100
5 Macroscopic Limits
senting this unitary representation in the standard fashion. The linear space g
∗ can be
identified with the group ˆ
g of characters by the bijection associating with any F ∈ g
∗
the character ξ → exp(i F(ξ)) on g. It is clear that the restriction of E
g on intervals
in g
∗ coincides with (5.1.125).
5.1.34 Lemma. All the nonzero projectors of the form E
g (F) := E g ({F}), F ∈ g
∗ ,
are minimal projectors in N
G and all minimal projectors in N
G are of this form.
Proof. Let q ∈ N
G be a minimal projector. Since q ∈ Z, there is a state ω ∈ S(A
),
the central projector of which is s ω ≤ q. Choose such an ω . Then ω(x) = ω(xs ω ) =
ω(xq) for all x ∈ A , and due to continuity properties of products in (A
)
∗∗ as well
as of the normal extension ω ∈ S ∗ ((A
)
∗∗
), the same is true for all x ∈ (A
)
∗∗
. The
minimality of q in N
G implies that one of the following possibilities (i) or (ii) is
valid
(i) q E
g (B) = q,
(ii) q E
g (B) = 0
(5.1.133)
for any Borel B ⊂ g
∗
. Let us define a probability Borel measure μ
ω
g on g
∗ corresponding to the ω ∈ S
g :
μ
ω
g (B) := p M ω(E
g (B)), for all Borel B ⊂ g
∗
.
(5.1.134)
We see from (5.1.133) that for the chosen ω the values of μ
ω
g lie in the two
point set {0, 1} ⊂ Z + . Each of the projection measures E ξ (ξ ∈ g) and E
g are σadditive, hence N
G is generated by those E
g (B) which correspond to bounded Borel
subsets B of g
∗ . Hence μ
ω
g is concentrated on a compact subset of g
∗ : μ
ω
g (B ◦ ) = 1
for some compact B ◦ . The σ-additivity of μ
ω
g implies then that μ
ω
g is concentrated on
a one-point set F ω ∈ B ◦ :
μ
ω
g ({F ω }) = p M ω(E
g (F ω )) = 1.
(5.1.135)
This implies s ω ≤ E
g (F ω ), and, due to (5.1.133) and due to our choice of s ω :
q ≤ E
g (F ω ).
(5.1.136)
According to the definition of N
G in 5.1.29, q can be approximated, in
σ(N
G , N
G∗ ) topology, by a net j → E
g (B j ), where F ω ∈ B j for all j, due to
(5.1.136). Coming to the Gel’fand representation C(N
) of N
G and considering
that clopen sets in the spectrum space N
form a basis of topology, e.g. 5.1.14
and [274], we see that the sets in N
corresponding to the projectors E
g (B j ) have in
their intersection exactly one point m q ∈ N
corresponding to the minimal projector
q. All of E
g (B j ) contain, however, also E
g (F ω ). This proves that q = E
g (F ω ).
5.1.35 Lemma. If ω ∈ S(A
) is pure or factor state, then also p M ω ∈ S ∗ (M
G ) is
pure.
5 Macroscopic Limits
senting this unitary representation in the standard fashion. The linear space g
∗ can be
identified with the group ˆ
g of characters by the bijection associating with any F ∈ g
∗
the character ξ → exp(i F(ξ)) on g. It is clear that the restriction of E
g on intervals
in g
∗ coincides with (5.1.125).
5.1.34 Lemma. All the nonzero projectors of the form E
g (F) := E g ({F}), F ∈ g
∗ ,
are minimal projectors in N
G and all minimal projectors in N
G are of this form.
Proof. Let q ∈ N
G be a minimal projector. Since q ∈ Z, there is a state ω ∈ S(A
),
the central projector of which is s ω ≤ q. Choose such an ω . Then ω(x) = ω(xs ω ) =
ω(xq) for all x ∈ A , and due to continuity properties of products in (A
)
∗∗ as well
as of the normal extension ω ∈ S ∗ ((A
)
∗∗
), the same is true for all x ∈ (A
)
∗∗
. The
minimality of q in N
G implies that one of the following possibilities (i) or (ii) is
valid
(i) q E
g (B) = q,
(ii) q E
g (B) = 0
(5.1.133)
for any Borel B ⊂ g
∗
. Let us define a probability Borel measure μ
ω
g on g
∗ corresponding to the ω ∈ S
g :
μ
ω
g (B) := p M ω(E
g (B)), for all Borel B ⊂ g
∗
.
(5.1.134)
We see from (5.1.133) that for the chosen ω the values of μ
ω
g lie in the two
point set {0, 1} ⊂ Z + . Each of the projection measures E ξ (ξ ∈ g) and E
g are σadditive, hence N
G is generated by those E
g (B) which correspond to bounded Borel
subsets B of g
∗ . Hence μ
ω
g is concentrated on a compact subset of g
∗ : μ
ω
g (B ◦ ) = 1
for some compact B ◦ . The σ-additivity of μ
ω
g implies then that μ
ω
g is concentrated on
a one-point set F ω ∈ B ◦ :
μ
ω
g ({F ω }) = p M ω(E
g (F ω )) = 1.
(5.1.135)
This implies s ω ≤ E
g (F ω ), and, due to (5.1.133) and due to our choice of s ω :
q ≤ E
g (F ω ).
(5.1.136)
According to the definition of N
G in 5.1.29, q can be approximated, in
σ(N
G , N
G∗ ) topology, by a net j → E
g (B j ), where F ω ∈ B j for all j, due to
(5.1.136). Coming to the Gel’fand representation C(N
) of N
G and considering
that clopen sets in the spectrum space N
form a basis of topology, e.g. 5.1.14
and [274], we see that the sets in N
corresponding to the projectors E
g (B j ) have in
their intersection exactly one point m q ∈ N
corresponding to the minimal projector
q. All of E
g (B j ) contain, however, also E
g (F ω ). This proves that q = E
g (F ω ).
5.1.35 Lemma. If ω ∈ S(A
) is pure or factor state, then also p M ω ∈ S ∗ (M
G ) is
pure.