5.1 Multiple Systems
85
5.1.13 Lemma. Let ξ j ( j = l, 2, . . . n := dim G) form a basis in g. For λ ∈ R
n let
F :=
j λ j F j ∈ g
∗ expressed in the corresponding dual basis {F j } ⊂ g
∗
. Let
E g (F) := E ξ 1 (λ 1 )E ξ 2 (λ 2 ) . . . E ξ n (λ n ) ∈ N G .
(5.1.56)
The projectors E g (F) (F ∈ g
∗
) do not depend on a specific choice of the basis in g
and they are all minimal projectors in N G . Here E ξ (λ) := E ξ ({λ}), and E g (F) :=
E g ({F}).
Proof. The restriction of the mapping ρ G to N G is a W
∗
−isomorphism of N G
into P G Z
#
⊂ B
#
. Let ∈ ρ G (E g (F))H . From linearity of the mapping ξ →
X ξ f or ξ =
τ j ξ j , we have
X ξ =
j
τ j X ξ j =
j
τ j λ j = F(ξ)).
(5.1.57)
The second equality is due to the definition of E ξ (λ j ) as the projector corresponding
to the eigenvalue λ j ∈ R of X ξ (we write λ j in the place of the one-point set {λ j }
for simplicity). The last equality in (5.1.57) is due to definition of the dual basis and
shows the stated independence of E g (F) on the choice of a basis.
Let
E
#
g (F) := ρ G (E g (F)) .
Any projector E
#
g (B) is a sum (uncountable—in general, see also [274, 1.13.4]) of
projectors E
#
g (F) (F(ξ) ∈ B). The algebra ρ G (N G ) is the double commutant of the
set
{E
#
g (F) : F ∈ g
∗
},
(5.1.58)
according to the bicommutant theorem by von Neumann taken in the algebra
L(P G H ) of bounded operators on P G H . All the projectors E
#
g (F) in (5.1.58)
are mutually orthogonal. The commutant of (5.1.58) contains all the orthogonal projectors p ≤ E
#
g (F). But any nonzero orthogonal projector q < E
#
g (F) (strict inequality!) cannot commute with all such p’s. Hence E
#
g (F) is minimal in ρ G (N G ) and
E G (F) is minimal in N G for any F ∈ g
∗ . Since E ξ (R) = s G (= the identity of N G )
is a sum of E g (F)’s and N G is commutative, the set of all the E g (F)’s exhausts the
set of all the minimal projectors in N G .
5.1.14 Any state ω ∈ S(A
) on the algebra of bounded observables of our system
has unique extension to a normal state on the algebra (A
)
∗∗ and its restriction to M G
is a normal state ω ∈ S(M G ). Any normal state on M G can be obtained in this way,
[274, 1.24.5]. Let M be the spectrum space of M G , i.e. the compact set of all pure
states on M G endowed with the induced topology from the w
∗ -topology of its dual
M
∗
G . Then M G is isomorphic (denoted by ∼) to the C
∗ -algebra C(M) of all complex
valued continuous functions on M (by a Gel’fand-Najmark theorem, cf. [223, 16.2
Thm.1],[53, Thm.2.1.11A]): x (∈ M G ) ↔ ˆ
x (∈ C(M)). An element x ∈ M G is
85
5.1.13 Lemma. Let ξ j ( j = l, 2, . . . n := dim G) form a basis in g. For λ ∈ R
n let
F :=
j λ j F j ∈ g
∗ expressed in the corresponding dual basis {F j } ⊂ g
∗
. Let
E g (F) := E ξ 1 (λ 1 )E ξ 2 (λ 2 ) . . . E ξ n (λ n ) ∈ N G .
(5.1.56)
The projectors E g (F) (F ∈ g
∗
) do not depend on a specific choice of the basis in g
and they are all minimal projectors in N G . Here E ξ (λ) := E ξ ({λ}), and E g (F) :=
E g ({F}).
Proof. The restriction of the mapping ρ G to N G is a W
∗
−isomorphism of N G
into P G Z
#
⊂ B
#
. Let ∈ ρ G (E g (F))H . From linearity of the mapping ξ →
X ξ f or ξ =
τ j ξ j , we have
X ξ =
j
τ j X ξ j =
j
τ j λ j = F(ξ)).
(5.1.57)
The second equality is due to the definition of E ξ (λ j ) as the projector corresponding
to the eigenvalue λ j ∈ R of X ξ (we write λ j in the place of the one-point set {λ j }
for simplicity). The last equality in (5.1.57) is due to definition of the dual basis and
shows the stated independence of E g (F) on the choice of a basis.
Let
E
#
g (F) := ρ G (E g (F)) .
Any projector E
#
g (B) is a sum (uncountable—in general, see also [274, 1.13.4]) of
projectors E
#
g (F) (F(ξ) ∈ B). The algebra ρ G (N G ) is the double commutant of the
set
{E
#
g (F) : F ∈ g
∗
},
(5.1.58)
according to the bicommutant theorem by von Neumann taken in the algebra
L(P G H ) of bounded operators on P G H . All the projectors E
#
g (F) in (5.1.58)
are mutually orthogonal. The commutant of (5.1.58) contains all the orthogonal projectors p ≤ E
#
g (F). But any nonzero orthogonal projector q < E
#
g (F) (strict inequality!) cannot commute with all such p’s. Hence E
#
g (F) is minimal in ρ G (N G ) and
E G (F) is minimal in N G for any F ∈ g
∗ . Since E ξ (R) = s G (= the identity of N G )
is a sum of E g (F)’s and N G is commutative, the set of all the E g (F)’s exhausts the
set of all the minimal projectors in N G .
5.1.14 Any state ω ∈ S(A
) on the algebra of bounded observables of our system
has unique extension to a normal state on the algebra (A
)
∗∗ and its restriction to M G
is a normal state ω ∈ S(M G ). Any normal state on M G can be obtained in this way,
[274, 1.24.5]. Let M be the spectrum space of M G , i.e. the compact set of all pure
states on M G endowed with the induced topology from the w
∗ -topology of its dual
M
∗
G . Then M G is isomorphic (denoted by ∼) to the C
∗ -algebra C(M) of all complex
valued continuous functions on M (by a Gel’fand-Najmark theorem, cf. [223, 16.2
Thm.1],[53, Thm.2.1.11A]): x (∈ M G ) ↔ ˆ
x (∈ C(M)). An element x ∈ M G is