6.3 Time Evolution in Generalized Mean-Field Theories
145
The state ϕ m ∈ S(A), ϕ m (y) := ω m (x
∗
m yx m ) is also supported by E g (B m ) with any
open B m ⊂ K containing F m . Hence the decomposition (5.1.146) of ω := ϕ m is
concentrated on the one point set {m}. This means that
ϕ m (E g ( f )) := ϕ m ( f (F m )) = 0,
(6.3.31)
hence E g ( f ) = 0 for any nonzero f ∈ C bs . This proves the isometry of E g , hence
E g is a C
∗ -isomorphism of C bs into E g (g
∗
)A
∗∗
= p G A
∗∗ . The remaining assertions
are clearly valid.
6.3.7 Lemma. Let f ∈ C bs , ω ∈ p G S(A). Then the function
(g; F) → ω
σ
−1
g ( f (F))
∈ C, (g; F) ∈ G × K ,
(6.3.32)
is jointly continuous on the topological product G × supp E g .
Proof. Let f := f 0 , cf. (6.3.7). Then
σ
−1
g ( f 0 (F)) = σ(g
−1
g 0 (F))(x) f (F),
(6.3.33)
and the joint continuity of the group operation
(g 1 ; g 2 ) (∈ G × G) → g
−1
1 g 2 ∈ G
(6.3.34)
gives the joint continuity in (6.3.32) with f := f 0 . It can be verified directly,
cf. e.g. (6.3.13), that the function in (6.3.33) is even s
∗ -continuous in the couple
(g; F) ∈ G × K . But the finite algebraic combinations as well as the uniform limits of
s
∗ -continuous bounded functions are s
∗ -continuous. Since C
G
bs is generated by functions of the form f 0 , we have proved that the functions
(g; F) → σ
−1
g ( f (F)) ∈ A, for all f ∈ C
G
bs ,
(6.3.35)
are even s
∗ -continuous.
6.3.8 Proposition. Let, with the notation of 6.1.3, be f ∈ C
G
bs , and for a fixed Q ∈
C
∞
(g
∗
, R) and for any t ∈ R, F ∈ K , let
f t (F) := σ(g
−1
Q (t, F))( f (ϕ
Q
t F)).
(6.3.36)
Then f t ∈ C
G
bs and the mappings f → f t form a one-parameter group of
∗ -automorphisms of C
G
bs : f t+s = ( f t ) s , for all t, s ∈ R.
Proof. From the continuity properties of g Q and ϕ
Q (g Q and ϕ
Q depend smoothly
on t and F), and from the s
∗ -continuity of functions (6.3.35), we have f t ∈ C bs for
any f ∈ C bs . The
∗ -morphism properties of the mapping f → f t are fulfilled due
to the morphism properties of the pull-back ϕ
∗ by any diffeomorphism ϕ of K,
145
The state ϕ m ∈ S(A), ϕ m (y) := ω m (x
∗
m yx m ) is also supported by E g (B m ) with any
open B m ⊂ K containing F m . Hence the decomposition (5.1.146) of ω := ϕ m is
concentrated on the one point set {m}. This means that
ϕ m (E g ( f )) := ϕ m ( f (F m )) = 0,
(6.3.31)
hence E g ( f ) = 0 for any nonzero f ∈ C bs . This proves the isometry of E g , hence
E g is a C
∗ -isomorphism of C bs into E g (g
∗
)A
∗∗
= p G A
∗∗ . The remaining assertions
are clearly valid.
6.3.7 Lemma. Let f ∈ C bs , ω ∈ p G S(A). Then the function
(g; F) → ω
σ
−1
g ( f (F))
∈ C, (g; F) ∈ G × K ,
(6.3.32)
is jointly continuous on the topological product G × supp E g .
Proof. Let f := f 0 , cf. (6.3.7). Then
σ
−1
g ( f 0 (F)) = σ(g
−1
g 0 (F))(x) f (F),
(6.3.33)
and the joint continuity of the group operation
(g 1 ; g 2 ) (∈ G × G) → g
−1
1 g 2 ∈ G
(6.3.34)
gives the joint continuity in (6.3.32) with f := f 0 . It can be verified directly,
cf. e.g. (6.3.13), that the function in (6.3.33) is even s
∗ -continuous in the couple
(g; F) ∈ G × K . But the finite algebraic combinations as well as the uniform limits of
s
∗ -continuous bounded functions are s
∗ -continuous. Since C
G
bs is generated by functions of the form f 0 , we have proved that the functions
(g; F) → σ
−1
g ( f (F)) ∈ A, for all f ∈ C
G
bs ,
(6.3.35)
are even s
∗ -continuous.
6.3.8 Proposition. Let, with the notation of 6.1.3, be f ∈ C
G
bs , and for a fixed Q ∈
C
∞
(g
∗
, R) and for any t ∈ R, F ∈ K , let
f t (F) := σ(g
−1
Q (t, F))( f (ϕ
Q
t F)).
(6.3.36)
Then f t ∈ C
G
bs and the mappings f → f t form a one-parameter group of
∗ -automorphisms of C
G
bs : f t+s = ( f t ) s , for all t, s ∈ R.
Proof. From the continuity properties of g Q and ϕ
Q (g Q and ϕ
Q depend smoothly
on t and F), and from the s
∗ -continuity of functions (6.3.35), we have f t ∈ C bs for
any f ∈ C bs . The
∗ -morphism properties of the mapping f → f t are fulfilled due
to the morphism properties of the pull-back ϕ
∗ by any diffeomorphism ϕ of K,