140
6 Dynamics of Quantum Mechanical Macroscopic Systems
(iv) For any σ(G)-invariant C
∗ -subalgebra A
J of A:
σ(g)(x) := σ g (x) ∈ A
J for all x ∈ A
J
, g ∈ G,
(6.3.6)
let C
J
bs := C bs (supp E g , A
J
) be defined equally as it was defined C bs in (iii) with the
replacement of A by A
J .
(v) Let C
G
bs (resp. C
G J
bs ) be the C
∗ -subalgebra (cf. 6.3.4) of C bs (resp. of C
J
bs ) generated
by all the functions f 0 ∈ C bs of the form
f 0 : F → σ g 0 (F) (x) f (F), f ∈ C b , g 0 ∈ C(supp E g , G),
(6.3.7)
with any x ∈ A (resp. any x ∈ A
J ). The set C(supp E g , G) consists of all continuous G-valued functions on supp E g .
(vi) We shall use also K := supp E g , resp. K ⊂ g
∗ will denote any Ad
∗
(G)-invariant
closed subset of the generalized classical phase space in more general cases. We
shall identify C b := C b (K , C) with the subset C bs (K , C id A ) of C bs in the canonical
way: f ∈ C b is identified with the function
f : F → id A f (F), F ∈ K , id A is the identity of A.
(6.3.8)
6.3.3 Proposition. The set C bs is a
∗ -algebra with respect to the natural (pointwise)
algebraic operations determined by the corresponding operations in the range A of
the elements f ∈ C bs :
( f 1 + λ f 2 )(F) := f 1 (F) + λ f 2 (F), ( f 1 f 2 )(F) := f 1 (F) f 2 (F),
f
∗
(F) := [ f (F)]
∗
, ∀F ∈ K , λ ∈ C, f j and f ∈ C bs ,
(6.3.9)
and it is a normed algebra with the norm f of f ∈ C bs given by (6.3.4). This
normed
∗ -algebra C bs is a C
∗ -algebra, and its subsets C
J
bs and C b endowed with the
induced algebraic operations and the norm are C
∗ -subalgebras of C bs .
Proof. The continuity properties of the product in A with respect to the s
∗ -topology
are given by Proposition 1.8.12 and Theorem 1.8.9 of [274]. Then the uniform boundedness of f ∈ C bs and the continuity of the ∗-operation in the s
∗ -topology gives the
invariance of C bs with respect to the algebraic operations (6.3.9). The norm properties of the function given in (6.3.4) are easily verified, and the C
∗ -property of the
norm:
f
2
= [sup
F
f (F)]
2
= sup
F
f (F)
2
= sup
F
f (F)
∗ f (F) = = f
∗ f ,
(6.3.10)
6 Dynamics of Quantum Mechanical Macroscopic Systems
(iv) For any σ(G)-invariant C
∗ -subalgebra A
J of A:
σ(g)(x) := σ g (x) ∈ A
J for all x ∈ A
J
, g ∈ G,
(6.3.6)
let C
J
bs := C bs (supp E g , A
J
) be defined equally as it was defined C bs in (iii) with the
replacement of A by A
J .
(v) Let C
G
bs (resp. C
G J
bs ) be the C
∗ -subalgebra (cf. 6.3.4) of C bs (resp. of C
J
bs ) generated
by all the functions f 0 ∈ C bs of the form
f 0 : F → σ g 0 (F) (x) f (F), f ∈ C b , g 0 ∈ C(supp E g , G),
(6.3.7)
with any x ∈ A (resp. any x ∈ A
J ). The set C(supp E g , G) consists of all continuous G-valued functions on supp E g .
(vi) We shall use also K := supp E g , resp. K ⊂ g
∗ will denote any Ad
∗
(G)-invariant
closed subset of the generalized classical phase space in more general cases. We
shall identify C b := C b (K , C) with the subset C bs (K , C id A ) of C bs in the canonical
way: f ∈ C b is identified with the function
f : F → id A f (F), F ∈ K , id A is the identity of A.
(6.3.8)
6.3.3 Proposition. The set C bs is a
∗ -algebra with respect to the natural (pointwise)
algebraic operations determined by the corresponding operations in the range A of
the elements f ∈ C bs :
( f 1 + λ f 2 )(F) := f 1 (F) + λ f 2 (F), ( f 1 f 2 )(F) := f 1 (F) f 2 (F),
f
∗
(F) := [ f (F)]
∗
, ∀F ∈ K , λ ∈ C, f j and f ∈ C bs ,
(6.3.9)
and it is a normed algebra with the norm f of f ∈ C bs given by (6.3.4). This
normed
∗ -algebra C bs is a C
∗ -algebra, and its subsets C
J
bs and C b endowed with the
induced algebraic operations and the norm are C
∗ -subalgebras of C bs .
Proof. The continuity properties of the product in A with respect to the s
∗ -topology
are given by Proposition 1.8.12 and Theorem 1.8.9 of [274]. Then the uniform boundedness of f ∈ C bs and the continuity of the ∗-operation in the s
∗ -topology gives the
invariance of C bs with respect to the algebraic operations (6.3.9). The norm properties of the function given in (6.3.4) are easily verified, and the C
∗ -property of the
norm:
f
2
= [sup
F
f (F)]
2
= sup
F
f (F)
2
= sup
F
f (F)
∗ f (F) = = f
∗ f ,
(6.3.10)