88
5 Macroscopic Limits
all {m F } (F ∈ g
∗
) is an open subset the closure of which is clopen, since M is a
Stonean space, see 5.1.14, and [274]. According to (i), it is the support of characteristic function corresponding to s G = c(s G g
∗
) = c(g
∗
). The projector s G is the unit
element in N G and the projector I − s G is minimal. This shows that the sum of the
characteristic functions corresponding to s G and I − s G is the characteristic function
of the whole M, i.e. M is the union of a one-point set {m ◦ } corresponding to I − s G
and of the closure of
N ∗ := M ∗ \{m ◦ } = {m F : F ∈ g
∗
},
(5.1.69)
where we set {m F } := ∅ := the empty set, if E g (F) = 0. This is (ii).
Notation: Let us introduce, for following usage, some further concepts. Let
μ
ω
ξ : B → μ
ω
ξ (B) := ω(E ξ (B)), for any ω ∈ S(M G ) and Borel B ⊂ R, (5.1.70)
be a finitely additive Borel measure on R. For mutually dual bases {ξ j : j =
1, 2, . . . n} in g and {F j : j = 1, 2, . . . n} in g
∗ define μ
ω
g on g
∗ by:
μ
ω
g (B) := ω(E ξ 1 (B 1 )E ξ 2 (B 2 ) . . . E ξ n (B n )) for B := {F ∈ g
∗
: F(ξ j ) ∈ B j }.
(5.1.71)
If ξ ∈ L
1
(μ
ω
g , g
∗
) with ξ ∈ (g
∗
)
∗
= g, then
ω(X ξ ) := μ
ω
g (ξ) =
λμ
ω
ξ (dλ).
(5.1.72)
5.1.17 Lemma. The image by the natural map defined in 5.1.14 of any factor state
ω ∈ S(A
) into S(M G ) is an equally denoted pure state ω ∈ M ∗ (:= the set of all
normal pure states on M G ).
Proof. The canonical cyclic representation {π ω , H ω , ϕ ω } of A
(here ϕ ω is the cyclic
vector in the Hilbert space H ω for the representation π ω such, that
ω(x) = (ϕ ω , π ω (x)ϕ ω )
(5.1.73)
for all x ∈ A ) corresponding to a factor state ω ∈ S(A
) has trivial center. Hence,
any projector in the center of the commutant π ω (A
)
is trivial. The canonical extension to (A
)
∗∗ (i.e. unique W
∗
−continuous) of π ω maps the bidual (A
)
∗∗ onto the
double commutant π ω (A
)
by which M G ⊂ Z is mapped into the center π ω (Z) of
this bicommutant. Since π ω (Z) ⊂ π ω (A
)
, any projector in π ω (M G ) is trivial. The
corresponding ω ∈ S(M G ) is expressed by (5.1.73) for x ∈ M G . This ω is normal:
ω ∈ S ∗ (M G ), hence there exists a unique projector s ω in (the center of) M G such,
that
ω(x) = ω(xs ω ), for all {x ∈ M G : ω(x
∗ x) = 0} ⇒ x = x(I − s ω ).
(5.1.74)
5 Macroscopic Limits
all {m F } (F ∈ g
∗
) is an open subset the closure of which is clopen, since M is a
Stonean space, see 5.1.14, and [274]. According to (i), it is the support of characteristic function corresponding to s G = c(s G g
∗
) = c(g
∗
). The projector s G is the unit
element in N G and the projector I − s G is minimal. This shows that the sum of the
characteristic functions corresponding to s G and I − s G is the characteristic function
of the whole M, i.e. M is the union of a one-point set {m ◦ } corresponding to I − s G
and of the closure of
N ∗ := M ∗ \{m ◦ } = {m F : F ∈ g
∗
},
(5.1.69)
where we set {m F } := ∅ := the empty set, if E g (F) = 0. This is (ii).
Notation: Let us introduce, for following usage, some further concepts. Let
μ
ω
ξ : B → μ
ω
ξ (B) := ω(E ξ (B)), for any ω ∈ S(M G ) and Borel B ⊂ R, (5.1.70)
be a finitely additive Borel measure on R. For mutually dual bases {ξ j : j =
1, 2, . . . n} in g and {F j : j = 1, 2, . . . n} in g
∗ define μ
ω
g on g
∗ by:
μ
ω
g (B) := ω(E ξ 1 (B 1 )E ξ 2 (B 2 ) . . . E ξ n (B n )) for B := {F ∈ g
∗
: F(ξ j ) ∈ B j }.
(5.1.71)
If ξ ∈ L
1
(μ
ω
g , g
∗
) with ξ ∈ (g
∗
)
∗
= g, then
ω(X ξ ) := μ
ω
g (ξ) =
λμ
ω
ξ (dλ).
(5.1.72)
5.1.17 Lemma. The image by the natural map defined in 5.1.14 of any factor state
ω ∈ S(A
) into S(M G ) is an equally denoted pure state ω ∈ M ∗ (:= the set of all
normal pure states on M G ).
Proof. The canonical cyclic representation {π ω , H ω , ϕ ω } of A
(here ϕ ω is the cyclic
vector in the Hilbert space H ω for the representation π ω such, that
ω(x) = (ϕ ω , π ω (x)ϕ ω )
(5.1.73)
for all x ∈ A ) corresponding to a factor state ω ∈ S(A
) has trivial center. Hence,
any projector in the center of the commutant π ω (A
)
is trivial. The canonical extension to (A
)
∗∗ (i.e. unique W
∗
−continuous) of π ω maps the bidual (A
)
∗∗ onto the
double commutant π ω (A
)
by which M G ⊂ Z is mapped into the center π ω (Z) of
this bicommutant. Since π ω (Z) ⊂ π ω (A
)
, any projector in π ω (M G ) is trivial. The
corresponding ω ∈ S(M G ) is expressed by (5.1.73) for x ∈ M G . This ω is normal:
ω ∈ S ∗ (M G ), hence there exists a unique projector s ω in (the center of) M G such,
that
ω(x) = ω(xs ω ), for all {x ∈ M G : ω(x
∗ x) = 0} ⇒ x = x(I − s ω ).
(5.1.74)