84
5 Macroscopic Limits
s G ∈ Z := the center of (A
)
∗∗
,
be the support of ρ, i.e. (I − s G )(A
)
∗∗ is the kernel of ρ (I is here the identity of
(A
)
∗∗ ). The restriction ρ G to s G (A
)
∗∗ of ρ is an isomorphism of W
∗ -algebras
(which is σ - σ continuous, see [274, 1.21.13+4.1.23]). Let S g ⊂ S(A
) consists
of such states ω, the central supports s ω ∈ Z of which are contained in s G , i.e.
s ω s G = s ω (the central support of a state is defined as the central support, equiv.
central cover—cf. [235, 3.8.1], [306], [274, 1.14.2], of the extension to (A
)
∗∗ of
the corresponding cyclic representation of A
). The set S g will play an important
role in the following.
The automorphisms σ g (g ∈ G), (5.1.25), have unique extensions to automorphisms of the W
∗ -algebra (A
)
∗∗ , which are σ-σ and also norm–norm continuous,
[274, 1.21.13]. The σ g can be understood also as an (uniquely defined) automorphism
of the von Neumann algebra B
# . Due to Proposition 5.1.9, it is
σ g (P G ) = P G for all g ∈ G,
(5.1.52)
hence also
σ g (s G ) = s G , g ∈ G.
(5.1.53)
Let us keep the notation X ξ (ξ ∈ g) for the closures of the restrictions to P G H
of operators denoted previously by the same symbols. According to (5.1.37), all the
X ξ ’s have in P G H a common complete orthonormal set (a basis) of eigenvectors
consisting of product vectors ∈ D (G), with real eigenvalues. Hence, they form
a set of mutually commuting selfadjoint operators on P G H . Let E
#
ξ (B) (B :=
any Borel subset of R) be projectors forming their spectral measures E
#
ξ . All these
projectors belong to P G Z
# , since any X ξ (ξ ∈ g) is a constant on each P
w
< P G .
Define
E ξ (B) := ρ
−1
G [E
#
ξ (B)] ∈ s G Z for all ξ ∈ g and Borel B ⊂ R.
(5.1.54)
Any E ξ (ξ ∈ g) is a resolution of identity in the W
∗ -algebra s G Z. Let us define
also
E
ξ (B) := E ξ (B), if B does not contain the zero 0 ∈ R,
(5.1.55)
:= E ξ (B) + I − s G , if 0 ∈ B.
Here I is the identity of Z. Then E
ξ (ξ ∈ g) is a resolution of identity in Z.
5.1.12 Definition. Let M G be the W
∗ -subalgebra of Z generated by projectors
E ξ (B) (ξ ∈ g, B - Borel in R) and by I. M G is called the algebra of G-macroscopic
observables of the system (A
, σ G ), or simply the (G-)macroscopic algebra. Let
N G := s G M G be the W
∗ -subalgebra of M G generated by projectors E ξ (B)
and called the algebra of G-definiteness of (A
, σ G ), or sometimes also the
(G-)macroscopic algebra, if there will be no confusion possible.
5 Macroscopic Limits
s G ∈ Z := the center of (A
)
∗∗
,
be the support of ρ, i.e. (I − s G )(A
)
∗∗ is the kernel of ρ (I is here the identity of
(A
)
∗∗ ). The restriction ρ G to s G (A
)
∗∗ of ρ is an isomorphism of W
∗ -algebras
(which is σ - σ continuous, see [274, 1.21.13+4.1.23]). Let S g ⊂ S(A
) consists
of such states ω, the central supports s ω ∈ Z of which are contained in s G , i.e.
s ω s G = s ω (the central support of a state is defined as the central support, equiv.
central cover—cf. [235, 3.8.1], [306], [274, 1.14.2], of the extension to (A
)
∗∗ of
the corresponding cyclic representation of A
). The set S g will play an important
role in the following.
The automorphisms σ g (g ∈ G), (5.1.25), have unique extensions to automorphisms of the W
∗ -algebra (A
)
∗∗ , which are σ-σ and also norm–norm continuous,
[274, 1.21.13]. The σ g can be understood also as an (uniquely defined) automorphism
of the von Neumann algebra B
# . Due to Proposition 5.1.9, it is
σ g (P G ) = P G for all g ∈ G,
(5.1.52)
hence also
σ g (s G ) = s G , g ∈ G.
(5.1.53)
Let us keep the notation X ξ (ξ ∈ g) for the closures of the restrictions to P G H
of operators denoted previously by the same symbols. According to (5.1.37), all the
X ξ ’s have in P G H a common complete orthonormal set (a basis) of eigenvectors
consisting of product vectors ∈ D (G), with real eigenvalues. Hence, they form
a set of mutually commuting selfadjoint operators on P G H . Let E
#
ξ (B) (B :=
any Borel subset of R) be projectors forming their spectral measures E
#
ξ . All these
projectors belong to P G Z
# , since any X ξ (ξ ∈ g) is a constant on each P
w
< P G .
Define
E ξ (B) := ρ
−1
G [E
#
ξ (B)] ∈ s G Z for all ξ ∈ g and Borel B ⊂ R.
(5.1.54)
Any E ξ (ξ ∈ g) is a resolution of identity in the W
∗ -algebra s G Z. Let us define
also
E
ξ (B) := E ξ (B), if B does not contain the zero 0 ∈ R,
(5.1.55)
:= E ξ (B) + I − s G , if 0 ∈ B.
Here I is the identity of Z. Then E
ξ (ξ ∈ g) is a resolution of identity in Z.
5.1.12 Definition. Let M G be the W
∗ -subalgebra of Z generated by projectors
E ξ (B) (ξ ∈ g, B - Borel in R) and by I. M G is called the algebra of G-macroscopic
observables of the system (A
, σ G ), or simply the (G-)macroscopic algebra. Let
N G := s G M G be the W
∗ -subalgebra of M G generated by projectors E ξ (B)
and called the algebra of G-definiteness of (A
, σ G ), or sometimes also the
(G-)macroscopic algebra, if there will be no confusion possible.