5.1 Multiple Systems
83
F : O → g
∗
, ω → F(ω) := F ω ,
(5.1.47)
maps the orbit O onto an orbit of Ad
∗
(G). Let
[ω] := F
−1
(F ω ), for ω ∈ O , be equivalence classes in O .
The corresponding factor space M is mapped by F (which is constant on classes [ω])
bijectively onto the orbit G · F . The last orbit is endowed by the Kirillov-Kostant
symplectic structure. The functions f ξ on M :
[ω] → f ξ (ω) := ω(X ξ ), ω ∈ O , ξ ∈ g,
(5.1.48)
are the Hamiltonian functions generating the flows
(t; [ω]) → [exp(tξ) · ω].
(5.1.49)
Corresponding Poisson brackets are:
{ f ξ , f η }([ω]) = −F ω ([ξ, η]), ξ, η ∈ g,
(5.1.50)
compare e.g. (3.2.2). Here it is assumed that M is endowed by the manifold structure
of the Ad
∗
(G)-orbit F(M ). We have obtained here classical phase spaces from
equivalence classes of states in S(A
) determined by the group action σ
∗
G . Although
the construction is formally parallel to that in the case of finite systems, there are
certain physically significant differences in the interpretation, as mentioned in 1.1.6.
5.1.11 Let P G be the orthogonal projector in L(H ) onto the subspace of H
spanned by all product vectors ∈ D (g). The operator P G is equal to the sum of all
mutually orthogonal projectors P
w
corresponding to the product vectors ∈ D (g),
as is seen from (5.1.35) and obvious commutativity of any A with all the U z ,
(5.1.18). Hence
P G ∈ Z
#
:= the center of B
#
,
B
#
:= (A
)
:= the weak operator closure of A
in L(H )
(commas denote here the double commutant). The mapping
ρ : A
→ P G B
#
, x → P G x,
(5.1.51)
is a *-representation of the C
∗ -algebra A
in the Hilbert space P G H .
The representation ρ can be uniquely extended to a W
∗ -representation of the W
∗ -
algebra (i.e. abstract von Neumann algebra) (A
)
∗∗
:= the double dual of A
, see
[274, 1.21.13]. (The unique extensions of mappings from a C
∗ -algebra to mappings
from its double dual will be usually denoted by the same symbols used for the original
mappings.) The image of (A
)
∗∗ under ρ is P G B
# . Let
83
F : O → g
∗
, ω → F(ω) := F ω ,
(5.1.47)
maps the orbit O onto an orbit of Ad
∗
(G). Let
[ω] := F
−1
(F ω ), for ω ∈ O , be equivalence classes in O .
The corresponding factor space M is mapped by F (which is constant on classes [ω])
bijectively onto the orbit G · F . The last orbit is endowed by the Kirillov-Kostant
symplectic structure. The functions f ξ on M :
[ω] → f ξ (ω) := ω(X ξ ), ω ∈ O , ξ ∈ g,
(5.1.48)
are the Hamiltonian functions generating the flows
(t; [ω]) → [exp(tξ) · ω].
(5.1.49)
Corresponding Poisson brackets are:
{ f ξ , f η }([ω]) = −F ω ([ξ, η]), ξ, η ∈ g,
(5.1.50)
compare e.g. (3.2.2). Here it is assumed that M is endowed by the manifold structure
of the Ad
∗
(G)-orbit F(M ). We have obtained here classical phase spaces from
equivalence classes of states in S(A
) determined by the group action σ
∗
G . Although
the construction is formally parallel to that in the case of finite systems, there are
certain physically significant differences in the interpretation, as mentioned in 1.1.6.
5.1.11 Let P G be the orthogonal projector in L(H ) onto the subspace of H
spanned by all product vectors ∈ D (g). The operator P G is equal to the sum of all
mutually orthogonal projectors P
w
corresponding to the product vectors ∈ D (g),
as is seen from (5.1.35) and obvious commutativity of any A with all the U z ,
(5.1.18). Hence
P G ∈ Z
#
:= the center of B
#
,
B
#
:= (A
)
:= the weak operator closure of A
in L(H )
(commas denote here the double commutant). The mapping
ρ : A
→ P G B
#
, x → P G x,
(5.1.51)
is a *-representation of the C
∗ -algebra A
in the Hilbert space P G H .
The representation ρ can be uniquely extended to a W
∗ -representation of the W
∗ -
algebra (i.e. abstract von Neumann algebra) (A
)
∗∗
:= the double dual of A
, see
[274, 1.21.13]. (The unique extensions of mappings from a C
∗ -algebra to mappings
from its double dual will be usually denoted by the same symbols used for the original
mappings.) The image of (A
)
∗∗ under ρ is P G B
# . Let