Let
(
A
,
G
,
τ
)
(A,G,\tau )
be a noncommutative dynamical system, i.e.
A
A
is a
C
∗
{C^{\ast } }
-algebra,
G
G
a topological group and
τ
\tau
an action of
G
G
on
A
A
by
∗
^{\ast }
-automorphisms, and let
(
M
α
)
({M_\alpha })
be an
M
M
-net on
G
G
. We characterize the set of
a
a
in
A
A
such that
M
α
a
{M_\alpha }a
converges in norm. We show that this set is intimately related to the problem of extensions of pure states of R. V. Kadison and I. M. Singer: if
B
B
is a maximal abelian subalgebra of
A
A
, we can associate a dynamical system
(
A
,
G
,
τ
)
(A,G,\tau )
such that
M
α
a
{M_\alpha }a
converges in norm if and only if all extensions to
A
A
, of a homomorphism of
B
B
, coincide on
a
a
. This result allows us to construct different examples of a
C
∗
{C^{\ast } }
-algebra
A
A
with maximal abelian subalgebra
B
B
(isomorphic to
C
(
R
/
Z
)
C({\mathbf {R}}/{\mathbf {Z}})
or
L
∞
[
0
,
1
]
)
{L^\infty }[0,1])
for which the property of unique pure state extension of homomorphisms is or is not verified.