5.1 Multiple Systems
87
are canonically mapped onto orbits of σ
∗
G in S(M G ) consisting of normal states on
M G . By this mapping orbits consisting of vector states ω
are
3 mapped onto orbits
in M. The functions
σ
∗
m : G → M, g → σ
∗
m (g) := σ
∗
g m, (m ∈ M)
(5.1.64)
are not continuous in the given topology on M, 5.1.14. The orbits of σ
∗
G consisting
of normal pure states on M G are, due to (5.1.62), bijective images of (some) orbits of
Ad
∗
(G) in g
∗ . It is also clear that the normal pure states on M G form a G-invariant
subset M ∗ of all states S(M G ) on M G :
σ
∗
G M ∗ = M ∗ , i.e. m ∈ M ∗ ⇒ σ
∗
g m ∈ M ∗ for all g ∈ G (σ
∗
e m ≡ m). (5.1.65)
5.1.16 Proposition. Let p = p
∗
= p
2
∈ M G be any projector and
pg
∗
:= {F ∈ g
∗
: 0 = E g (F) ≤ p}.
(5.1.66)
Let J ⊂ g
∗ be a finite set and let by p J be denoted
p J :=
F∈J
E g (F), for any finite J ⊂ g
∗
.
(5.1.67)
Denote further for any subset K ⊂ g
∗ :
c(K ) := l.u.b.{ p J : J ⊂ K , J finite}.
(5.1.68)
Assume ps G = p.
Then the following assertions are fulfilled:
(i) p = c( pg
∗
), and (ii) M = M ∗ := the closure of M ∗ .
Proof. The projector s G is constructed in such a way that ρ G (s G ) = P G and P G =
E
#
ξ (R) for any ξ ∈ g. Since ρ G is an isomorphism of N G = s G M G into Z
# , 5.1.11, it
is s G = c(s G g
∗
). Hence, for any projector q = qs G in M G , there is a nonzero minimal
projector E g (F ◦ ) = E g (F ◦ )q, if q is nonzero. Let q := p − c( pg
∗
) (≥ 0, according
to the definition (5.1.68)) and assume that q = 0. Let 0 = E g (F ◦ ) = q E g (F ◦ ). But
E g (F ◦ ) ≤ p, hence E g (F ◦ ))c( pg
∗
) = E g (F ◦ ). This ia a contradiction, since q is
orthogonal to c( pg
∗
). Hence q = 0, what proves (i).
Any projector in M G is represented in C(M) by the characteristic function of a
clopen set, and conversely, the characteristic function of a clopen set in M represents
by Gel’fand isomorphism a projector in M G , 5.1.14. The minimal projector E g (F)
corresponds to the one-point clopen set {m F } containing m F ∈ M ∗ . The union of
3 Where ∈ H such that there is an F ∈ g ∗ satisfying: E #
g (F)) = .
87
are canonically mapped onto orbits of σ
∗
G in S(M G ) consisting of normal states on
M G . By this mapping orbits consisting of vector states ω
are
3 mapped onto orbits
in M. The functions
σ
∗
m : G → M, g → σ
∗
m (g) := σ
∗
g m, (m ∈ M)
(5.1.64)
are not continuous in the given topology on M, 5.1.14. The orbits of σ
∗
G consisting
of normal pure states on M G are, due to (5.1.62), bijective images of (some) orbits of
Ad
∗
(G) in g
∗ . It is also clear that the normal pure states on M G form a G-invariant
subset M ∗ of all states S(M G ) on M G :
σ
∗
G M ∗ = M ∗ , i.e. m ∈ M ∗ ⇒ σ
∗
g m ∈ M ∗ for all g ∈ G (σ
∗
e m ≡ m). (5.1.65)
5.1.16 Proposition. Let p = p
∗
= p
2
∈ M G be any projector and
pg
∗
:= {F ∈ g
∗
: 0 = E g (F) ≤ p}.
(5.1.66)
Let J ⊂ g
∗ be a finite set and let by p J be denoted
p J :=
F∈J
E g (F), for any finite J ⊂ g
∗
.
(5.1.67)
Denote further for any subset K ⊂ g
∗ :
c(K ) := l.u.b.{ p J : J ⊂ K , J finite}.
(5.1.68)
Assume ps G = p.
Then the following assertions are fulfilled:
(i) p = c( pg
∗
), and (ii) M = M ∗ := the closure of M ∗ .
Proof. The projector s G is constructed in such a way that ρ G (s G ) = P G and P G =
E
#
ξ (R) for any ξ ∈ g. Since ρ G is an isomorphism of N G = s G M G into Z
# , 5.1.11, it
is s G = c(s G g
∗
). Hence, for any projector q = qs G in M G , there is a nonzero minimal
projector E g (F ◦ ) = E g (F ◦ )q, if q is nonzero. Let q := p − c( pg
∗
) (≥ 0, according
to the definition (5.1.68)) and assume that q = 0. Let 0 = E g (F ◦ ) = q E g (F ◦ ). But
E g (F ◦ ) ≤ p, hence E g (F ◦ ))c( pg
∗
) = E g (F ◦ ). This ia a contradiction, since q is
orthogonal to c( pg
∗
). Hence q = 0, what proves (i).
Any projector in M G is represented in C(M) by the characteristic function of a
clopen set, and conversely, the characteristic function of a clopen set in M represents
by Gel’fand isomorphism a projector in M G , 5.1.14. The minimal projector E g (F)
corresponds to the one-point clopen set {m F } containing m F ∈ M ∗ . The union of
3 Where ∈ H such that there is an F ∈ g ∗ satisfying: E #
g (F)) = .