6.2 Spin Systems with Polynomial Local Hamiltonians Q N
137
as well as of the compactness of supp E
π
g and of the continuity of f .
(v) The continuity in (6.2.81) is a consequence of (iv). Let us consider τ
π
t (t∈R)
as a family of representations of A := π u (A) ⊂ A
∗∗ in the subalgebra s π A
∗∗ of A
∗∗ .
The unique σ(A
∗∗
, A
∗
) − σ(s π A
∗∗
, s π A
∗
)-continuous extensions of these representations to A
∗∗ , [274, 1.21.13], will be denoted by τ
Q
t (resp. τ
π
t for s π < p G ). From
Proposition 6.2.10 and from its proof one can see
τ
π
t (s π ) = τ
π
t (E
π
g (g
∗
)) = s π .
(6.2.88)
We have also τ
π
t (id A − s π ) = 0, and the restrictions of τ
π
t
(t ∈ R)
to the τ
π
t -invariant subalgebra s π A
∗∗ form a family of
∗ -automorphisms which
are automatically σ-continuous, [274, 4.1.23]. The definition τ
π with a he1p of
strong limits, cf. (6.2.23), shows that the restriction of τ
π
t to C π coincides with the
above defined τ
π
t ∈
∗ - Aut C π . This proves the normality of ω ◦ τ
π
t for any normal ω
(i.e. ω ∈ s π A
∗ ), hence (6.2.82).
(vi) The automorphism group τ
π of C π is determined uniquely by the determination
of τ
π
t (x) for all x ∈ B
N (cf. 6.2.6 and 6.2.2(iii) for notation), for |t| ≤ κ N , N ∈ ;
this is clear from 6.2.5 and from its consequences. The series in the formula
d
dt
τ
π
t (x) = i
∞
m=0
(it)
m
m!
s
∗ - lim
K
[Q
K
, [Q
K
, x]
(m)
]
(6.2.89)
converges uniformly in the disc |t| ≤ κ N (x ∈ B
N
), hence the equality (6.2.89) is
valid. Considerations similar to those used in the dealing with (6.2.58) lead to the
equalities:
s- lim
K
[i Q
K
, [i Q
K
, x]
(m)
] = s- lim
K
s- lim
L
[i Q
L
, [i Q
K
, x]
(m)
]
= s- lim
K
δ π ([i Q
K
, x]
(m)
),
(6.2.90)
where for all N ∈ :
δ π (x) := s- lim
L
i[Q
L
, x], for all x ∈ B
N
:= s π π u (B
N
).
(6.2.91)
The derivation property of commutators and the polynomial form of Q together with
(6.2.20) lead to the expression (6.2.83a) for δ π in (6.2.91). Setting t = 0 in (6.2.89),
we see that so defined δ π (x) is the value of the generator δ π of τ
π on x ∈ A
N
(N ∈ ).
By the differentiation of (6.2.32) with f ∈ C
1
(supp E
π
g ) we obtain (6.2.83b), cf.
(6.2.43) and notes in [53] above 3.2.29. From the continuity properties of τ
π and
the corresponding closedness of δ π , cf. [53, 3.1.6], we obtain by the repeated use of
(6.2.90):
137
as well as of the compactness of supp E
π
g and of the continuity of f .
(v) The continuity in (6.2.81) is a consequence of (iv). Let us consider τ
π
t (t∈R)
as a family of representations of A := π u (A) ⊂ A
∗∗ in the subalgebra s π A
∗∗ of A
∗∗ .
The unique σ(A
∗∗
, A
∗
) − σ(s π A
∗∗
, s π A
∗
)-continuous extensions of these representations to A
∗∗ , [274, 1.21.13], will be denoted by τ
Q
t (resp. τ
π
t for s π < p G ). From
Proposition 6.2.10 and from its proof one can see
τ
π
t (s π ) = τ
π
t (E
π
g (g
∗
)) = s π .
(6.2.88)
We have also τ
π
t (id A − s π ) = 0, and the restrictions of τ
π
t
(t ∈ R)
to the τ
π
t -invariant subalgebra s π A
∗∗ form a family of
∗ -automorphisms which
are automatically σ-continuous, [274, 4.1.23]. The definition τ
π with a he1p of
strong limits, cf. (6.2.23), shows that the restriction of τ
π
t to C π coincides with the
above defined τ
π
t ∈
∗ - Aut C π . This proves the normality of ω ◦ τ
π
t for any normal ω
(i.e. ω ∈ s π A
∗ ), hence (6.2.82).
(vi) The automorphism group τ
π of C π is determined uniquely by the determination
of τ
π
t (x) for all x ∈ B
N (cf. 6.2.6 and 6.2.2(iii) for notation), for |t| ≤ κ N , N ∈ ;
this is clear from 6.2.5 and from its consequences. The series in the formula
d
dt
τ
π
t (x) = i
∞
m=0
(it)
m
m!
s
∗ - lim
K
[Q
K
, [Q
K
, x]
(m)
]
(6.2.89)
converges uniformly in the disc |t| ≤ κ N (x ∈ B
N
), hence the equality (6.2.89) is
valid. Considerations similar to those used in the dealing with (6.2.58) lead to the
equalities:
s- lim
K
[i Q
K
, [i Q
K
, x]
(m)
] = s- lim
K
s- lim
L
[i Q
L
, [i Q
K
, x]
(m)
]
= s- lim
K
δ π ([i Q
K
, x]
(m)
),
(6.2.90)
where for all N ∈ :
δ π (x) := s- lim
L
i[Q
L
, x], for all x ∈ B
N
:= s π π u (B
N
).
(6.2.91)
The derivation property of commutators and the polynomial form of Q together with
(6.2.20) lead to the expression (6.2.83a) for δ π in (6.2.91). Setting t = 0 in (6.2.89),
we see that so defined δ π (x) is the value of the generator δ π of τ
π on x ∈ A
N
(N ∈ ).
By the differentiation of (6.2.32) with f ∈ C
1
(supp E
π
g ) we obtain (6.2.83b), cf.
(6.2.43) and notes in [53] above 3.2.29. From the continuity properties of τ
π and
the corresponding closedness of δ π , cf. [53, 3.1.6], we obtain by the repeated use of
(6.2.90):