86
5 Macroscopic Limits
an orthogonal projector iff the corresponding element ˆ
x ∈ C(M) is characteristic
function of some Borel subset B of M, i.e.
ˆ
x(m) = χ B (m) for all m ∈ M.
A pure state m ∈ M is normal, iff the characteristic function χ {m} of the one-point
set {m} is continuous, χ {m} ∈ C(M). This means, that normal pure states on M G
are just the isolated points of M. The corresponding projectors χ {m} are minimal
projectors in M G ∼ C(M). The spectrum space M is Hausdorff and the family of
clopen (i.e. closed and open) sets forms a basis of the topology of M, cf. [274].
Hence, any minimal projector in C(M) is of the form χ {m} .
Any state ω ∈ S(M G ) is represented by a probability Baire (i.e. regular Borel)
measure on M and any such measure μ ω represents a state on M G : ω(x) = μ ω ( ˆ
x),
where x in the left hand side is an element of the abstract algebra M G and ˆ
x in
the right hand side denotes the corresponding function ˆ
x ∈ C(M). Any pure state
m ∈ M corresponds to the Dirac measure δ m .
5.1.15 The algebra M G (and also N G ) is σ G −invariant:
σ g x ∈ N G for all g ∈ G and any x ∈ N G .
(5.1.59)
This is a consequence of the relation (compare the proof of 5.1.9)
U (g)X ξ U (g
−1
) = X Ad(g)ξ , (g ∈ G, ξ ∈ g),
(5.1.60)
what implies
σ g [E g (B)] = E Ad(g)ξ (B) (g ∈ G and Borel B ⊂ R),
(5.1.61)
due to uniqueness of spectral measures of selfadjoint operators and also due to
continuity properties of the used mappings. From (5.1.61), we obtain immediately
(by calculation of the eigenvalues of X ξ ):
σ g [E g (F)] = E g (Ad
∗
(g)F), (g ∈ G, F ∈ g
∗
).
(5.1.62)
This specifies, according to 5.1.13 and 5.1.14, the action of G on the set of all normal pure states on N G . The remaining normal pure state on M G corresponds to
the σ G -invariant minimal projector I − s G . Hence, σ G acts on M G as a group of
W
∗
−automorphisms and σ
∗
G acts on M (resp. on S(M G )) as a group of homeomorphisms (resp. a group of continuous affine transformations). As a consequence, the
orbits
O ω := {σ
∗
g ω : g ∈ G} ⊂ S(A
)
(5.1.63)
5 Macroscopic Limits
an orthogonal projector iff the corresponding element ˆ
x ∈ C(M) is characteristic
function of some Borel subset B of M, i.e.
ˆ
x(m) = χ B (m) for all m ∈ M.
A pure state m ∈ M is normal, iff the characteristic function χ {m} of the one-point
set {m} is continuous, χ {m} ∈ C(M). This means, that normal pure states on M G
are just the isolated points of M. The corresponding projectors χ {m} are minimal
projectors in M G ∼ C(M). The spectrum space M is Hausdorff and the family of
clopen (i.e. closed and open) sets forms a basis of the topology of M, cf. [274].
Hence, any minimal projector in C(M) is of the form χ {m} .
Any state ω ∈ S(M G ) is represented by a probability Baire (i.e. regular Borel)
measure on M and any such measure μ ω represents a state on M G : ω(x) = μ ω ( ˆ
x),
where x in the left hand side is an element of the abstract algebra M G and ˆ
x in
the right hand side denotes the corresponding function ˆ
x ∈ C(M). Any pure state
m ∈ M corresponds to the Dirac measure δ m .
5.1.15 The algebra M G (and also N G ) is σ G −invariant:
σ g x ∈ N G for all g ∈ G and any x ∈ N G .
(5.1.59)
This is a consequence of the relation (compare the proof of 5.1.9)
U (g)X ξ U (g
−1
) = X Ad(g)ξ , (g ∈ G, ξ ∈ g),
(5.1.60)
what implies
σ g [E g (B)] = E Ad(g)ξ (B) (g ∈ G and Borel B ⊂ R),
(5.1.61)
due to uniqueness of spectral measures of selfadjoint operators and also due to
continuity properties of the used mappings. From (5.1.61), we obtain immediately
(by calculation of the eigenvalues of X ξ ):
σ g [E g (F)] = E g (Ad
∗
(g)F), (g ∈ G, F ∈ g
∗
).
(5.1.62)
This specifies, according to 5.1.13 and 5.1.14, the action of G on the set of all normal pure states on N G . The remaining normal pure state on M G corresponds to
the σ G -invariant minimal projector I − s G . Hence, σ G acts on M G as a group of
W
∗
−automorphisms and σ
∗
G acts on M (resp. on S(M G )) as a group of homeomorphisms (resp. a group of continuous affine transformations). As a consequence, the
orbits
O ω := {σ
∗
g ω : g ∈ G} ⊂ S(A
)
(5.1.63)