110
5 Macroscopic Limits
it is bounded: E L ( f ) ≤ ≤ f := sup{| f (F)| : F ∈ g
∗
}. Due to continuity of the
product in the strong topology, the mapping E L is a C
∗ -homomorphism of the commutative C
∗ -algebra B(g
∗ ) into Z. This implies the σ-additivity of the set-function
E L : B → E L (B) := E L (χ B ), B ∈ G , hence, E L is a projection measure. Since
σ g ∈
∗ - Aut Z, and any automorphism of a W
∗ -algebra is σ-σ-continuous, E L is a
G-measure. Clearly E L ≥ E, ∀E ∈ L.
5.2.8 Proposition. The directed set of classes of purely n-dimensional G-measures
has a maximal element.
7
Proof. Let J be any linearly ordered subset of the directed set; cf. (5.2.12). We
shall prove that it is possible to choose E ∈ [E] in any [E] ∈ J in such a way that
[E] ≺ [E
] iff E
= Es E
. Then the result will follow from the Lemma 5.2.7 and
from the Zorn lemma. It is clear that the choice E ∈ [E] of the desired kind can be
made in any finite subset K ◦ ⊂ J, [E] ∈ K ◦ .
The desired choice (it will be called a ‘consistent choice’) can be made in the subset
K E := {[E
] ∈ J : [E
] ≺ [E]} of J by E
:= p E E for any [E] ∈ J , with any fixed
E ∈ [E]. We have to prove existence of a consistent choice on the whole J . Let J ◦
be a well ordered cofinal subset of J (the well ordering of J ◦ is that one induced by
the ordering of J –it is possible by the axiom of choice, and cofinality means that for
any [E] ∈ J there is an [E j ] ∈ J ◦ : [E] ≺ [E j ]). Now we can choose E j ∈ [E j ] (for
all [E j ] ∈ J ◦ ) in a consistent way: For the successor [E j+1 ] of [E j ] in J ◦ we shall
choose E j+1 := E j s E
j+1 with any E
j+1 ∈ [E j+1 ], if [E j ] has been defined before.
If [E j ] is not a successor in J ◦ , put E
◦
j := l.u.b.{E k : [E j ] ] [E k ] ∈ J ◦ , all E k (∈
[E k ]) are mutually consistent}, according to the Lemma 5.2.7, and choose E j :=
E
◦
j s E
j with any E
j ∈ [E j ]. Then we can ‘to fill gaps’ by setting E := p E E j for all
[E] ≺ [E j ] ∈ J ◦ . This provides a consistent choice E ∈ [E] for all [E] ∈ J , if J ◦ is
considered as an initial segment of the set of all ordinals.
Note: The same proof applies to purely n-dimensional measures E of the form
E = q E for any fixed G−invariant projector q ∈ Z.
5.2.9 Let [E]
◦
G be the maximal element of classes of purely n G -dimensional
G-measures and let p
◦
G := p E for E ∈ [E]
◦
G . Let [E]
k
G be the maximal element
of classes of purely (n G − 2k)-dimensional G-measures of the form E = (I −
k−1
j=0 p
j
G )E, and for E ∈ [E]
k
G let p
k
G := p E , k = 1, 2, . . . ,
n G
2
. Define now the
class [E] G of maximal G-measures by
[E] G :=
n G
2
k=0
[E]
k
G , with E ∈ [E] G iff E =
n G
2
k=0
E k , E k ∈ [E]
k
G ,
and the sum of mutually orthogonal G-measures is defined in (5.2.8). The choice
of measures E ∈ [E] G for the realization of macroscopic limits corresponds to a
7 The present author was informed about some important set-theoretical concepts connected with
this Proposition by the late colleague Ivan Korec (1943–1998).
5 Macroscopic Limits
it is bounded: E L ( f ) ≤ ≤ f := sup{| f (F)| : F ∈ g
∗
}. Due to continuity of the
product in the strong topology, the mapping E L is a C
∗ -homomorphism of the commutative C
∗ -algebra B(g
∗ ) into Z. This implies the σ-additivity of the set-function
E L : B → E L (B) := E L (χ B ), B ∈ G , hence, E L is a projection measure. Since
σ g ∈
∗ - Aut Z, and any automorphism of a W
∗ -algebra is σ-σ-continuous, E L is a
G-measure. Clearly E L ≥ E, ∀E ∈ L.
5.2.8 Proposition. The directed set of classes of purely n-dimensional G-measures
has a maximal element.
7
Proof. Let J be any linearly ordered subset of the directed set; cf. (5.2.12). We
shall prove that it is possible to choose E ∈ [E] in any [E] ∈ J in such a way that
[E] ≺ [E
] iff E
= Es E
. Then the result will follow from the Lemma 5.2.7 and
from the Zorn lemma. It is clear that the choice E ∈ [E] of the desired kind can be
made in any finite subset K ◦ ⊂ J, [E] ∈ K ◦ .
The desired choice (it will be called a ‘consistent choice’) can be made in the subset
K E := {[E
] ∈ J : [E
] ≺ [E]} of J by E
:= p E E for any [E] ∈ J , with any fixed
E ∈ [E]. We have to prove existence of a consistent choice on the whole J . Let J ◦
be a well ordered cofinal subset of J (the well ordering of J ◦ is that one induced by
the ordering of J –it is possible by the axiom of choice, and cofinality means that for
any [E] ∈ J there is an [E j ] ∈ J ◦ : [E] ≺ [E j ]). Now we can choose E j ∈ [E j ] (for
all [E j ] ∈ J ◦ ) in a consistent way: For the successor [E j+1 ] of [E j ] in J ◦ we shall
choose E j+1 := E j s E
j+1 with any E
j+1 ∈ [E j+1 ], if [E j ] has been defined before.
If [E j ] is not a successor in J ◦ , put E
◦
j := l.u.b.{E k : [E j ] ] [E k ] ∈ J ◦ , all E k (∈
[E k ]) are mutually consistent}, according to the Lemma 5.2.7, and choose E j :=
E
◦
j s E
j with any E
j ∈ [E j ]. Then we can ‘to fill gaps’ by setting E := p E E j for all
[E] ≺ [E j ] ∈ J ◦ . This provides a consistent choice E ∈ [E] for all [E] ∈ J , if J ◦ is
considered as an initial segment of the set of all ordinals.
Note: The same proof applies to purely n-dimensional measures E of the form
E = q E for any fixed G−invariant projector q ∈ Z.
5.2.9 Let [E]
◦
G be the maximal element of classes of purely n G -dimensional
G-measures and let p
◦
G := p E for E ∈ [E]
◦
G . Let [E]
k
G be the maximal element
of classes of purely (n G − 2k)-dimensional G-measures of the form E = (I −
k−1
j=0 p
j
G )E, and for E ∈ [E]
k
G let p
k
G := p E , k = 1, 2, . . . ,
n G
2
. Define now the
class [E] G of maximal G-measures by
[E] G :=
n G
2
k=0
[E]
k
G , with E ∈ [E] G iff E =
n G
2
k=0
E k , E k ∈ [E]
k
G ,
and the sum of mutually orthogonal G-measures is defined in (5.2.8). The choice
of measures E ∈ [E] G for the realization of macroscopic limits corresponds to a
7 The present author was informed about some important set-theoretical concepts connected with
this Proposition by the late colleague Ivan Korec (1943–1998).