Let
(
B
,
R
,
α
)
(B,\mathbf {R},\alpha )
be a
C
∗
C^{*}
- dynamical system and let
A
=
B
α
(
[
0
,
∞
)
)
A=B^\alpha ([0,\infty ))
be the analytic subalgebra of
B
B
. We extend the work of Loebl and the first author that relates the invariant subspace structure of
π
(
A
)
,
\pi (A),
for a
C
∗
C^{*}
-representation
π
\pi
on a Hilbert space
H
π
\mathcal {H}_\pi
, to the possibility of implementing
α
\alpha
on
H
π
.
\mathcal {H}_\pi .
We show that if
π
\pi
is irreducible and if lat
π
(
A
)
\pi (A)
is trivial, then
π
(
A
)
\pi (A)
is ultraweakly dense in
L
(
H
π
)
.
\mathcal {L(H}_\pi ).
We show, too, that if
A
A
satisfies what we call the strong Dirichlet condition, then the ultraweak closure of
π
(
A
)
\pi (A)
is a nest algebra for each irreducible representation
π
.
\pi .
Our methods give a new proof of a “density” theorem of Kaftal, Larson, and Weiss and they sharpen earlier results of ours on the representation theory of certain subalgebras of groupoid
C
∗
C^{*}
-algebras.