Suppose that
(
A
,
G
,
α
)
(A,G,\alpha )
is a
C
∗
C^*
-dynamical system such that
G
G
is of polynomial growth. If
A
A
is finite dimensional, we show that any element in
K
(
G
;
A
)
K(G;A)
has slow growth and that
L
1
(
G
,
A
)
L^1(G, A)
is
∗
*
-regular. Furthermore, if
G
G
is discrete and
π
\pi
is a “nice representation” of
A
A
, we define a new Banach
∗
*
-algebra
l
π
1
(
G
,
A
)
l^1_{\pi }(G, A)
which coincides with
l
1
(
G
;
A
)
l^1(G;A)
when
A
A
is finite dimensional. We also show that any element in
K
(
G
;
A
)
K(G;A)
has slow growth and
l
π
1
(
G
,
A
)
l^1_{\pi }(G, A)
is
∗
*
-regular.