Let
A
\mathcal {A}
be a commutative Banach algebra. Denote the spectral radius of an element
a
a
in
A
\mathcal {A}
by
ρ
A
(
a
)
{\rho _\mathcal {A}}(a)
. An extension of
A
\mathcal {A}
is a Banach algebra
B
\mathcal {B}
such that
A
\mathcal {A}
is algebraically, but not necessarily continuously, embedded in
B
\mathcal {B}
. We view
A
\mathcal {A}
as a subalgebra of
B
\mathcal {B}
. If
B
\mathcal {B}
is an extension of
A
\mathcal {A}
then
S
p
B
(
a
)
∪
{
0
}
⊆
S
p
A
(
a
)
∪
{
0
}
S{p_\mathcal {B}}(a) \cup \{ 0\} \subseteq S{p_\mathcal {A}}(a) \cup \{ 0\}
and thus
ρ
B
(
a
)
≤
ρ
A
(
a
)
,
∀
a
∈
A
{\rho _\mathcal {B}}(a) \leq {\rho _\mathcal {A}}(a),\forall a \in \mathcal {A}
. Let us say that
A
\mathcal {A}
has the spectral extension property if
ρ
B
(
a
)
=
ρ
A
(
a
)
{\rho _\mathcal {B}}(a) = {\rho _\mathcal {A}}(a)
for all
a
∈
A
a \in \mathcal {A}
and all extensions
B
\mathcal {B}
of
A
\mathcal {A}
, that
A
\mathcal {A}
has the strong spectral extension property if
S
p
B
(
a
)
∪
{
0
}
=
S
p
A
(
a
)
∪
{
0
}
S{p_\mathcal {B}}(a) \cup \{ 0\} = S{p_\mathcal {A}}(a) \cup \{ 0\}
for all
a
∈
A
a \in \mathcal {A}
and all extensions
B
\mathcal {B}
of
A
\mathcal {A}
, and that
A
\mathcal {A}
has the multiplicative Hahn-Banach property if every multiplicative linear functional
χ
\chi
on
A
\mathcal {A}
has a multiplicative linear extension to every commutative extension
B
\mathcal {B}
of
A
\mathcal {A}
. We give characterizations of these properties for semisimple commutative Banach algebras.