1.4 Quantum Theory of Large Systems
17
subalgebra of A corresponding to v ⊂ V . If v 1 ⊂ v 2 ⊂ V , then A v 1 ⊂ A v 2 . All the
A v (v ∈ B(V )) have common unit ≡ the unit e := id A of A, and
v∈B(V )
A v = A,
(1.4.1)
where the over-bar denotes the uniform closure. We assume further that π( p)(A v ) =
A p·v , where p · v := {λ
∈ V : λ
= p · λ, λ ∈ v}, and p · λ denotes the action of p ∈
on the point λ ∈ V . This action is supposed continuous and bounded: p · B(V ) ⊂
B(V ) for all p ∈ . We can assume (for simplicity) that for mutually disjoint v, u ∈
B(V ), v ∩ u = ∅, we have
[x, y] = 0, for all x ∈ A v , y ∈ A u .
(1.4.2)
(The anticommutativity of Fermi systems can also be included, cf. [53, Sect. 2.6]).
11
We shall characterize this situation by saying that the algebra A is quasilocal with
respect to the action of the group . We shall use another technical assumption, that
all the local subalgebras A v are W
∗ -algebras: A W
∗ -algebra A is such a C
∗ -algebra
which is (isomorphic to a) Banach space topological dual of another B-space A ∗
called the predual of A ; such an A is always unital and generated by its projectors.
W
∗ -algebras were introduced originally as weakly closed symmetric subalgebras of
bounded operators in a Hilbert space containing identity and named von Neumann
algebras after their originator.
1.4.3 Mathematically defined states on a C
∗ -algebra A are any positive normalized
linear functionals ω on A, i.e. such ω ∈ A
∗ (:= the dual of A), that
ω(x
∗ x) ≥ 0, ω = 1 (= ω(id A )).
(1.4.3)
Not all mathematical states, however, can be used as adequate descriptions of
physical situations. As physical states on a quasilocal algebra A are usually used
locally normal states, i.e. such states ω on A, the restrictions of which to all the
local W
∗ -subalgebras A v (v ∈ B(V )) are σ(A v , (A v ) ∗ )-continuous (here (A v ) ∗ is
the predual Banach space of A v ); the local normality of ω means that the restriction
of ω to any A v is expressible by a density matrix in a faithful W
∗
−representation
of A v . We shall denote by S(A) the set of all mathematical states on A and by
S ph := S ph (A) the set of (properly defined) physical states of the system. The subset
S ph (A) ⊂ S(A) has to satisfy some natural requirements, e.g. invariance with respect
to transformations of physical symmetries (cf. below), convexity, local normality
and (eventually) to form a stable face (see [53, Sect. 4.1]).
The set S(A) is convex and compact in the w
∗ -topology of A
∗ (i.e. in σ(A
∗
, A)topology). The set ES(A) of extreme points of S(A) consists of pure states on
11 Our formalism is built for the nonrelativistic situations. If the space V was the Minkowski space
and our considerations were Einstein-Lorentz–relativistic, the condition for the commutativity in
(1.4.2) would be the space–like separation instead of the disjointness of the domains u, v ⊂ V .
17
subalgebra of A corresponding to v ⊂ V . If v 1 ⊂ v 2 ⊂ V , then A v 1 ⊂ A v 2 . All the
A v (v ∈ B(V )) have common unit ≡ the unit e := id A of A, and
v∈B(V )
A v = A,
(1.4.1)
where the over-bar denotes the uniform closure. We assume further that π( p)(A v ) =
A p·v , where p · v := {λ
∈ V : λ
= p · λ, λ ∈ v}, and p · λ denotes the action of p ∈
on the point λ ∈ V . This action is supposed continuous and bounded: p · B(V ) ⊂
B(V ) for all p ∈ . We can assume (for simplicity) that for mutually disjoint v, u ∈
B(V ), v ∩ u = ∅, we have
[x, y] = 0, for all x ∈ A v , y ∈ A u .
(1.4.2)
(The anticommutativity of Fermi systems can also be included, cf. [53, Sect. 2.6]).
11
We shall characterize this situation by saying that the algebra A is quasilocal with
respect to the action of the group . We shall use another technical assumption, that
all the local subalgebras A v are W
∗ -algebras: A W
∗ -algebra A is such a C
∗ -algebra
which is (isomorphic to a) Banach space topological dual of another B-space A ∗
called the predual of A ; such an A is always unital and generated by its projectors.
W
∗ -algebras were introduced originally as weakly closed symmetric subalgebras of
bounded operators in a Hilbert space containing identity and named von Neumann
algebras after their originator.
1.4.3 Mathematically defined states on a C
∗ -algebra A are any positive normalized
linear functionals ω on A, i.e. such ω ∈ A
∗ (:= the dual of A), that
ω(x
∗ x) ≥ 0, ω = 1 (= ω(id A )).
(1.4.3)
Not all mathematical states, however, can be used as adequate descriptions of
physical situations. As physical states on a quasilocal algebra A are usually used
locally normal states, i.e. such states ω on A, the restrictions of which to all the
local W
∗ -subalgebras A v (v ∈ B(V )) are σ(A v , (A v ) ∗ )-continuous (here (A v ) ∗ is
the predual Banach space of A v ); the local normality of ω means that the restriction
of ω to any A v is expressible by a density matrix in a faithful W
∗
−representation
of A v . We shall denote by S(A) the set of all mathematical states on A and by
S ph := S ph (A) the set of (properly defined) physical states of the system. The subset
S ph (A) ⊂ S(A) has to satisfy some natural requirements, e.g. invariance with respect
to transformations of physical symmetries (cf. below), convexity, local normality
and (eventually) to form a stable face (see [53, Sect. 4.1]).
The set S(A) is convex and compact in the w
∗ -topology of A
∗ (i.e. in σ(A
∗
, A)topology). The set ES(A) of extreme points of S(A) consists of pure states on
11 Our formalism is built for the nonrelativistic situations. If the space V was the Minkowski space
and our considerations were Einstein-Lorentz–relativistic, the condition for the commutativity in
(1.4.2) would be the space–like separation instead of the disjointness of the domains u, v ⊂ V .