In the first three sections the question of when a pure state g on a
C
∗
{C^{\ast }}
-subalgebra B of a
C
∗
{C^{\ast }}
-algebra A has a unique state extension is studied. It is shown that an extension f is unique if and only if inf
‖
b
(
a
−
f
(
a
)
1
)
b
‖
=
0
\left \| {b\left ( {a\, - \,f\left ( a \right )1} \right )b} \right \|\, = \,0
for each a in A, where the inf is taken over those b in B such that
0
⩽
b
⩽
1
0\, \leqslant \,b\, \leqslant \,1
and
g
(
b
)
=
1
g(b) = 1
. The special cases where B is maximal abelian and/or
A
=
B
(
H
)
A\, = \,B\left ( H \right )
are treated in more detail. In the remaining sections states of the form
T
↦
lim
u
(
T
x
α
,
x
α
)
T \mapsto \lim \limits _{\mathcal {u}} \left ( {T{x_\alpha },\,{x_\alpha }} \right )
, where
{
x
α
}
α
∈
κ
\left \{ {{x_\alpha }} \right \}{\,_{\alpha \, \in \,\kappa }}
is a set of unit vectors in H and
u
\mathcal {u}
is an ultrafilter are studied.