Let
M
M
be a von Neumann algebra and let
φ
\varphi
be a normal linear functional on a strongly closed
C
∗
{C^{\ast }}
-subalgebra
N
N
of
M
M
. Denote by
F
φ
{\mathcal {F}_\varphi }
the set of normal linear functionals
ψ
\psi
on
M
M
extending
φ
\varphi
with
|
|
ψ
|
|
=
|
|
φ
|
|
||\psi || = ||\varphi ||
. It is shown that there exists a partial isometry
v
v
in
N
N
such that
\[
φ
=
|
φ
|
(
v
⋅
)
,
|
φ
|
=
φ
(
v
∗
⋅
)
,
|
|
φ
|
|
=
|
|
|
φ
|
|
|
\varphi = |\varphi |(v \cdot ),\qquad |\varphi | = \varphi ({v^{\ast }} \cdot ),\qquad ||\varphi || = |||\varphi |||
\]
and
\[
ψ
=
|
ψ
|
(
v
⋅
)
,
|
ψ
|
=
ψ
(
v
∗
⋅
)
,
|
|
ψ
|
|
=
|
|
|
ψ
|
|
|
\psi = |\psi |(v\cdot ),\qquad |\psi | = \psi ({v^{\ast }}\cdot ),\qquad ||\psi || = |||\psi |||
\]
for all
ψ
\psi
in
F
φ
{\mathcal {F}_\varphi }
, where
|
φ
|
|\varphi |
and
|
ψ
|
|\psi |
denote the absolute values of
φ
\varphi
and
ψ
\psi
respectively. Let
A
A
be a
C
∗
{C^{\ast }}
-algebra and let
B
B
be a
C
∗
{C^{\ast }}
-subalgebra of
A
A
. As a consequence of this result, we obtain that every state on
B
B
has a unique state extension to
A
A
if and only if every bounded linear functional on
B
B
has a unique norm-preserving extension to a bounded linear functional on
A
A
.