142
6 Dynamics of Quantum Mechanical Macroscopic Systems
where 1(F) := 1 for all F ∈ g
∗ . The classical (macroscopic) observables are embedded into C
G
bs according to the formula (6.3.8), where the classical observables are
represented by functions belonging to C b (K , C). We can (and we shall) consider C
G
bs ,
or C bs , as the (extended) C
∗ -algebra of observables of the systems with ‘mean-field’
dynamics. It might be useful, however, to embed this new algebra of observables in a
canonical way into the W
∗ -algebra A
∗∗ , since there is a canonical bijection between
the set of all states ω ∈ S(A) and the set of all normal states ω ∈ S ∗ (A
∗∗
) on the
double dual A
∗∗ of A: any ω ∈ S(A) corresponds to its (equally denoted) canonical
normal extension ω ∈ S ∗ (A
∗∗
). Hence, after obtaining an embedding of C bs into
A
∗∗ such that A ⊂ C bs is mapped onto π u (A) ⊂ A
∗∗ or onto its subrepresentation,
we shall obtain a certain canonical extension of any state ω ∈ S(A) (or of any state
ω ∈ p G S(A), where p G ∈ Z is the projector onto the above mentioned subrepresentation of π u ) to a state on A
∗∗ (resp. on p G A
∗∗ ), and this in turn gives to us a certain
canonical extension of states on A to states on C bs . Such an embedding is given in
the following proposition.
6.3.6 Proposition. Let us consider the integral decomposition of any ω ∈ p G S ∗ (A
∗∗
)
[where A is simple] given by the formula 5.1.146 according to Theorem 5.2.11, and
let F g : M → ˙
g
∗ be given as in 5.1.39. There is a C
∗ -isomorphism of C bs into p G A
∗∗
formally written in the form
E g : f (∈ C bs ) → E g ( f ) :=
f (F) E g (dF),
(6.3.15)
where E g denotes the G-measure (as before) as well as the presently introduced
isomorphism. The isomorphism E g is uniquely determined by the formula
2
ω(E g ( f )) :=
ω m ( f (F m )) μ ω (dm), ∀ω ∈ p G S ∗ (A
∗∗
),
(6.3.16)
where the decomposition (5.1.146) was used, and (cf. 5.1.39)
F g : m → F m := F g (m), m ∈ N ⊂ M,
(6.3.17)
is defined on the spectrum space N of the (commutative) subalgebra N(E g ) of
p G A
∗∗ , cf. 5.2.3.
The mapping E g leaves A := p G π u (A) invariant and maps C b onto a
C
∗ -subalgebra N
c of N(E g ) =: N G , see also 5.2.10.
Proof. Let B ⊂ N be any Borel set and χ B is its characteristic function. The functions
m → ω m (x) χ B (m), x ∈ A
∗∗
,
(6.3.18)
2 Note: For noncompact supp E g , the integral is a limit of integrals over bounded subsets B ⊂ g ∗ :
· · · := lim B↑g ∗
ω m (E g (B) f (F m )) μ ω (dm).
6 Dynamics of Quantum Mechanical Macroscopic Systems
where 1(F) := 1 for all F ∈ g
∗ . The classical (macroscopic) observables are embedded into C
G
bs according to the formula (6.3.8), where the classical observables are
represented by functions belonging to C b (K , C). We can (and we shall) consider C
G
bs ,
or C bs , as the (extended) C
∗ -algebra of observables of the systems with ‘mean-field’
dynamics. It might be useful, however, to embed this new algebra of observables in a
canonical way into the W
∗ -algebra A
∗∗ , since there is a canonical bijection between
the set of all states ω ∈ S(A) and the set of all normal states ω ∈ S ∗ (A
∗∗
) on the
double dual A
∗∗ of A: any ω ∈ S(A) corresponds to its (equally denoted) canonical
normal extension ω ∈ S ∗ (A
∗∗
). Hence, after obtaining an embedding of C bs into
A
∗∗ such that A ⊂ C bs is mapped onto π u (A) ⊂ A
∗∗ or onto its subrepresentation,
we shall obtain a certain canonical extension of any state ω ∈ S(A) (or of any state
ω ∈ p G S(A), where p G ∈ Z is the projector onto the above mentioned subrepresentation of π u ) to a state on A
∗∗ (resp. on p G A
∗∗ ), and this in turn gives to us a certain
canonical extension of states on A to states on C bs . Such an embedding is given in
the following proposition.
6.3.6 Proposition. Let us consider the integral decomposition of any ω ∈ p G S ∗ (A
∗∗
)
[where A is simple] given by the formula 5.1.146 according to Theorem 5.2.11, and
let F g : M → ˙
g
∗ be given as in 5.1.39. There is a C
∗ -isomorphism of C bs into p G A
∗∗
formally written in the form
E g : f (∈ C bs ) → E g ( f ) :=
f (F) E g (dF),
(6.3.15)
where E g denotes the G-measure (as before) as well as the presently introduced
isomorphism. The isomorphism E g is uniquely determined by the formula
2
ω(E g ( f )) :=
ω m ( f (F m )) μ ω (dm), ∀ω ∈ p G S ∗ (A
∗∗
),
(6.3.16)
where the decomposition (5.1.146) was used, and (cf. 5.1.39)
F g : m → F m := F g (m), m ∈ N ⊂ M,
(6.3.17)
is defined on the spectrum space N of the (commutative) subalgebra N(E g ) of
p G A
∗∗ , cf. 5.2.3.
The mapping E g leaves A := p G π u (A) invariant and maps C b onto a
C
∗ -subalgebra N
c of N(E g ) =: N G , see also 5.2.10.
Proof. Let B ⊂ N be any Borel set and χ B is its characteristic function. The functions
m → ω m (x) χ B (m), x ∈ A
∗∗
,
(6.3.18)
2 Note: For noncompact supp E g , the integral is a limit of integrals over bounded subsets B ⊂ g ∗ :
· · · := lim B↑g ∗
ω m (E g (B) f (F m )) μ ω (dm).