6.2 Spin Systems with Polynomial Local Hamiltonians Q N
123
(i) The sums
τ
K
t (x) := e
it Q
K x e
−it Q
K =
∞
m=0
(it)
m
m!
[Q
K
, x]
(m)
, K ∈ ,
(6.2.17)
are convergent in the norm-topology of A, and this convergence is uniform on
{K : K ∈ } × {t : |t| ≤ κ N } × {x : x ∈ B
N
0 , x ≤ a} for any a ∈ R + .
(ii) The following limits exist in s G A
∗∗ :
τ
Q
t (x) := s
∗ - lim
|K |→∞
τ
K
t (x),
(6.2.18)
where the convergence is understood in the s
∗
(s G A
∗∗
, s G A
∗
)-topology generated
by the seminorms ˆ
p ω and ˆ
p
∗
ω for all ω ∈ S ∗ (s G A
∗∗
):
ˆ
p ω : x → ˆ
p ω (x) :=
ω(x ∗ x), ˆ
p
∗
ω : x → ˆ
p
∗
ω (x) :=
ω(x x ∗ ).
(6.2.19)
Proof. The estimates (6.2.9) are independent of K ∈ and the corresponding
majorizing power series for (6.2.17) is uniformly convergent on the product of the
disc {t : |t| ≤ κ N , t ∈ C} and the ball {x : x ∈ B
N
0 , x ≤ a} for any nonnegative a.
This proves (i). The definition of s G in 5.1.11 implies the existence of the limits
X ξ := s
∗ - lim
|K |→∞
X ξ K = E g ( f ξ ) ∈ s G A
∗∗
, ξ ∈ g,
(6.2.20)
what implies, in turn, together with the uniform boundedness in K ∈ of the multip1e commutators in (6.2.17), the existence of the limits
s
∗ - lim
|K |→∞
[Q
K
, x]
(m)
∈ s G A
∗∗
.
(6.2.21)
The statement (i) together with these facts imply (ii).
6.2.6 Lemma. Let B
N be the C
∗ -subalgebra of A generated by B
N
0 . Each of
the mappings τ
Q
t : B
N
0 → s G A
∗∗
(|t| ≤ κ N ) can be extended to a unique
∗ -homomorphism of the C
∗ -algebra B
N into s G A
∗∗ .
Proof. The mappings τ
K
t are inner automorphisms of A, and their canonical extensions to A
∗∗ leave the center Z elementwise invariant. Hence, we can consider τ
K
t as
(inner) automorphisms of s G A
∗∗ :
τ
K
t ∈
∗ -Aut s G A
∗∗
, for all t ∈ R, K ⊂ .
(6.2.22)
The properties of the s
∗ -limit imply that τ
Q
t
(|t| ≤ κ N , t ∈ R) are
∗ -homomorphisms of the symmetric set B
N
0 into s G A
∗∗ , as well as they are
123
(i) The sums
τ
K
t (x) := e
it Q
K x e
−it Q
K =
∞
m=0
(it)
m
m!
[Q
K
, x]
(m)
, K ∈ ,
(6.2.17)
are convergent in the norm-topology of A, and this convergence is uniform on
{K : K ∈ } × {t : |t| ≤ κ N } × {x : x ∈ B
N
0 , x ≤ a} for any a ∈ R + .
(ii) The following limits exist in s G A
∗∗ :
τ
Q
t (x) := s
∗ - lim
|K |→∞
τ
K
t (x),
(6.2.18)
where the convergence is understood in the s
∗
(s G A
∗∗
, s G A
∗
)-topology generated
by the seminorms ˆ
p ω and ˆ
p
∗
ω for all ω ∈ S ∗ (s G A
∗∗
):
ˆ
p ω : x → ˆ
p ω (x) :=
ω(x ∗ x), ˆ
p
∗
ω : x → ˆ
p
∗
ω (x) :=
ω(x x ∗ ).
(6.2.19)
Proof. The estimates (6.2.9) are independent of K ∈ and the corresponding
majorizing power series for (6.2.17) is uniformly convergent on the product of the
disc {t : |t| ≤ κ N , t ∈ C} and the ball {x : x ∈ B
N
0 , x ≤ a} for any nonnegative a.
This proves (i). The definition of s G in 5.1.11 implies the existence of the limits
X ξ := s
∗ - lim
|K |→∞
X ξ K = E g ( f ξ ) ∈ s G A
∗∗
, ξ ∈ g,
(6.2.20)
what implies, in turn, together with the uniform boundedness in K ∈ of the multip1e commutators in (6.2.17), the existence of the limits
s
∗ - lim
|K |→∞
[Q
K
, x]
(m)
∈ s G A
∗∗
.
(6.2.21)
The statement (i) together with these facts imply (ii).
6.2.6 Lemma. Let B
N be the C
∗ -subalgebra of A generated by B
N
0 . Each of
the mappings τ
Q
t : B
N
0 → s G A
∗∗
(|t| ≤ κ N ) can be extended to a unique
∗ -homomorphism of the C
∗ -algebra B
N into s G A
∗∗ .
Proof. The mappings τ
K
t are inner automorphisms of A, and their canonical extensions to A
∗∗ leave the center Z elementwise invariant. Hence, we can consider τ
K
t as
(inner) automorphisms of s G A
∗∗ :
τ
K
t ∈
∗ -Aut s G A
∗∗
, for all t ∈ R, K ⊂ .
(6.2.22)
The properties of the s
∗ -limit imply that τ
Q
t
(|t| ≤ κ N , t ∈ R) are
∗ -homomorphisms of the symmetric set B
N
0 into s G A
∗∗ , as well as they are