104
5 Macroscopic Limits
L
∞
( ˆ
μ ω ) with p ω M
G imply that the restriction of the mapping r M onto supp ˆ
μ ω is a
bijection onto
supp p ω := {m ∈ M
: m( p ω ) = 1}.
(5.1.149)
Denote ω m := ϕ iff r M ϕ = m ∈ M
. Let μ ω be the image of ˆ
μ ω under r M :
μ ω := ˆ
μ ω ◦ r
−1
M
is a regular Borel measure on M
.
(5.1.150)
Since p M ω = r M ˜
ω, μ ω is the measure specified in (v). The measurability of
the function ˆ
ω : m → ω m defined on supp p ω = supp μ ω is clear, compare [53,
4.1.36]. The integral in (5.1.146) is then another form of (5.1.147). This concludes the
proof.
5.1.39 Note. Let r A : S((A
)
∗∗
) → S(A
) be the restriction mapping. Let e ∗ :
S(A
) → S ∗ ((A
)
∗∗
) be the normal extension, e ∗ ω = ˜
ω. Then p M = r M ◦ e ∗ . For
a general ϕ ∈ S((A
)
∗∗
), it is
r M ϕ = ( p M ◦ r A )ϕ.
(5.1.151)
Since ω m ∈ supp ˆ
μ ω need not be normal, the inequality (5.1.151) holds also for
ϕ = ω m in general. The open question is, however, whether (under some conditions)
( p M ◦ r A )ω m ∈ M
= ES(M
G ) or, at least, when the canonical measure corresponding to ( p M ◦ r A )ω m ∈ S(C(M
)) is concentrated on a set F
−1
g (F) for some
F ∈ M
G , where
F g : M
→ ˙
M
G is the natural mapping defined according to (5.1.87) ∧ (5.1.93),
and ˙
M
G is the one-point compactification of M
G . Let us write down the definition of F g explicitly (see also proof of 5.1.19):
(*) Let E
g be the projection-valued measure defined on Borel subsets of g
∗ with values in Z, as determined in 5.1.29 and in 5.1.33. Let ˙
g
∗
:= g
∗
∪ {∞} be the one-point
compactification of g
∗ and ˙
M
G
:= ˙
g
∗
∪ {m ◦ }, where m ◦ is an isolated point. Let
M
:= ES(M
G ) be the spectrum space of the algebra M
G = C(M
) generated
by projectors E
g (B) (Borel B ⊂ g
∗ ), i.e. by continuous functions m → m(E
g (B)),
m ∈ Z := ES(Z). Define the (continuous) mapping F g : M
→ ˙
M
G by
(i) F g (m) ∈ g
∗ iff there is a bounded Borel B ⊂ g
∗ such that
ω m (E
g (B)) ≡ m(E
g (B)) = 1,
(5.1.152)
and, in this case, F g (m)(ξ) := m(X ξ E
g (B)) for all ξ ∈ g. Here X ξ are
defined in 5.1.29. The character property of m ensures independency of F g (m)
on B satisfying (5.1.152).
5 Macroscopic Limits
L
∞
( ˆ
μ ω ) with p ω M
G imply that the restriction of the mapping r M onto supp ˆ
μ ω is a
bijection onto
supp p ω := {m ∈ M
: m( p ω ) = 1}.
(5.1.149)
Denote ω m := ϕ iff r M ϕ = m ∈ M
. Let μ ω be the image of ˆ
μ ω under r M :
μ ω := ˆ
μ ω ◦ r
−1
M
is a regular Borel measure on M
.
(5.1.150)
Since p M ω = r M ˜
ω, μ ω is the measure specified in (v). The measurability of
the function ˆ
ω : m → ω m defined on supp p ω = supp μ ω is clear, compare [53,
4.1.36]. The integral in (5.1.146) is then another form of (5.1.147). This concludes the
proof.
5.1.39 Note. Let r A : S((A
)
∗∗
) → S(A
) be the restriction mapping. Let e ∗ :
S(A
) → S ∗ ((A
)
∗∗
) be the normal extension, e ∗ ω = ˜
ω. Then p M = r M ◦ e ∗ . For
a general ϕ ∈ S((A
)
∗∗
), it is
r M ϕ = ( p M ◦ r A )ϕ.
(5.1.151)
Since ω m ∈ supp ˆ
μ ω need not be normal, the inequality (5.1.151) holds also for
ϕ = ω m in general. The open question is, however, whether (under some conditions)
( p M ◦ r A )ω m ∈ M
= ES(M
G ) or, at least, when the canonical measure corresponding to ( p M ◦ r A )ω m ∈ S(C(M
)) is concentrated on a set F
−1
g (F) for some
F ∈ M
G , where
F g : M
→ ˙
M
G is the natural mapping defined according to (5.1.87) ∧ (5.1.93),
and ˙
M
G is the one-point compactification of M
G . Let us write down the definition of F g explicitly (see also proof of 5.1.19):
(*) Let E
g be the projection-valued measure defined on Borel subsets of g
∗ with values in Z, as determined in 5.1.29 and in 5.1.33. Let ˙
g
∗
:= g
∗
∪ {∞} be the one-point
compactification of g
∗ and ˙
M
G
:= ˙
g
∗
∪ {m ◦ }, where m ◦ is an isolated point. Let
M
:= ES(M
G ) be the spectrum space of the algebra M
G = C(M
) generated
by projectors E
g (B) (Borel B ⊂ g
∗ ), i.e. by continuous functions m → m(E
g (B)),
m ∈ Z := ES(Z). Define the (continuous) mapping F g : M
→ ˙
M
G by
(i) F g (m) ∈ g
∗ iff there is a bounded Borel B ⊂ g
∗ such that
ω m (E
g (B)) ≡ m(E
g (B)) = 1,
(5.1.152)
and, in this case, F g (m)(ξ) := m(X ξ E
g (B)) for all ξ ∈ g. Here X ξ are
defined in 5.1.29. The character property of m ensures independency of F g (m)
on B satisfying (5.1.152).