6.3 Time Evolution in Generalized Mean-Field Theories
143
are Borel functions on N for any ω ∈ p G S ∗ (A
∗∗
). Since the function F g in (6.3.17)
is continuous, the measurability of the functions
m → ω m ( f ◦ F g (m)), f ∈ C bs ,
(6.3.19)
can be proved with a help of a sequence F
(n)
g of functions from N into the one point
compactification ˙
g
∗ of g
∗ assuming each only a finite number of values and pointwise
converging to F g in the natural topology of ˙
g
∗ . Then the functions
m → ω m ( f ◦ F
(n)
g (m)), f ∈ C bs , ω ∈ p G S ∗ (A
∗∗
)
(6.3.20)
are finite sums of functions of the form (6.3.18), hence the functions (6.3.20) are
measurable. The s
∗ -continuity of f implies then the pointwise convergence of the
functions (6.3.20) to the function (6.3.19) for n → ∞. According to a known theorem
in measure theory, cf. e.g. [223, 6.l0.VII.], the pointwise limit of uniformly bounded
measurable functions is measurable, hence (6.3.19) are Borel functions. We have
proved the existence of the integrals in (6.3.16) for any f ∈ C bs . The function
E g ( f ) : ω (∈ S(A)) → ω(E g ( f )) ∈ C,
(6.3.21)
is affine: The extension mapping e ∗ of S(A) onto S ∗ (A
∗∗ ) is affine, and the association
of subcentral (hence orthogonal, hence regular Borel) measures to the states ω ∈
S ∗ (A
∗∗
) defined by (cf. 5.1.147)
ˆ
μ : ω → ˆ
μ ω ∈ {probability measures on S(A
∗∗
)},
(6.3.22)
where the measure ˆ
μ ω corresponds to the decomposition of ω ∈ S(A
∗∗
) given by the
commutative subalgebra π ω (M G )
in L(H ω ) (cf. [53, 4.1.25.], and for the definition
of M G see 5.2.10), is also affine. The affinity of (6.3.22) can be proved on the basis of
the fact that all the measures in (6.3.22) are obtained from the same algebra M G ⊂ Z
by considering ˆ
μ ω and λ 1 ˆ
μ ω 1 + λ 2 ˆ
μ ω 2 (with ω := λ 1 ω 1 + λ 2 ω 2 ) as limits of the nets
of measures which correspond to the net of finite dimensional subalgebras of M G ,
compare Lemma 4.1.26 in [53]:
ω(x) =
j
ω( p j x) = λ 1
j
ω 1 ( p j x) + λ 2
j
ω 2 ( p j x),
j
p j = id A ,
(6.3.23)
for any finite set of mutually orthogonal projectors p j ∈ M G . Hence, (6.3.22) is an
affine mapping:
ˆ
μ ω = λ 1 ˆ
μ ω 1 + λ 2 ˆ
μ ω 2 , for ω := λ 1 ω 1 + λ 2 ω 2 .
(6.3.24)
The relation (6.3.24) has a unique extension to all ω j ∈ A
∗
(λ j ∈ C). Writing
for ω ∈ p G S(A):
143
are Borel functions on N for any ω ∈ p G S ∗ (A
∗∗
). Since the function F g in (6.3.17)
is continuous, the measurability of the functions
m → ω m ( f ◦ F g (m)), f ∈ C bs ,
(6.3.19)
can be proved with a help of a sequence F
(n)
g of functions from N into the one point
compactification ˙
g
∗ of g
∗ assuming each only a finite number of values and pointwise
converging to F g in the natural topology of ˙
g
∗ . Then the functions
m → ω m ( f ◦ F
(n)
g (m)), f ∈ C bs , ω ∈ p G S ∗ (A
∗∗
)
(6.3.20)
are finite sums of functions of the form (6.3.18), hence the functions (6.3.20) are
measurable. The s
∗ -continuity of f implies then the pointwise convergence of the
functions (6.3.20) to the function (6.3.19) for n → ∞. According to a known theorem
in measure theory, cf. e.g. [223, 6.l0.VII.], the pointwise limit of uniformly bounded
measurable functions is measurable, hence (6.3.19) are Borel functions. We have
proved the existence of the integrals in (6.3.16) for any f ∈ C bs . The function
E g ( f ) : ω (∈ S(A)) → ω(E g ( f )) ∈ C,
(6.3.21)
is affine: The extension mapping e ∗ of S(A) onto S ∗ (A
∗∗ ) is affine, and the association
of subcentral (hence orthogonal, hence regular Borel) measures to the states ω ∈
S ∗ (A
∗∗
) defined by (cf. 5.1.147)
ˆ
μ : ω → ˆ
μ ω ∈ {probability measures on S(A
∗∗
)},
(6.3.22)
where the measure ˆ
μ ω corresponds to the decomposition of ω ∈ S(A
∗∗
) given by the
commutative subalgebra π ω (M G )
in L(H ω ) (cf. [53, 4.1.25.], and for the definition
of M G see 5.2.10), is also affine. The affinity of (6.3.22) can be proved on the basis of
the fact that all the measures in (6.3.22) are obtained from the same algebra M G ⊂ Z
by considering ˆ
μ ω and λ 1 ˆ
μ ω 1 + λ 2 ˆ
μ ω 2 (with ω := λ 1 ω 1 + λ 2 ω 2 ) as limits of the nets
of measures which correspond to the net of finite dimensional subalgebras of M G ,
compare Lemma 4.1.26 in [53]:
ω(x) =
j
ω( p j x) = λ 1
j
ω 1 ( p j x) + λ 2
j
ω 2 ( p j x),
j
p j = id A ,
(6.3.23)
for any finite set of mutually orthogonal projectors p j ∈ M G . Hence, (6.3.22) is an
affine mapping:
ˆ
μ ω = λ 1 ˆ
μ ω 1 + λ 2 ˆ
μ ω 2 , for ω := λ 1 ω 1 + λ 2 ω 2 .
(6.3.24)
The relation (6.3.24) has a unique extension to all ω j ∈ A
∗
(λ j ∈ C). Writing
for ω ∈ p G S(A):