on X is a mapping α : G→ L(X) satisfying the following properties: (a) Group property: αe = IX the identity operator on X and αs+t = αsαt for all s, t ∈ G; (b) Boundedness: ‖α‖ := supt∈G ‖αt‖ <∞. Moreover, α is called strongly (resp. weakly) continuous if for each x ∈ X the mapping t 7→ αtx is norm (resp. weakly) continuous. We need a further notion. Definition 1.1. A bounded group representation α : G → L(X) of G on X is called integrable if there exists a subspace X∗ ⊂ X∗ satisfying the following Received by the editors September 1, 1997. 1991 Mathematics Subject Classification. Primary 47A67, 47A10.