5.1 Multiple Systems
93
where sp( f ) denotes the spectrum of f . For any finite subset J ⊂ sp( f ) define
x J :=
ι∈J
ι p ι ∈ M G .
(5.1.100)
The finite subsets J of sp( f ) are directed by inclusion and the net {x J : finite J ⊂
sp( f )} is increasing. Any increasing net of selfadjoint elements of a W
∗ -algebra M
converges to its least upper bound in M, [274, 1.7.4]. Let x f ∈ M G be the limit of
{x J }. We claim that the function ˜
f ∈ C(M), ˜
f (m) := m(x f ) coincides with f on
M ∗ .
Let m ∈ M ∗ . Then, due to normality of m,
˜
f (m) = l.u.b.{m(x J ) : finite J ⊂ sp( f )} = l.u.b. {
ι∈J
ι m( p ι ) : finite J ⊂ sp( f )}.
(5.1.101)
But m ∈ M ∗ lies in support of the characteristic function m → m( p ι ) iff f (m) =
ι, compare 5.1.16. Hence ˜
f (m) = f (m), what we intended to prove.
5.1.23 Lemma. For a finitely additive probability measure μ on g
∗ (without any
specification of a -algebra of measurable subsets in g
∗ ) supported by s G g
∗ , (5.1.66),
the following assertions are equivalent:
(i) F
∗
g μ = μ ω on N ∗ for some ω ∈ S ∗ (N G ), i.e. μ = μ ω ◦ F
−1
g .
(ii) μ is supported by a countable subset of g
∗
.
If these conditions are fulfilled, μ is σ-additive. Any μ ω (ω ∈ S ∗ (N G )) is of the form
F
∗
g μ for some σ-additive probability Borel measure μ on g
∗ with at most countable
supporting set in s G g
∗
.
Proof. N ∗ is mapped bijectively by F g onto s G g
∗ and μ ω is supported by N ∗ for
all ω ∈ S ∗ (N G ). Hence (i) is fulfilled for μ := μ ω ◦ F
−1
g . Complete additivity of μ ω
(what is a consequence of normality of ω ) leads then to the expression
μ ω =
m∈N ∗
ω(E g (F m )) δ m , (δ m := Dirac measure at m).
(5.1.102)
Hence at most countable number of coefficients ω(E g (F m )) = 0. This proves (i) ⇒
(ii) as well as the last assertion of the Lemma. Let
μ =
j∈Z +
λ j δ F j , with λ j ≥ 0,
j
λ j = 1, F j ∈ s G g
∗
.
(5.1.103)
Then F
∗
g μ := μ ◦ F g is a Baire measure on N ∗ , hence represents a (normal) state
ω on N G . The σ-additivity is clear.
5.1.24 Lemma. Let, for ω ∈ S(N G ), μ
ω
g be the additive function of Borel subsets of
g
∗ defined in (5.1.71). Then μ
ω
g has a unique extension to a finitely additive probability
93
where sp( f ) denotes the spectrum of f . For any finite subset J ⊂ sp( f ) define
x J :=
ι∈J
ι p ι ∈ M G .
(5.1.100)
The finite subsets J of sp( f ) are directed by inclusion and the net {x J : finite J ⊂
sp( f )} is increasing. Any increasing net of selfadjoint elements of a W
∗ -algebra M
converges to its least upper bound in M, [274, 1.7.4]. Let x f ∈ M G be the limit of
{x J }. We claim that the function ˜
f ∈ C(M), ˜
f (m) := m(x f ) coincides with f on
M ∗ .
Let m ∈ M ∗ . Then, due to normality of m,
˜
f (m) = l.u.b.{m(x J ) : finite J ⊂ sp( f )} = l.u.b. {
ι∈J
ι m( p ι ) : finite J ⊂ sp( f )}.
(5.1.101)
But m ∈ M ∗ lies in support of the characteristic function m → m( p ι ) iff f (m) =
ι, compare 5.1.16. Hence ˜
f (m) = f (m), what we intended to prove.
5.1.23 Lemma. For a finitely additive probability measure μ on g
∗ (without any
specification of a -algebra of measurable subsets in g
∗ ) supported by s G g
∗ , (5.1.66),
the following assertions are equivalent:
(i) F
∗
g μ = μ ω on N ∗ for some ω ∈ S ∗ (N G ), i.e. μ = μ ω ◦ F
−1
g .
(ii) μ is supported by a countable subset of g
∗
.
If these conditions are fulfilled, μ is σ-additive. Any μ ω (ω ∈ S ∗ (N G )) is of the form
F
∗
g μ for some σ-additive probability Borel measure μ on g
∗ with at most countable
supporting set in s G g
∗
.
Proof. N ∗ is mapped bijectively by F g onto s G g
∗ and μ ω is supported by N ∗ for
all ω ∈ S ∗ (N G ). Hence (i) is fulfilled for μ := μ ω ◦ F
−1
g . Complete additivity of μ ω
(what is a consequence of normality of ω ) leads then to the expression
μ ω =
m∈N ∗
ω(E g (F m )) δ m , (δ m := Dirac measure at m).
(5.1.102)
Hence at most countable number of coefficients ω(E g (F m )) = 0. This proves (i) ⇒
(ii) as well as the last assertion of the Lemma. Let
μ =
j∈Z +
λ j δ F j , with λ j ≥ 0,
j
λ j = 1, F j ∈ s G g
∗
.
(5.1.103)
Then F
∗
g μ := μ ◦ F g is a Baire measure on N ∗ , hence represents a (normal) state
ω on N G . The σ-additivity is clear.
5.1.24 Lemma. Let, for ω ∈ S(N G ), μ
ω
g be the additive function of Borel subsets of
g
∗ defined in (5.1.71). Then μ
ω
g has a unique extension to a finitely additive probability