Let
G
G
be a locally compact group,
A
(
G
)
A(G)
the Fourier algebra of
G
G
,
B
(
G
)
B(G)
the Fourier-Stieltjes algebra of
G
G
and
VN
(
G
)
{\text {VN}}(G)
the von Neumann algebra generated by the left regular representation
λ
\lambda
of
G
G
. Then
A
(
G
)
A(G)
is the predual of
VN
(
G
)
{\text {VN}}(G)
;
VN
(
G
)
{\text {VN}}(G)
is a
B
(
G
)
B(G)
-module and
A
(
G
)
A(G)
is a closed ideal of
B
(
G
)
B(G)
. Let
AP
(
G
^
)
=
{
T
∈
VN
(
G
)
:
u
↦
u
⋅
T
{\text {AP}}(\hat G) = \{ T \in {\text {VN}}(G):u \mapsto u \cdot T
is a compact operator from
A
(
G
)
A(G)
into
VN
(
G
)
}
{\text {VN}}(G)\}
, the space of almost periodic operators in
VN
(
G
)
{\text {VN}}(G)
. Let
C
δ
∗
(
G
)
C_\delta ^*(G)
be the
C
∗
{C^*}
-algebra generated by
{
λ
(
x
)
:
x
∈
G
}
\{ \lambda (x):x \in G\}
. Then
C
δ
∗
(
G
)
⊂
AP
(
G
^
)
C_\delta ^*(G) \subset {\text {AP}}(\hat G)
. For a compact
G
G
, let
E
E
be the rank one operator on
L
2
(
G
)
{L^2}(G)
that sends
h
∈
L
2
(
G
)
h \in {L^2}(G)
to the constant function
∫
h
(
x
)
d
x
\int {h(x)dx}
. We have the following results: (1) There exists a compact group
G
G
such that
E
∈
AP
(
G
^
)
∖
C
δ
∗
(
G
)
E \in \text {AP}(\hat G)\backslash C_\delta ^*(G)
. (2) For a compact Lie group
G
G
,
E
∈
AP(
G
^
)
⇔
E
∈
C
δ
∗
(
G
)
⇔
L
∞
(
G
)
E \in {\text {AP(}}\hat G{\text {)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G)
has a unique left invariant mean
⇔
G
\Leftrightarrow G
is semisimple. (3) If
G
G
is an extension of a locally compact abelian group by an amenable discrete group then
AP
(
G
^
)
=
C
δ
∗
(
G
)
{\text {AP}}(\hat G) = C_\delta ^*(G)
. (4) Let
G
=
F
r
G = {{\mathbf {F}}_r}
, the free group with
r
r
generators,
1
>
r
>
∞
1 > r > \infty
. If
T
∈
VN
(
G
)
T \in {\text {VN}}(G)
and
u
↦
u
⋅
T
u \mapsto u \cdot T
is a compact operator from
B
(
G
)
B(G)
into
VN
(
G
)
{\text {VN}}(G)
then
T
∈
C
δ
∗
(
G
)
T \in C_\delta ^*(G)
.