5.1 Multiple Systems
99
The last considerations show to us that M G and M
G are different from one
another. The W
∗ -subalgebra of M
G generated by all E
g (F) := E
g ({F}) (F ∈ g
∗
)
is naturally isomorphic to M G . Hence M
G is larger than M G which can be injected
into M
G via the last mentioned isomorphism.
5.1.32 Let us now introduce the mapping p M :
p M : S(A
) → S ∗ (M
G ), ω → p M ω,
(5.1.128)
where p M ω is the restriction to M
G of the canonical extension of the state ω ∈
S(A
) to the normal state on (A
)
∗∗ . Any state ω ∈ S(A
) (resp. ω ∈ S(M
G )) can
be uniquely decomposed as
ω = ω( p G ) p G ω + ω(I − p G ) ω ◦ ,
(5.1.129)
where the symbols p G ω(x) and ω ◦ (x) are given by
p G ω(x) :=
1
ω( p G )
ω(x p G ), ω ◦ (x) :=
1
ω(I − p G )
ω(x(I − p G )).
Hence for ω(I − p G ) = 0 it is p M ω ◦ = m ◦ := the pure state in S(M
G ) sup −
ported by the minimal projector I − p G ∈ M
G .
Let
S
g := {ω ∈ S(A
) : ω( p G ) = 1}.
(5.1.130)
In other words: S
g = p G S(A
). For ω ∈ S
g one has p M ω ∈ S ∗ (N
G ). Conversely,
each state in S ∗ (N
G ) is of the form p M ω for some states ω ∈ S
g .
5.1.33 Lemma. The projector-valued additive function of intervals in g
∗ introduced
in (5.1.125) can be extended to a unique projector-valued measure E
g : B →
E
g (B) defined on all Borel sets B in g
∗ .
Proof. The mapping
w : g → L( p G H u ), ξ → w(ξ) := exp(i X ξ ) p G ,
(5.1.131)
see (5.1.120), is strongly continuous unitary representation of the abelian group
g (group multiplication is here the vector addition) in the subspace p G H u of the
Hilbert space H u of the universal representation of A
. This can be seen with a help
of linearity of the mapping
ξ → X ξ , ξ ∈ g,
(5.1.132)
According to the SNAG-theorem ([266, Chap. X], [262, Theorem VIII.12], [120,
Chap. IV]), there is unique projection measure E
g on the dual group ˆ
g of g repre-
99
The last considerations show to us that M G and M
G are different from one
another. The W
∗ -subalgebra of M
G generated by all E
g (F) := E
g ({F}) (F ∈ g
∗
)
is naturally isomorphic to M G . Hence M
G is larger than M G which can be injected
into M
G via the last mentioned isomorphism.
5.1.32 Let us now introduce the mapping p M :
p M : S(A
) → S ∗ (M
G ), ω → p M ω,
(5.1.128)
where p M ω is the restriction to M
G of the canonical extension of the state ω ∈
S(A
) to the normal state on (A
)
∗∗ . Any state ω ∈ S(A
) (resp. ω ∈ S(M
G )) can
be uniquely decomposed as
ω = ω( p G ) p G ω + ω(I − p G ) ω ◦ ,
(5.1.129)
where the symbols p G ω(x) and ω ◦ (x) are given by
p G ω(x) :=
1
ω( p G )
ω(x p G ), ω ◦ (x) :=
1
ω(I − p G )
ω(x(I − p G )).
Hence for ω(I − p G ) = 0 it is p M ω ◦ = m ◦ := the pure state in S(M
G ) sup −
ported by the minimal projector I − p G ∈ M
G .
Let
S
g := {ω ∈ S(A
) : ω( p G ) = 1}.
(5.1.130)
In other words: S
g = p G S(A
). For ω ∈ S
g one has p M ω ∈ S ∗ (N
G ). Conversely,
each state in S ∗ (N
G ) is of the form p M ω for some states ω ∈ S
g .
5.1.33 Lemma. The projector-valued additive function of intervals in g
∗ introduced
in (5.1.125) can be extended to a unique projector-valued measure E
g : B →
E
g (B) defined on all Borel sets B in g
∗ .
Proof. The mapping
w : g → L( p G H u ), ξ → w(ξ) := exp(i X ξ ) p G ,
(5.1.131)
see (5.1.120), is strongly continuous unitary representation of the abelian group
g (group multiplication is here the vector addition) in the subspace p G H u of the
Hilbert space H u of the universal representation of A
. This can be seen with a help
of linearity of the mapping
ξ → X ξ , ξ ∈ g,
(5.1.132)
According to the SNAG-theorem ([266, Chap. X], [262, Theorem VIII.12], [120,
Chap. IV]), there is unique projection measure E
g on the dual group ˆ
g of g repre-