Let
S
S
be a topological semigroup,
US
(
S
)
{\text {US}}(S)
the set of all bounded uniformly continuous functions on
S
,
WAP(
S
)
S,{\text {WAP(}}S)
the set of all (bounded) weakly almost periodic functions on
S
,
E
0
(
S
)
:=
{
f
∈
UC(
S
)
:
m
(
|
f
|
)
=
0
S,{E_0}(S): = \{ f \in {\text {UC(}}S):m(|f|) = 0
for each left and right invariant mean
m
m
on
UC(
S
)
}
{\text {UC(}}S)\}
and
W
0
(
S
)
:=
{
f
∈
WAP
(
S
)
:
m
(
|
f
|
)
=
0
{W_0}(S): = \{ f \in {\text {WAP}}(S):\:m(|f|) = 0
for each left and right invariant mean
m
m
on
WAP(
S
)
}
{\text {WAP(}}S)\}
. Among other results, for a large class of noncompact locally compact topological semigroups
S
S
, we show that the quotient space
E
0
(
S
)
/
W
0
(
S
)
{E_0}(S)/{W_0}(S)
contains a linear isometric copy of
l
∞
{l^\infty }
and so is nonseparable.