5.1 Multiple Systems
95
Any state on M G can be, on the other hand, extended to some states on (A
)
∗∗
(not normal—in general) and these determine their restrictions to A
considered as a
subalgebra of its bidual. In this way, we can obtain also those states on P G A
which
are not expressible by density matrices. Hence to general finitely additive probability
measures on N ∗ ‘correspond’, in some many-to-many way, arbitrary states on P G A
.
We intend now to change our ascription of states on M G to arbitrary states on A
in such a way, that any state on P G A
will be mapped into S(N G ) (and not onto
m ◦ ∈ M ∗ as before).
5.1.27 Quasilocal structure of A
: The algebra A
has a natural quasilocal structure in the sense of 1.4.2. It is generated by local algebras A v := A
N
(N ∈ ),
(1.4.1), where, in the notations of 5.1.3, A
N is generated by π j (y) (y ∈ L(H), j =
1, 2, . . . N ) and is isomorphic to L(H N ). H N is here the N −fold tensor product
of the Hilbert space H, (5.1.10). Denote by A L the set of all finite linear combinations of finite products of arbitrary elements y ∈ A
N for any finite N . The algebra
A L := ∪ finite N A
N is called the ‘local algebra’ and its elements are ‘local observables’. The norm closure of A L is A
= the algebra of quasilocal observables of our
system.
A locally normal state ω ∈ S(A
), i.e. a state the restriction of which to any
local subalgebra A
N is normal (cf. 1.4.3), can be calculated (with a use of natural
isomorphisms) on all the elements x ∈ A
N
⊂ A
with the help of density matrices
ρ
N
ω on H N (N = 1, 2, . . . ) via the usual formula
ω(x) = T r(ρ
N
ω x), x ∈ L(H N ),
(5.1.108)
where we have identified A
N with L(H N ). Let S L (A
) =: S L denotes the set of all
locally normal states on A
. The states expressible (globally) by a density matrix
in the defining representation of A
in H are locally normal. A
is simple, [53,
2.6.20].
5.1.28 Example. We shall illustrate here the fact that a strongly continuous one
parameter group of unitaries exp(it P) acting on a Hilbert space H need not be
continuous in certain other representations of L(H).
Let A:= L(H) be the considered W
∗ -algebra, H:= L
2
(R), and Q (resp. P) be
the selfadjoint operator on H defined on ϕ ∈ C
1
0 (R) by (Qϕ)(λ) := λϕ(λ) (resp.
(Pϕ)(λ) := −i
d
dλ
ϕ(λ)), λ ∈ R. Let M be the maximal commutative W
∗ -algebra in
L(H) generated by exp(it Q), t ∈ R. Let χ λ be the pure state on M determined by
χ λ (exp(it Q)) := exp(itλ), t ∈ R.
(5.1.109)
Let ω λ be an extension of χ λ onto the whole W
∗ -algebra A. We claim that the function
t → ω λ (exp(it P)), t ∈ R,
(5.1.110)
is discontinuous, hence the group π λ (exp(it P)) of unitaries in the cyclic representation π λ of A corresponding to the state ω λ ∈ S(A) is not strongly continuous. Since
95
Any state on M G can be, on the other hand, extended to some states on (A
)
∗∗
(not normal—in general) and these determine their restrictions to A
considered as a
subalgebra of its bidual. In this way, we can obtain also those states on P G A
which
are not expressible by density matrices. Hence to general finitely additive probability
measures on N ∗ ‘correspond’, in some many-to-many way, arbitrary states on P G A
.
We intend now to change our ascription of states on M G to arbitrary states on A
in such a way, that any state on P G A
will be mapped into S(N G ) (and not onto
m ◦ ∈ M ∗ as before).
5.1.27 Quasilocal structure of A
: The algebra A
has a natural quasilocal structure in the sense of 1.4.2. It is generated by local algebras A v := A
N
(N ∈ ),
(1.4.1), where, in the notations of 5.1.3, A
N is generated by π j (y) (y ∈ L(H), j =
1, 2, . . . N ) and is isomorphic to L(H N ). H N is here the N −fold tensor product
of the Hilbert space H, (5.1.10). Denote by A L the set of all finite linear combinations of finite products of arbitrary elements y ∈ A
N for any finite N . The algebra
A L := ∪ finite N A
N is called the ‘local algebra’ and its elements are ‘local observables’. The norm closure of A L is A
= the algebra of quasilocal observables of our
system.
A locally normal state ω ∈ S(A
), i.e. a state the restriction of which to any
local subalgebra A
N is normal (cf. 1.4.3), can be calculated (with a use of natural
isomorphisms) on all the elements x ∈ A
N
⊂ A
with the help of density matrices
ρ
N
ω on H N (N = 1, 2, . . . ) via the usual formula
ω(x) = T r(ρ
N
ω x), x ∈ L(H N ),
(5.1.108)
where we have identified A
N with L(H N ). Let S L (A
) =: S L denotes the set of all
locally normal states on A
. The states expressible (globally) by a density matrix
in the defining representation of A
in H are locally normal. A
is simple, [53,
2.6.20].
5.1.28 Example. We shall illustrate here the fact that a strongly continuous one
parameter group of unitaries exp(it P) acting on a Hilbert space H need not be
continuous in certain other representations of L(H).
Let A:= L(H) be the considered W
∗ -algebra, H:= L
2
(R), and Q (resp. P) be
the selfadjoint operator on H defined on ϕ ∈ C
1
0 (R) by (Qϕ)(λ) := λϕ(λ) (resp.
(Pϕ)(λ) := −i
d
dλ
ϕ(λ)), λ ∈ R. Let M be the maximal commutative W
∗ -algebra in
L(H) generated by exp(it Q), t ∈ R. Let χ λ be the pure state on M determined by
χ λ (exp(it Q)) := exp(itλ), t ∈ R.
(5.1.109)
Let ω λ be an extension of χ λ onto the whole W
∗ -algebra A. We claim that the function
t → ω λ (exp(it P)), t ∈ R,
(5.1.110)
is discontinuous, hence the group π λ (exp(it P)) of unitaries in the cyclic representation π λ of A corresponding to the state ω λ ∈ S(A) is not strongly continuous. Since