5.1 Multiple Systems
91
m(E ξ (B n ) f (X ξ )) = f (F m (ξ)) for m ∈ M ξn ,
(5.1.88)
if for λ = F m (ξ) the value f (λ) is defined. From (5.1.81), one sees that the functions
m → χ ξn (m) f (F m (ξ)) are in L
1
(M, μ ω ). By an application of the Beppo-Levi theorem to their absolute values, we obtain:
The functions F
∗
ξ f ∈ L
1
(M, μ ω ); here it is
F
∗
ξ f := f ◦ F ξ : M ξ → R, m → f (F ξ (m)) = f (F m (ξ)).
(5.1.89)
We have used here (5.1.80) and (5.1.81). After a subsequent application of the
Lebesgue dominated convergence theorem we arrive at:
ω( f (X ξ )) = μ ω (F
∗
ξ f ) :=
M
f (F m (ξ)) μ ω (dm).
(5.1.90)
The relation (5.1.77) is valid due to (i). This means that the functions f 1 (λ) :=
λ, f 2 (λ) := λ
2
, (λ ∈ R), are both in L
1
(R, μ
ω
ξ ) for all ξ ∈ g and for f := f j ( j =
1, 2) (5.1.90) is valid. Hence F ξ ∈ L
2
(M, μ ω ) for all ξ ∈ g and, due to (5.1.77), we
have
(F ξ , F ξ ) = (F ξ , 1)(1, F ξ ), for all ξ ∈ g.
(5.1.91)
The brackets denote here here the scalar product in L
2
(M, μ ω ) and 1 ∈ L
2
(M, μ ω )
is the function identically equal to one: 1(m) := 1 for all m ∈ M. Applying the
Schwarz inequality to (5.1.91), we obtain:
F ξ = const. = (1, F ξ )1 = F ω (ξ)1, μ ω -a.e. for all ξ ∈ g.
(5.1.92)
This means that the function
F g : M g → g
∗
, m → F m ,
(5.1.93)
is constant μ ω -almost everywhere, too. The restriction of F g to the set of normal
states N ∗ separates points in N ∗ according to 5.1.13 and (5.1.87). Hence the set
F
−1
g (F ω ) ⊂ M g contains at most one m ∈ N ∗ . Due to continuity of F g , the set
F
−1
g (F ω ) is closed in M g = M
◦
g (:= the interior of M g ), what implies measurability
of F
−1
g (F ω ). Due to (5.1.92):
μ ω (F
−1
g (F ω )) = μ ω (M) = 1.
(5.1.94)
It is known, see e.g. [274], that for any ω ∈ S ∗ (M G ) there is a unique projector
s ω ∈ M G such that ω(x) = ω(xs ω ) for all x ∈ M G and ω(x
∗ x) = 0 implies xs ω = 0.
The characteristic function in C(M) corresponding to s ω is supported by the clopen
set supp μ ω ⊂ M. Since it is nonempty, it contains some m ∈ M ∗ \{m ◦ } = N ∗ , and
91
m(E ξ (B n ) f (X ξ )) = f (F m (ξ)) for m ∈ M ξn ,
(5.1.88)
if for λ = F m (ξ) the value f (λ) is defined. From (5.1.81), one sees that the functions
m → χ ξn (m) f (F m (ξ)) are in L
1
(M, μ ω ). By an application of the Beppo-Levi theorem to their absolute values, we obtain:
The functions F
∗
ξ f ∈ L
1
(M, μ ω ); here it is
F
∗
ξ f := f ◦ F ξ : M ξ → R, m → f (F ξ (m)) = f (F m (ξ)).
(5.1.89)
We have used here (5.1.80) and (5.1.81). After a subsequent application of the
Lebesgue dominated convergence theorem we arrive at:
ω( f (X ξ )) = μ ω (F
∗
ξ f ) :=
M
f (F m (ξ)) μ ω (dm).
(5.1.90)
The relation (5.1.77) is valid due to (i). This means that the functions f 1 (λ) :=
λ, f 2 (λ) := λ
2
, (λ ∈ R), are both in L
1
(R, μ
ω
ξ ) for all ξ ∈ g and for f := f j ( j =
1, 2) (5.1.90) is valid. Hence F ξ ∈ L
2
(M, μ ω ) for all ξ ∈ g and, due to (5.1.77), we
have
(F ξ , F ξ ) = (F ξ , 1)(1, F ξ ), for all ξ ∈ g.
(5.1.91)
The brackets denote here here the scalar product in L
2
(M, μ ω ) and 1 ∈ L
2
(M, μ ω )
is the function identically equal to one: 1(m) := 1 for all m ∈ M. Applying the
Schwarz inequality to (5.1.91), we obtain:
F ξ = const. = (1, F ξ )1 = F ω (ξ)1, μ ω -a.e. for all ξ ∈ g.
(5.1.92)
This means that the function
F g : M g → g
∗
, m → F m ,
(5.1.93)
is constant μ ω -almost everywhere, too. The restriction of F g to the set of normal
states N ∗ separates points in N ∗ according to 5.1.13 and (5.1.87). Hence the set
F
−1
g (F ω ) ⊂ M g contains at most one m ∈ N ∗ . Due to continuity of F g , the set
F
−1
g (F ω ) is closed in M g = M
◦
g (:= the interior of M g ), what implies measurability
of F
−1
g (F ω ). Due to (5.1.92):
μ ω (F
−1
g (F ω )) = μ ω (M) = 1.
(5.1.94)
It is known, see e.g. [274], that for any ω ∈ S ∗ (M G ) there is a unique projector
s ω ∈ M G such that ω(x) = ω(xs ω ) for all x ∈ M G and ω(x
∗ x) = 0 implies xs ω = 0.
The characteristic function in C(M) corresponding to s ω is supported by the clopen
set supp μ ω ⊂ M. Since it is nonempty, it contains some m ∈ M ∗ \{m ◦ } = N ∗ , and