5.1 Multiple Systems
105
(ii) F g (m) := m ◦ iff m(I − p G ) = 1, i.e. iff m = m ◦ ∈ M
, 5.1.32.
(iii) F g (m) := (∞) iff m(I − p G ) = 0 and (5.1.152) is false for all bounded Borel
subsets B ⊂ g
∗
: m(E
g (B)) = 0.
The same definition applied to all m ∈ Z leads to the mapping
F g ◦ r M : Z → ˙
M
G
, Z := C(Z),
(5.1.153)
which is continuous on the whole Z.
The mapping F g ◦ r M determines the projectors E
g (B). For bounded B we have
(B := closur e, B
◦
:= interior)
5
m(E
g (B)) = 1 iff m ∈ [(F g ◦ r M ) −1 (B)] ◦ = [(F g ◦ r M )
−1
(B)]
◦
.
(5.1.154)
If we extend the Ad
∗
(G) to the whole ˙
M
G by the requirement of Ad
∗
(G)invariance of the points m ◦ and (∞) we see, that F g is G-equivariant:
F g (σ
∗
g m) = Ad
∗
(g)F g (m), for all m ∈ M
, g ∈ G.
(5.1.155)
5.1.40 The projection measure E
g on g
∗ together with the Ad
∗ -action of G determine a macroscopic limit of the system (A
, σ G ). This formulation together with
the mapping p M of S(A
) into the classical macroscopic states of the system will
enable us to generalize the notion of the macroscopic limit to much more general
situations. We shall investigate also the dynamics of the system (A
, σ G ) (resp. of
its generalizations) if the time evolution were not included in the action σ G as the
action of a one parameter subgroup of G. The action σ G of the ‘kinematical group’ G
allows us, as we shall show, to introduce rather wide class of ‘mean-field-type’ time
evolutions connected with noncompact groups G—at least for a large σ G -invariant
subset of states in S(A
). Also automorphic time evolutions τ : t → τ t ∈
∗ - Aut A
of a system (A, σ G , τ R ) will be considered.
5.2 Generalized Macroscopic Limits
5.2.1 We have considered, in the preceding section, a macroscopic limit of the
system (A
, σ G ). This system was of a rather special type: the algebra A
was the
infinite tensor product of identical copies A j ( j ∈ Z + =: ) of a C
∗ -algebra A 0 and
the automorphism group σ G left each of the copies A j invariant: σ g x ∈ A j for each
x ∈ A j , for all g ∈ G and any j ∈ Z + ≡ . We shall now generalize the procedure
of obtaining a macroscopic limit to much more general situations. We shall ignore
here possible quasilocal structures of the considered C
∗ -algebra A; the usage of the
5 The relation (5.1.154) has been proved in the assumption that any projector p ∈ M
G is of the form
p = E
g (B) for some B ⊂ g ∗ , if p(I − p G ) = 0.
105
(ii) F g (m) := m ◦ iff m(I − p G ) = 1, i.e. iff m = m ◦ ∈ M
, 5.1.32.
(iii) F g (m) := (∞) iff m(I − p G ) = 0 and (5.1.152) is false for all bounded Borel
subsets B ⊂ g
∗
: m(E
g (B)) = 0.
The same definition applied to all m ∈ Z leads to the mapping
F g ◦ r M : Z → ˙
M
G
, Z := C(Z),
(5.1.153)
which is continuous on the whole Z.
The mapping F g ◦ r M determines the projectors E
g (B). For bounded B we have
(B := closur e, B
◦
:= interior)
5
m(E
g (B)) = 1 iff m ∈ [(F g ◦ r M ) −1 (B)] ◦ = [(F g ◦ r M )
−1
(B)]
◦
.
(5.1.154)
If we extend the Ad
∗
(G) to the whole ˙
M
G by the requirement of Ad
∗
(G)invariance of the points m ◦ and (∞) we see, that F g is G-equivariant:
F g (σ
∗
g m) = Ad
∗
(g)F g (m), for all m ∈ M
, g ∈ G.
(5.1.155)
5.1.40 The projection measure E
g on g
∗ together with the Ad
∗ -action of G determine a macroscopic limit of the system (A
, σ G ). This formulation together with
the mapping p M of S(A
) into the classical macroscopic states of the system will
enable us to generalize the notion of the macroscopic limit to much more general
situations. We shall investigate also the dynamics of the system (A
, σ G ) (resp. of
its generalizations) if the time evolution were not included in the action σ G as the
action of a one parameter subgroup of G. The action σ G of the ‘kinematical group’ G
allows us, as we shall show, to introduce rather wide class of ‘mean-field-type’ time
evolutions connected with noncompact groups G—at least for a large σ G -invariant
subset of states in S(A
). Also automorphic time evolutions τ : t → τ t ∈
∗ - Aut A
of a system (A, σ G , τ R ) will be considered.
5.2 Generalized Macroscopic Limits
5.2.1 We have considered, in the preceding section, a macroscopic limit of the
system (A
, σ G ). This system was of a rather special type: the algebra A
was the
infinite tensor product of identical copies A j ( j ∈ Z + =: ) of a C
∗ -algebra A 0 and
the automorphism group σ G left each of the copies A j invariant: σ g x ∈ A j for each
x ∈ A j , for all g ∈ G and any j ∈ Z + ≡ . We shall now generalize the procedure
of obtaining a macroscopic limit to much more general situations. We shall ignore
here possible quasilocal structures of the considered C
∗ -algebra A; the usage of the
5 The relation (5.1.154) has been proved in the assumption that any projector p ∈ M
G is of the form
p = E
g (B) for some B ⊂ g ∗ , if p(I − p G ) = 0.