Let
Γ
\Gamma
be a discrete abelian group, and
E
⊂
Γ
E \subset \Gamma
. For
F
⊂
E
F \subset E
, we say that
F
∈
P
(
E
)
F \in \mathcal {P}(E)
, if for all
Λ
\Lambda
, finite subsets of
Γ
,
0
∉
Λ
,
Λ
+
F
∩
F
\Gamma ,0 \notin \Lambda ,\Lambda + F \cap F
is finite. Having defined the Banach algebra,
A
~
(
E
)
=
c
(
E
)
∩
B
(
E
)
\tilde A(E) = c(E) \cap B(E)
, we prove the following: (i)
E
⊂
Γ
E \subset \Gamma
is a Sidon set if and only if every
F
∈
P
(
E
)
F \in \mathcal {P}(E)
is a Sidon set; (ii)
E
∈
P
(
Γ
)
E \in \mathcal {P}(\Gamma )
is a Sidon set if and only if
A
~
(
E
)
=
A
(
E
)
\tilde A(E) = A(E)
.