98
5 Macroscopic Limits
it the identity I of Z; it will be called the G-macroscopic algebra of (A
, σ G ).
The relation between M
G and the previously introduced M G , 5.1.12, is clear without
any proof:
5.1.30 Lemma. N G = s G N
G = s G M
G = s G M G , where the projector s G ∈ Z was
introduced in 5.1.11.
5.1.31 We shall use concepts and notations connected with the usage of M
G in
analogy to those connected with M G , as they were introduced above. Let e.g., B ⊂ g
∗
be a Borel set (with respect to the usual topology of a finite dimensional vector space),
and ξ j ( j = 1, 2, . . . n) form a basis of g. Let
ξ j B := {λ ∈ R : λ = F(ξ j ), F ∈ B}
(5.1.123)
be the projection of B onto the j-th coordinate axis of the dual frame. If B has the
form
B = {F ∈ g
∗
: F(ξ j ) ∈ ξ j B, ∀ j ∈ {1, 2, . . . n := dim G}},
(5.1.124)
then we set
E
g (B) := E ξ 1 (ξ 1 B)E ξ 2 (ξ 2 B) . . . E ξ n (ξ n B).
(5.1.125)
The W
∗ -algebra M
G is generated by the projectors E
g (B) from (5.1.125) and
(5.1.124), and by the unit I ∈ Z.
The algebra M
G , contrary to M G , cannot be built from projectors (5.1.125)
corresponding to one point sets B := {F} (F ∈ g
∗
) only. This can be seen as follows:
Choose a probability Borel measure μ on s G g
∗ , in the old notation from 5.1.16, such
that any point (Dirac) measure is singular with respect to it: μ({F}) = 0 for all
F ∈ g
∗ . Choose a product vector (F) ∈ E
#
g (F)P G H , one and only one for each
such F ∈ g
∗ , for which E
#
g (F) = 0. Denote by ω
F
:= ω
(F) the corresponding state
on A
. Assume, that all the functions
F → ω
F
(x), x ∈ A
,
(5.1.126)
are μ−measurable. This last assumption is trivially fulfilled, if μ is concentrated on
an Ad
∗
(G) orbit G · F ⊂ g
∗ and ω
g·F
:= σ
∗
g ω
F . Define then the state ω μ ∈ S(A
)
by
ω μ (x) :=
g ∗
ω
F
(x) μ(dF).
(5.1.127)
In this way, we can construct states ω μ the central supports s μ ∈ Z of which are
contained in p G , s μ p G = s μ , but s μ s G = 0, as well as s μ E
g (F) = 0 for all F ∈ g
∗
,
in M
G .
5 Macroscopic Limits
it the identity I of Z; it will be called the G-macroscopic algebra of (A
, σ G ).
The relation between M
G and the previously introduced M G , 5.1.12, is clear without
any proof:
5.1.30 Lemma. N G = s G N
G = s G M
G = s G M G , where the projector s G ∈ Z was
introduced in 5.1.11.
5.1.31 We shall use concepts and notations connected with the usage of M
G in
analogy to those connected with M G , as they were introduced above. Let e.g., B ⊂ g
∗
be a Borel set (with respect to the usual topology of a finite dimensional vector space),
and ξ j ( j = 1, 2, . . . n) form a basis of g. Let
ξ j B := {λ ∈ R : λ = F(ξ j ), F ∈ B}
(5.1.123)
be the projection of B onto the j-th coordinate axis of the dual frame. If B has the
form
B = {F ∈ g
∗
: F(ξ j ) ∈ ξ j B, ∀ j ∈ {1, 2, . . . n := dim G}},
(5.1.124)
then we set
E
g (B) := E ξ 1 (ξ 1 B)E ξ 2 (ξ 2 B) . . . E ξ n (ξ n B).
(5.1.125)
The W
∗ -algebra M
G is generated by the projectors E
g (B) from (5.1.125) and
(5.1.124), and by the unit I ∈ Z.
The algebra M
G , contrary to M G , cannot be built from projectors (5.1.125)
corresponding to one point sets B := {F} (F ∈ g
∗
) only. This can be seen as follows:
Choose a probability Borel measure μ on s G g
∗ , in the old notation from 5.1.16, such
that any point (Dirac) measure is singular with respect to it: μ({F}) = 0 for all
F ∈ g
∗ . Choose a product vector (F) ∈ E
#
g (F)P G H , one and only one for each
such F ∈ g
∗ , for which E
#
g (F) = 0. Denote by ω
F
:= ω
(F) the corresponding state
on A
. Assume, that all the functions
F → ω
F
(x), x ∈ A
,
(5.1.126)
are μ−measurable. This last assumption is trivially fulfilled, if μ is concentrated on
an Ad
∗
(G) orbit G · F ⊂ g
∗ and ω
g·F
:= σ
∗
g ω
F . Define then the state ω μ ∈ S(A
)
by
ω μ (x) :=
g ∗
ω
F
(x) μ(dF).
(5.1.127)
In this way, we can construct states ω μ the central supports s μ ∈ Z of which are
contained in p G , s μ p G = s μ , but s μ s G = 0, as well as s μ E
g (F) = 0 for all F ∈ g
∗
,
in M
G .