5.1 Multiple Systems
97
ascription from 5.1.14. In the notations from 5.1.3 and 5.1.4, let
X ξ N :=
1
N
N
j=1
π j (X ξ ), N = 1, 2, . . . , ξ ∈ g.
(5.1.118)
Then the elements exp(it X ξ N ) ∈ A
N
(t ∈ R) are represented in the defining representation of A
in H by strongly continuous groups converging with N → ∞ in
the strong operator topology on the G−invariant subspace P G H of H to strongly
continuous central subgroups of P G B
# ,
s- lim
N
exp(it X ξ N )P G = exp(it X ξ )P G ,
(5.1.119)
see 5.1.7, 5.1.8 and 5.1.11. The algebra M G of macroscopic observables was built
from spectral projectors E
#
ξ of X ξ ’s mapped into the center Z of the bidual (A
)
∗∗ .
We want to generalize this construction. We shall identify the bidual (A
)
∗∗ with
the weak closure of the universal representation of A
(cf. [274, Def.1.16.5], [235,
3.7.6]). Let p G be the l.u.b. of all such projectors p ∈ Z, for which the limits in
σ((A
)
∗∗
, (A
)
∗
)-topology:
exp(it X ξ ) p G := σ- lim
N
exp(it X ξ N ) p G , ∀ξ ∈ g,
(5.1.120)
exist (with p G → p). The symbol X ξ denotes here a selfadjoint operator acting
on the subspace p G H u of the space H u – the space of universal representation of
A
. Here it is assumed, of course, that the groups t → exp(it X ξ N ) p G are strongly
continuous for all N ∈ . It is clear from the definitions of X ξ N and σ G , 5.1.5, that
σ G ( p G ) = p G .
(5.1.121)
The convergence in (5.1.120) means the convergence X ξ N → X ξ of selfadjoint
operators on p G H u in the strong-resolvent sense, [262]. From
[exp(it X ξ N ), y] =
⎛
⎝ exp
⎛
⎝ it
N
K
j=1
π j (X ξ )
⎞
⎠ y exp
−
it
N
K
k=1
π k (X ξ )
− y
⎞
⎠ e
it X ξ N ,
(5.1.122)
which is valid for all y ∈ A
K
(K ∈ ) and ξ ∈ g, t ∈ R, as well as from the assumed
continuity of p G exp(it X ξ N ) we conclude that the limit p G exp(it X ξ ) ∈ (A
)
∗∗
belongs to the center Z of (A
)
∗∗ . Let now the -macroscopic algebra of G -
definiteness of (A
, σ G ) be defined as the von Neumann subalgebra N
G of the
center Z generated by all the spectral projectors E ξ (B) (Borel B ⊂ R and ξ ∈ g)
of operators X ξ in p G H u (we hope that no confusion arises from the keeping an old
notation for new objects!). The algebra M
G is obtained from N
G by adjoining to
97
ascription from 5.1.14. In the notations from 5.1.3 and 5.1.4, let
X ξ N :=
1
N
N
j=1
π j (X ξ ), N = 1, 2, . . . , ξ ∈ g.
(5.1.118)
Then the elements exp(it X ξ N ) ∈ A
N
(t ∈ R) are represented in the defining representation of A
in H by strongly continuous groups converging with N → ∞ in
the strong operator topology on the G−invariant subspace P G H of H to strongly
continuous central subgroups of P G B
# ,
s- lim
N
exp(it X ξ N )P G = exp(it X ξ )P G ,
(5.1.119)
see 5.1.7, 5.1.8 and 5.1.11. The algebra M G of macroscopic observables was built
from spectral projectors E
#
ξ of X ξ ’s mapped into the center Z of the bidual (A
)
∗∗ .
We want to generalize this construction. We shall identify the bidual (A
)
∗∗ with
the weak closure of the universal representation of A
(cf. [274, Def.1.16.5], [235,
3.7.6]). Let p G be the l.u.b. of all such projectors p ∈ Z, for which the limits in
σ((A
)
∗∗
, (A
)
∗
)-topology:
exp(it X ξ ) p G := σ- lim
N
exp(it X ξ N ) p G , ∀ξ ∈ g,
(5.1.120)
exist (with p G → p). The symbol X ξ denotes here a selfadjoint operator acting
on the subspace p G H u of the space H u – the space of universal representation of
A
. Here it is assumed, of course, that the groups t → exp(it X ξ N ) p G are strongly
continuous for all N ∈ . It is clear from the definitions of X ξ N and σ G , 5.1.5, that
σ G ( p G ) = p G .
(5.1.121)
The convergence in (5.1.120) means the convergence X ξ N → X ξ of selfadjoint
operators on p G H u in the strong-resolvent sense, [262]. From
[exp(it X ξ N ), y] =
⎛
⎝ exp
⎛
⎝ it
N
K
j=1
π j (X ξ )
⎞
⎠ y exp
−
it
N
K
k=1
π k (X ξ )
− y
⎞
⎠ e
it X ξ N ,
(5.1.122)
which is valid for all y ∈ A
K
(K ∈ ) and ξ ∈ g, t ∈ R, as well as from the assumed
continuity of p G exp(it X ξ N ) we conclude that the limit p G exp(it X ξ ) ∈ (A
)
∗∗
belongs to the center Z of (A
)
∗∗ . Let now the -macroscopic algebra of G -
definiteness of (A
, σ G ) be defined as the von Neumann subalgebra N
G of the
center Z generated by all the spectral projectors E ξ (B) (Borel B ⊂ R and ξ ∈ g)
of operators X ξ in p G H u (we hope that no confusion arises from the keeping an old
notation for new objects!). The algebra M
G is obtained from N
G by adjoining to