92
5 Macroscopic Limits
all these m’s are contained in F
−1
g (F ω ) due to (5.1.85). Hence the clopen set supp μ ω
contains exactly one point of M ∗ which means, according to Proposition 5.1.16, that
supp μ ω is a one point subset of M ∗ and s ω = E g (F ω ). This proves the implication
(i) ⇒ (ii).
5.1.20 Corollary. σ
∗
G E g = E g , i.e. E g is σ
∗
G -invariant (:= ‘G-invariant’).
Proof. According to (5.1.62) and Lemma 5.1.13, the set N ∗ is G-invariant. The
action of G (via σ
∗
G ) commutes with the mapping ω (∈ S(A
)) → ω (∈ S ∗ (M G )).
Then the result is immediate after an application of 5.1.19.
5.1.21 Proposition. For any ω ∈ S ∗ (M G ), the corresponding probability Radon
measure μ ω on M is supported by M ∗ :
μ ω (M ∗ ) = μ ω (M) = 1.
(5.1.95)
Proof. We can assume that s G s ω = s ω for the support projector s ω of ω. We have,
according to 5.1.16 (i), s ω = c(s ω g
∗
). Due to normality of ω, it is
1 = ω(s ω ) = l.u.b.{ω( p J ) : p J :=
F∈J
E g (F), finite J ⊂ s ω g
∗
}.
(5.1.96)
Let m F ∈ M ∗ (F ∈ g
∗
, E g (F) = 0) be defined by m F (E g (F)) = 1. For any
subset K ⊂ g
∗
, the open set (which is clopen for finite K )
M(K ) := {m F ∈ M ∗ : F ∈ K }, m F is void if E g (F) = 0,
(5.1.97)
is μ ω −measurable. But ω( p J ) = μ ω (M(J )), and μ ω is regular. Hence,
1 = l.u.b.{μ ω (M(J )) : J ⊂ s ω g
∗ finite} ≤ μ ω (M(s ω g
∗
)) ≤ μ ω (M ∗ ) ≤ 1,
(5.1.98)
what proves (5.1.95).
5.1.22 Lemma. Any uniformly bounded function on M ∗ with values in C can be
uniquely extended to a continuous function on M, i.e. the spectrum space M of M G
is the Stone- ˇ
Cech compactification of the discrete space M ∗ of normal pure states
on M G .
Proof. Since M ∗ is discrete, C(M ∗ ) consists of all bounded complex valued functions on M ∗ . The Stone- ˇ
Cech compactification of a normal topological space S is a
compact Hausdorff space S
and a homeomorphism τ of S into S
such, that τ (S) is
dense in S
and any f ∈ C(S) can be continued to some ˜
f ∈ C(S
). It is clear, that
the continuation ˜
f is uniquely determined by f .
Let f ∈ C(M ∗ ), f ≥ 0. For any ι ∈ [0, f ] (:= closed interval in R) define (cf.
(5.1.68)) p ι := 0 for ι /
∈ sp( f ) and (let f (m ◦ ) = 0):
p ι := c({F m ∈ g
∗
: f (m) = ι, m ∈ M ∗ }), ι ∈ sp( f ),
(5.1.99)
5 Macroscopic Limits
all these m’s are contained in F
−1
g (F ω ) due to (5.1.85). Hence the clopen set supp μ ω
contains exactly one point of M ∗ which means, according to Proposition 5.1.16, that
supp μ ω is a one point subset of M ∗ and s ω = E g (F ω ). This proves the implication
(i) ⇒ (ii).
5.1.20 Corollary. σ
∗
G E g = E g , i.e. E g is σ
∗
G -invariant (:= ‘G-invariant’).
Proof. According to (5.1.62) and Lemma 5.1.13, the set N ∗ is G-invariant. The
action of G (via σ
∗
G ) commutes with the mapping ω (∈ S(A
)) → ω (∈ S ∗ (M G )).
Then the result is immediate after an application of 5.1.19.
5.1.21 Proposition. For any ω ∈ S ∗ (M G ), the corresponding probability Radon
measure μ ω on M is supported by M ∗ :
μ ω (M ∗ ) = μ ω (M) = 1.
(5.1.95)
Proof. We can assume that s G s ω = s ω for the support projector s ω of ω. We have,
according to 5.1.16 (i), s ω = c(s ω g
∗
). Due to normality of ω, it is
1 = ω(s ω ) = l.u.b.{ω( p J ) : p J :=
F∈J
E g (F), finite J ⊂ s ω g
∗
}.
(5.1.96)
Let m F ∈ M ∗ (F ∈ g
∗
, E g (F) = 0) be defined by m F (E g (F)) = 1. For any
subset K ⊂ g
∗
, the open set (which is clopen for finite K )
M(K ) := {m F ∈ M ∗ : F ∈ K }, m F is void if E g (F) = 0,
(5.1.97)
is μ ω −measurable. But ω( p J ) = μ ω (M(J )), and μ ω is regular. Hence,
1 = l.u.b.{μ ω (M(J )) : J ⊂ s ω g
∗ finite} ≤ μ ω (M(s ω g
∗
)) ≤ μ ω (M ∗ ) ≤ 1,
(5.1.98)
what proves (5.1.95).
5.1.22 Lemma. Any uniformly bounded function on M ∗ with values in C can be
uniquely extended to a continuous function on M, i.e. the spectrum space M of M G
is the Stone- ˇ
Cech compactification of the discrete space M ∗ of normal pure states
on M G .
Proof. Since M ∗ is discrete, C(M ∗ ) consists of all bounded complex valued functions on M ∗ . The Stone- ˇ
Cech compactification of a normal topological space S is a
compact Hausdorff space S
and a homeomorphism τ of S into S
such, that τ (S) is
dense in S
and any f ∈ C(S) can be continued to some ˜
f ∈ C(S
). It is clear, that
the continuation ˜
f is uniquely determined by f .
Let f ∈ C(M ∗ ), f ≥ 0. For any ι ∈ [0, f ] (:= closed interval in R) define (cf.
(5.1.68)) p ι := 0 for ι /
∈ sp( f ) and (let f (m ◦ ) = 0):
p ι := c({F m ∈ g
∗
: f (m) = ι, m ∈ M ∗ }), ι ∈ sp( f ),
(5.1.99)