We establish the following result.
Theorem. Let
α
:
G
→
L
(
X
)
\alpha :G\to {\mathcal L}(X)
be a
σ
(
X
,
X
∗
)
\sigma (X,X_*)
integrable bounded group representation whose Arveson spectrum
Sp
(
α
)
\operatorname {Sp}(\alpha )
is scattered. Then the subspace generated by all eigenvectors of the dual representation
α
∗
\alpha ^*
is
w
∗
w^*
dense in
X
∗
.
X^*.
Moreover, the
σ
(
X
,
X
∗
)
\sigma (X,X_*)
closed subalgebra
W
α
W_\alpha
generated by the operators
α
t
\alpha _t
(
t
∈
G
t\in G
) is semisimple.
If, in addition,
X
X
does not contain any copy of
c
0
,
c_0,
then the subspace spanned by all eigenvectors of
α
\alpha
is
σ
(
X
,
X
∗
)
\sigma (X,X_*)
dense in
X
.
X.
Hence, the representation
α
\alpha
is almost periodic whenever it is strongly continuous.