5.1 Multiple Systems
103
(v) Let μ ω be the canonical measure on the spectrum space M
of M
G =
C(M
) corresponding to the state p M ω ∈ S ∗ (M
G ), ω ∈ S(A
). Then there is
a canonically defined μ ω −measurable function ˆ
ω(m) =: ω m (spaces are taken
with their w
∗ -topologies) such that the restriction r M ω m = m ∈ ES(M
G ):
r M : (A
)
∗∗∗
→ (M
G )
∗ is the natural restriction
and
ω(x) =
M
ω m (x)μ ω (dm) for any x ∈ (A
)
∗∗
[⊃ A
].
(5.1.146)
Proof. (i) is clear from the definition of p M , compare also [53, 4.1.36]. (ii) is a
rephrasing of (5.1.138). Since δ F corresponds to a pure state on M
G and p M is affine,
S F is a face. Closedness of S F follows from the continuity of p M , and convexity is
clear. The rest of (iii) is contained in the concluding remark of 5.1.36 which implies
also the first statement of (iv). A proof of the second statement of (iv) is an easy
adaptation of that of 5.1.17 for the case of factor states. It remains to prove (v):
Let ˜
ω ∈ S ∗ ((A
)
∗∗
) be the unique normal extension of ω ∈ S(A
) and
(π ω , H ω , , ω ) be the corresponding cyclic representation of (A
)
∗∗ . Denote by ˆ
μ ω
the orthogonal measure (cf. [53, 4.1.20]) on S((A
)
∗∗
) corresponding to the canonical decomposition of ˜
ω with respect to the subalgebra π ω (M
G ) of the center of
π ω ((A
)
∗∗
), compare [53, 4.1.25]:
˜
ω(x) =
ϕ(x) ˆ
μ ω (dϕ), for all x ∈ (A
)
∗∗
.
(5.1.147)
The mapping
y (∈ M
G ) → ˆ
y (∈ C(S((A
)
∗∗
))), ˆ
y(ϕ) := ϕ(y) ,
restricted to the subalgebra p ω M
G (which is isomorphic to π ω (M
G ) for the uniquely
determined projector p ω ∈ M
G ) provides an isomorphism of the W
∗ -algebras
p ω M
G and L
∞
( ˆ
μ ω ), [118, Chap. I.9] and [53, 4.1.22]. Hence, for y j ∈ p ω M
G ( j =
1, 2) we have
(y 1 y 2 )(ϕ) = ˆ
y 1 (ϕ) ˆ
y 2 (ϕ), for ϕ ∈ supp ˆ
μ ω .
(5.1.148)
Clearly, ϕ(y) = 0 for y ∈ (I − p ϕ )M
G and ϕ ∈ supp ˆ
μ ω . This fact together
with (5.1.148) implies that the restriction r M ϕ is a pure state on M
G for ϕ ∈
supp ˆ
μ ω , r M ϕ =: m ϕ ∈ M
. The w
∗ -topology of the state space is Hausdorff and
the clopen sets form a basis of the topology of M
. This and the isomorphism of
103
(v) Let μ ω be the canonical measure on the spectrum space M
of M
G =
C(M
) corresponding to the state p M ω ∈ S ∗ (M
G ), ω ∈ S(A
). Then there is
a canonically defined μ ω −measurable function ˆ
ω(m) =: ω m (spaces are taken
with their w
∗ -topologies) such that the restriction r M ω m = m ∈ ES(M
G ):
r M : (A
)
∗∗∗
→ (M
G )
∗ is the natural restriction
and
ω(x) =
M
ω m (x)μ ω (dm) for any x ∈ (A
)
∗∗
[⊃ A
].
(5.1.146)
Proof. (i) is clear from the definition of p M , compare also [53, 4.1.36]. (ii) is a
rephrasing of (5.1.138). Since δ F corresponds to a pure state on M
G and p M is affine,
S F is a face. Closedness of S F follows from the continuity of p M , and convexity is
clear. The rest of (iii) is contained in the concluding remark of 5.1.36 which implies
also the first statement of (iv). A proof of the second statement of (iv) is an easy
adaptation of that of 5.1.17 for the case of factor states. It remains to prove (v):
Let ˜
ω ∈ S ∗ ((A
)
∗∗
) be the unique normal extension of ω ∈ S(A
) and
(π ω , H ω , , ω ) be the corresponding cyclic representation of (A
)
∗∗ . Denote by ˆ
μ ω
the orthogonal measure (cf. [53, 4.1.20]) on S((A
)
∗∗
) corresponding to the canonical decomposition of ˜
ω with respect to the subalgebra π ω (M
G ) of the center of
π ω ((A
)
∗∗
), compare [53, 4.1.25]:
˜
ω(x) =
ϕ(x) ˆ
μ ω (dϕ), for all x ∈ (A
)
∗∗
.
(5.1.147)
The mapping
y (∈ M
G ) → ˆ
y (∈ C(S((A
)
∗∗
))), ˆ
y(ϕ) := ϕ(y) ,
restricted to the subalgebra p ω M
G (which is isomorphic to π ω (M
G ) for the uniquely
determined projector p ω ∈ M
G ) provides an isomorphism of the W
∗ -algebras
p ω M
G and L
∞
( ˆ
μ ω ), [118, Chap. I.9] and [53, 4.1.22]. Hence, for y j ∈ p ω M
G ( j =
1, 2) we have
(y 1 y 2 )(ϕ) = ˆ
y 1 (ϕ) ˆ
y 2 (ϕ), for ϕ ∈ supp ˆ
μ ω .
(5.1.148)
Clearly, ϕ(y) = 0 for y ∈ (I − p ϕ )M
G and ϕ ∈ supp ˆ
μ ω . This fact together
with (5.1.148) implies that the restriction r M ϕ is a pure state on M
G for ϕ ∈
supp ˆ
μ ω , r M ϕ =: m ϕ ∈ M
. The w
∗ -topology of the state space is Hausdorff and
the clopen sets form a basis of the topology of M
. This and the isomorphism of