6.2 Spin Systems with Polynomial Local Hamiltonians Q N
131
with L ≥ K ≥ 1. Then the continuity of the product of elements of a W
∗ -algebra in
the s-topology leads to:
s- lim
K →∞
τ
Q
t 1 (τ
K
t 2
(x)) = s- lim
K
∞
k,m=0
(it 1 )
k
k!
(it 2 )
m
m!
[Q
K
, [Q
K
, x]
(m)
]
(k)
= s- lim
K
∞
p=0
(t 1 + t 2 )
p
p!
[i Q
K
, x]
( p)
= s- lim
K
τ
K
t 1 +t 2
(x) = τ
Q
t 1 +t 2 (x).
(6.2.60)
The relations (6.2.56) and (6.2.60) give the desired group property (6.2.52) for
all sufficiently small nonzero t 1 , t 2 , hence τ
Q
t ∈
∗ - Aut C
N (due to the consequent
invertibility of τ
Q
t on C
N ), and τ
Q is a one-parameter group of automorphisms of
C
N (for any given N ∈ ).
6.2.16 Note. We have worked in this section in the framework of the subalgebra
s G A
∗∗ of the von Neumann algebra A
∗∗ . The only properties of the projector
s G ∈ Z we have used in the previous considerations was the existence of the limits
X ξ := s
∗ - lim N s G X ξ N for all ξ ∈ g (here the elements X ξ N ∈ A are identified with
π u (X ξ N ), cf. 6.2.1) as well as the σ(G)-invariance: σ(g)(s G ) = s G for all g ∈ G.
Any projector s π ∈ Z with these two properties, i.e. s π such that:
(i) the limits s
∗ - lim N X ξ N s π exist in s
∗
(A
∗∗
, A
∗
)-topology for all ξ ∈ g,
(ii) s π is σ(G)-invariant: σ(g)(s π ) = s π for all g ∈ G,
could be used instead of s G in the considerations of this section. Such projectors
form a lattice in Z with the maximal element p G defined in 5.1.29 . The G -measure
corresponding to p G was introduced in 5.1.33 and denoted by E
g . Then the G
-measure used up to now in this section was E g = s G E
g , and the G-measure E
π
G
corresponding to another projector s π ∈ Z satisfying (i) and (ii) equals to s π E
g .
The algebra N
c
π := E
π
g (C b (g
∗
, C)) corresponding to the projector s π , hence also the
quasilocal algebra C π := A ⊗ N
c
π , depend nontrivially on the choice of s π . If, however, s G ≤ s π ≤ p G , then N
c
π is isomorphic to N
c . This is an immediate consequence
of the Proposition 6.2.9 as well as of the following 6.2.17.
6.2.17 Lemma. Let the projector s π ∈ Z (:= the center of A
∗∗ ) satisfy 6.2.16 (i)+(ii).
Let s G ≤ s π ≤ p G , and let E
π
g := s π E
g . Then supp E
π
g = supp E
g (= supp E g ,
consequently).
Proof. Let sp(X ξ ) ⊂ R (ξ ∈ g) be spectrum of the bounded selfadjoint operator
X ξ ∈ L(H). Let conv(B) be the convex hull of the subset B of a linear space. We
131
with L ≥ K ≥ 1. Then the continuity of the product of elements of a W
∗ -algebra in
the s-topology leads to:
s- lim
K →∞
τ
Q
t 1 (τ
K
t 2
(x)) = s- lim
K
∞
k,m=0
(it 1 )
k
k!
(it 2 )
m
m!
[Q
K
, [Q
K
, x]
(m)
]
(k)
= s- lim
K
∞
p=0
(t 1 + t 2 )
p
p!
[i Q
K
, x]
( p)
= s- lim
K
τ
K
t 1 +t 2
(x) = τ
Q
t 1 +t 2 (x).
(6.2.60)
The relations (6.2.56) and (6.2.60) give the desired group property (6.2.52) for
all sufficiently small nonzero t 1 , t 2 , hence τ
Q
t ∈
∗ - Aut C
N (due to the consequent
invertibility of τ
Q
t on C
N ), and τ
Q is a one-parameter group of automorphisms of
C
N (for any given N ∈ ).
6.2.16 Note. We have worked in this section in the framework of the subalgebra
s G A
∗∗ of the von Neumann algebra A
∗∗ . The only properties of the projector
s G ∈ Z we have used in the previous considerations was the existence of the limits
X ξ := s
∗ - lim N s G X ξ N for all ξ ∈ g (here the elements X ξ N ∈ A are identified with
π u (X ξ N ), cf. 6.2.1) as well as the σ(G)-invariance: σ(g)(s G ) = s G for all g ∈ G.
Any projector s π ∈ Z with these two properties, i.e. s π such that:
(i) the limits s
∗ - lim N X ξ N s π exist in s
∗
(A
∗∗
, A
∗
)-topology for all ξ ∈ g,
(ii) s π is σ(G)-invariant: σ(g)(s π ) = s π for all g ∈ G,
could be used instead of s G in the considerations of this section. Such projectors
form a lattice in Z with the maximal element p G defined in 5.1.29 . The G -measure
corresponding to p G was introduced in 5.1.33 and denoted by E
g . Then the G
-measure used up to now in this section was E g = s G E
g , and the G-measure E
π
G
corresponding to another projector s π ∈ Z satisfying (i) and (ii) equals to s π E
g .
The algebra N
c
π := E
π
g (C b (g
∗
, C)) corresponding to the projector s π , hence also the
quasilocal algebra C π := A ⊗ N
c
π , depend nontrivially on the choice of s π . If, however, s G ≤ s π ≤ p G , then N
c
π is isomorphic to N
c . This is an immediate consequence
of the Proposition 6.2.9 as well as of the following 6.2.17.
6.2.17 Lemma. Let the projector s π ∈ Z (:= the center of A
∗∗ ) satisfy 6.2.16 (i)+(ii).
Let s G ≤ s π ≤ p G , and let E
π
g := s π E
g . Then supp E
π
g = supp E
g (= supp E g ,
consequently).
Proof. Let sp(X ξ ) ⊂ R (ξ ∈ g) be spectrum of the bounded selfadjoint operator
X ξ ∈ L(H). Let conv(B) be the convex hull of the subset B of a linear space. We