94
5 Macroscopic Limits
measure on the set of all subsets of s G g
∗ . Conversely, any finitely additive probability
measure on s G g
∗ is of the form μ
ω
g for some ω ∈ S(N G ).
Proof. For any subset K ⊂ g
∗ define, (5.1.68),
E g (K ) := c(K ) :=
F∈K
E g (F).
(5.1.104)
Then E g (K ) is a projector in N G and we can define
μ
ω
g (K ) := ω(E g (K )) for any K ⊂ g
∗ and any ω ∈ S(N G ).
(5.1.105)
It is easily to see that μ
ω
g in (5.1.105) is the desired unique extension. For the proof
of the second assertion, choose any finitely additive probability measure μ on s G g
∗
,
μ defined on all subsets K of s G g
∗ . Define a positive linear functional on N G , ω μ ,
by its values on all projectors:
ω μ (E g (K )) := μ(K ),
(5.1.106)
compare 5.1.16. The von Neumann algebra N G is generated by the set of all its
projectors and (5.1.106) defines uniquely a state on N G .
5.1.25 Let us look what measures μ
ω
g correspond to pure states ω ∈ M, which are
not normal. From the character property of pure states we have ω(E g (K 1 ∩ K 2 )) =
ω(E g (K 1 )E g (K 2 )) = ω(E g (K 1 ))ω(E g (K 2 )) what together with finite additivity
gives:
K ⊂ g
∗
⇒ μ
ω
g (K ) ∈ {0, 1}.
(5.1.107)
Remember that supp μ
ω
g ⊂ s G g
∗
. Any finitely additive measure μ on s G g
∗ satisfying
(5.1.107) corresponds to a pure state ω μ ∈ M. It determines also an ultrafilter on
s G g
∗ consisting of all subsets K for which it is μ(K ) = 1. This is clearly a bijection
between the set of all ultrafilters on s G g
∗ and the set of pure states
ES(N G ) = N .
Remember that to any m ∈ M corresponds the Dirac measure δ m on M which
is concentrated at a point m. For a nonnormal m the measure μ
m
g is not concentrated
at any point in g
∗ .
5.1.26 Let us keep in mind that we have associated with any state ω ∈ S(A
) a
state (equally denoted) ω ∈ S(M G ) which is the restriction to M G of the unique w
∗ -
continuous extension to (A
)
∗∗ of ω ∈ S(A ). Such an ω ∈ S(M G ) is necessarily
normal: ω ∈ S ∗ (M G ), and the corresponding measure μ
ω
g := μ ω ◦ F
−1
g
is purely
atomic, 5.1.23. This reflects that fact that the described procedure maps into S(N G )
only such states on (A
)
∗∗ which are describable by density matrices in L(P G H ).
5 Macroscopic Limits
measure on the set of all subsets of s G g
∗ . Conversely, any finitely additive probability
measure on s G g
∗ is of the form μ
ω
g for some ω ∈ S(N G ).
Proof. For any subset K ⊂ g
∗ define, (5.1.68),
E g (K ) := c(K ) :=
F∈K
E g (F).
(5.1.104)
Then E g (K ) is a projector in N G and we can define
μ
ω
g (K ) := ω(E g (K )) for any K ⊂ g
∗ and any ω ∈ S(N G ).
(5.1.105)
It is easily to see that μ
ω
g in (5.1.105) is the desired unique extension. For the proof
of the second assertion, choose any finitely additive probability measure μ on s G g
∗
,
μ defined on all subsets K of s G g
∗ . Define a positive linear functional on N G , ω μ ,
by its values on all projectors:
ω μ (E g (K )) := μ(K ),
(5.1.106)
compare 5.1.16. The von Neumann algebra N G is generated by the set of all its
projectors and (5.1.106) defines uniquely a state on N G .
5.1.25 Let us look what measures μ
ω
g correspond to pure states ω ∈ M, which are
not normal. From the character property of pure states we have ω(E g (K 1 ∩ K 2 )) =
ω(E g (K 1 )E g (K 2 )) = ω(E g (K 1 ))ω(E g (K 2 )) what together with finite additivity
gives:
K ⊂ g
∗
⇒ μ
ω
g (K ) ∈ {0, 1}.
(5.1.107)
Remember that supp μ
ω
g ⊂ s G g
∗
. Any finitely additive measure μ on s G g
∗ satisfying
(5.1.107) corresponds to a pure state ω μ ∈ M. It determines also an ultrafilter on
s G g
∗ consisting of all subsets K for which it is μ(K ) = 1. This is clearly a bijection
between the set of all ultrafilters on s G g
∗ and the set of pure states
ES(N G ) = N .
Remember that to any m ∈ M corresponds the Dirac measure δ m on M which
is concentrated at a point m. For a nonnormal m the measure μ
m
g is not concentrated
at any point in g
∗ .
5.1.26 Let us keep in mind that we have associated with any state ω ∈ S(A
) a
state (equally denoted) ω ∈ S(M G ) which is the restriction to M G of the unique w
∗ -
continuous extension to (A
)
∗∗ of ω ∈ S(A ). Such an ω ∈ S(M G ) is necessarily
normal: ω ∈ S ∗ (M G ), and the corresponding measure μ
ω
g := μ ω ◦ F
−1
g
is purely
atomic, 5.1.23. This reflects that fact that the described procedure maps into S(N G )
only such states on (A
)
∗∗ which are describable by density matrices in L(P G H ).