128
6 Dynamics of Quantum Mechanical Macroscopic Systems
6.2.11 Lemma. The mappings τ
Q
t (|t| ≤ κ N ) defined in 6.2.5 (ii) map the C
∗ -algebra
A
N into the C
∗ -algebra C
N (which is generated in s G A
∗∗ by A
N and N
c ).
Proof. We can write the definition of τ
Q
t (|t| ≤ κ N ) on A
N , (6.2.17) and (6.2.18),
in the form
τ
Q
t (x) :=
∞
m=0
t
m
m!
s
∗ - lim
K
[i Q
K
, x]
(m)
, x ∈ A
N
.
(6.2.45)
Each multiple commutator in (6.2.45) can be expressed in the form of a polynomial
in the variables X ξ K and some of the variables y s of the form, cf. also (6.2.10) and
(6.2.11):
y s := [X
K
j 1
, [X
K
j 2
, . . . [X
K
j s
, x] . . . ]] ∈ A
N
, K ∈ ,
(6.2.46)
with the coefficients independent of K . Due to (6.2.20) and the independence of any
y s of K , the strong limits in (6.2.45) are elements of C
N . The norm convergence of
the sum in right hand side of (6.2.45) and the closeness of C
N in the norm-topology
give then the result.
6.2.12 Lemma. For any x ∈ A and any z ∈ N
c , the equality x z = 0 implies the
validity of x · ·z = 0.
Proof. For z = 0, we have z = E g ( f ) with | f (F 0 )| = 0 for some f ∈ C(supp E g )
and some F 0 ∈ supp E g . Let, for the definiteness, be f (F 0 ) > 0. Then there is a
subset B 0 ⊂ g
∗ such that E g (B 0 ) = 0 and f (F) >
1
2
f (F 0 ) for all F ∈ B 0 . Since
N
c is in the commutant of A in s G A
∗∗ , the product of the positive (i.e. nonnegative)
operator x
∗ x ∈ A with the positive operator (E g ( f ) −
1
2
f (F 0 ))E g (B 0 ) ∈ N
c is a
nonnegative operator in C. Then xz = 0 implies
0 ≤ x
∗ x (E g ( f ) −
1
2
f (F 0 ))E g (B 0 ) = −
1
2
f (F 0 ) x
∗ x E g (B 0 ).
(6.2.47)
Hence we have x E g (B 0 ) = 0. The mapping: x → x E g (B 0 ) is a nonzero (nondegenerate) representation of the simple C
∗ -algebra A in A
∗∗ , hence x = 0.
6.2.13 Lemma. Let A
N
⊗ N
c and A ⊗ N
c be the C
∗ -products (uniquely defined,
since N
c is abelian, [274, 1.22.5.]), with the canonical inclusion A
N
⊗ N
c
⊂
A ⊗ N
c . Let λ
−1
0 be the homomorphism of A ⊗ N
c into C, 6.2.2, determined by
the association:
λ
−1
0 :
j
x j ⊗ z j →
j
x j z j ∈ C, x j ∈ A, z j ∈ N
c
.
(6.2.48)
Then λ
−1
0
can be extended to a unique
∗ -isomorphism λ
−1
0 =: (λ 0 )
−1 of the
C
∗ -algebra A ⊗ N
c onto C, the restrictions of which to the subalgebras A
N
⊗
N
c
(N ∈ ) are
∗ -isomorphisms onto C
N
(N ∈ ), cf. 6.2.2.
6 Dynamics of Quantum Mechanical Macroscopic Systems
6.2.11 Lemma. The mappings τ
Q
t (|t| ≤ κ N ) defined in 6.2.5 (ii) map the C
∗ -algebra
A
N into the C
∗ -algebra C
N (which is generated in s G A
∗∗ by A
N and N
c ).
Proof. We can write the definition of τ
Q
t (|t| ≤ κ N ) on A
N , (6.2.17) and (6.2.18),
in the form
τ
Q
t (x) :=
∞
m=0
t
m
m!
s
∗ - lim
K
[i Q
K
, x]
(m)
, x ∈ A
N
.
(6.2.45)
Each multiple commutator in (6.2.45) can be expressed in the form of a polynomial
in the variables X ξ K and some of the variables y s of the form, cf. also (6.2.10) and
(6.2.11):
y s := [X
K
j 1
, [X
K
j 2
, . . . [X
K
j s
, x] . . . ]] ∈ A
N
, K ∈ ,
(6.2.46)
with the coefficients independent of K . Due to (6.2.20) and the independence of any
y s of K , the strong limits in (6.2.45) are elements of C
N . The norm convergence of
the sum in right hand side of (6.2.45) and the closeness of C
N in the norm-topology
give then the result.
6.2.12 Lemma. For any x ∈ A and any z ∈ N
c , the equality x z = 0 implies the
validity of x · ·z = 0.
Proof. For z = 0, we have z = E g ( f ) with | f (F 0 )| = 0 for some f ∈ C(supp E g )
and some F 0 ∈ supp E g . Let, for the definiteness, be f (F 0 ) > 0. Then there is a
subset B 0 ⊂ g
∗ such that E g (B 0 ) = 0 and f (F) >
1
2
f (F 0 ) for all F ∈ B 0 . Since
N
c is in the commutant of A in s G A
∗∗ , the product of the positive (i.e. nonnegative)
operator x
∗ x ∈ A with the positive operator (E g ( f ) −
1
2
f (F 0 ))E g (B 0 ) ∈ N
c is a
nonnegative operator in C. Then xz = 0 implies
0 ≤ x
∗ x (E g ( f ) −
1
2
f (F 0 ))E g (B 0 ) = −
1
2
f (F 0 ) x
∗ x E g (B 0 ).
(6.2.47)
Hence we have x E g (B 0 ) = 0. The mapping: x → x E g (B 0 ) is a nonzero (nondegenerate) representation of the simple C
∗ -algebra A in A
∗∗ , hence x = 0.
6.2.13 Lemma. Let A
N
⊗ N
c and A ⊗ N
c be the C
∗ -products (uniquely defined,
since N
c is abelian, [274, 1.22.5.]), with the canonical inclusion A
N
⊗ N
c
⊂
A ⊗ N
c . Let λ
−1
0 be the homomorphism of A ⊗ N
c into C, 6.2.2, determined by
the association:
λ
−1
0 :
j
x j ⊗ z j →
j
x j z j ∈ C, x j ∈ A, z j ∈ N
c
.
(6.2.48)
Then λ
−1
0
can be extended to a unique
∗ -isomorphism λ
−1
0 =: (λ 0 )
−1 of the
C
∗ -algebra A ⊗ N
c onto C, the restrictions of which to the subalgebras A
N
⊗
N
c
(N ∈ ) are
∗ -isomorphisms onto C
N
(N ∈ ), cf. 6.2.2.