Let
X
X
be a compact Hausdorff space and
A
⊂
C
(
X
)
\mathcal {A}\subset C(X)
a function algebra. Assume that
X
X
is the maximal ideal space of
A
\mathcal A
. Denoting by
σ
(
f
)
\sigma (f)
the spectrum of an
f
∈
A
f\in \mathcal {A}
, which in this case coincides with the range of
f
f
, a result of Molnár is generalized by our Main Theorem: If
Φ
:
A
→
A
\Phi :\mathcal {A} \rightarrow \mathcal {A}
is a surjective map with the property
σ
(
f
g
)
=
σ
(
Φ
(
f
)
Φ
(
g
)
)
\sigma (fg)=\sigma (\Phi (f)\Phi (g))
for every pair of functions
f
,
g
∈
A
f,g\in \mathcal {A}
, then there exists a homeomorphism
Λ
:
X
→
X
\Lambda :X\rightarrow X
such that
\[
Φ
(
f
)
(
Λ
(
x
)
)
=
τ
(
x
)
f
(
x
)
\Phi (f)(\Lambda (x))=\tau (x)f(x)
\]
for every
x
∈
X
x\in X
and every
f
∈
A
f\in \mathcal {A}
with
τ
=
Φ
(
1
)
\tau =\Phi (1)
.