Let
G
G
be a locally compact group,
A
p
=
A
p
(
G
)
A_{p}=A_{p}(G)
the Banach algebra defined by Herz; thus
A
2
(
G
)
=
A
(
G
)
A_{2}(G)=A(G)
is the Fourier algebra of
G
G
. Let
P
M
p
=
A
p
∗
PM_{p}=A^{*}_{p}
the dual,
J
⊂
A
p
J \subset A_{p}
a closed ideal, with zero set
F
=
Z
(
J
)
F=Z(J)
, and
P
=
(
A
p
/
J
)
∗
\mathbb {P} = (A_{p}/J)^{*}
. We consider the set
T
I
M
P
(
x
)
⊂
P
∗
TIM_{\mathbb {P}}(x) \subset {\mathbb {P}}^{*}
of topologically invariant means on
P
\mathbb {P}
at
x
∈
F
x\in F
, where
F
F
is “thin.” We show that in certain cases card
T
I
M
P
(
x
)
≥
2
c
TIM_{\mathbb {P}}(x) \geq 2^{c}
and
T
I
M
P
(
x
)
TIM_{\mathbb {P}}(x)
does not have the WRNP, i.e. is far from being weakly compact in
P
∗
\mathbb {P}^{*}
. This implies the non-Arens regularity of the algebra
A
p
/
J
A_{p}/J
.