5.1 Multiple Systems
79
i.e. irreducibility of the action of A
in each H
. The weak closure of A
in L(H )
consists of all elements x ∈ L(H ) satisfying (5.1.22). The action of A
in H
is a
representation of this C
∗ -algebra. Such representations (all irreducible and faithful)
for two product vectors are unitarily equivalent iff these vectors are weakly equivalent. The center of the weak closure of A
in L(H ) is generated by the projectors
P
w
. Denote this weak closure by B
# and by Z
# its center: x ∈ Z
#
⊂ B
# iff [x, y] = 0
for all y ∈ B
# .
5.1.5 Proposition. The mapping
σ : G →
∗ -Aut A
, g → σ g ,
(5.1.24)
defined by (see (5.1.13))
σ g (x) := U (g)xU (g
−1
), ∀x ∈ A
, g ∈ G,
(5.1.25)
is a group homomorphism of G into the group
∗ - Aut A
of
∗ -automorphisms of the
C
∗ -algebra A
. For any normalized vector ∈ H define the vector state ω
on
A
by
ω
: x → ω
(x) := ((, x).
(5.1.26)
The functions
g → ω
(σ g (x))
(5.1.27)
for any x ∈ A
and any ∈ H are continuous functions from G to C.
Proof. The mapping A → U (g)AU (g
−1
) is a
∗ -automorphism of L(H), A ∈ L(H).
Since A
is generated by elements x := π j (A) ( j ∈ , A ∈ L(H)) defined in
(5.1.12) (i.e. A
is the norm-closure of finite linear combinations of finite products of such elements), the first statement follows from the definition (5.1.13) of U .
The functions (5.1.27) are continuous for all x = π j (A) and for all product states
ω
(i.e. states corresponding via (5.1.26) to product vectors of the form (5.1.11)).
The set of product vectors is total in H and any
∗ -automorphism of a C
∗ -algebra
is norm-continuous. These facts imply by standard considerations validity of the last
statement.
5.1.6 Note. Due to weak discontinuity of U , the second statement of 5.1.5 is not
valid if A
would be replaced by its weak closure B
# in L(H ). This can be seen
by setting :=
a from (5.1.19) with a j := 0 (for all j ∈ ) and with a choice
ϕ
0
∈ H such that it is not an eigenvector of the generator X ξ of U (G) for some
ξ ∈ g. Then, setting x := P
w
∈ B
# in (5.1.26), the function
t → ω
(σ exp(tξ) (P
w
))
(5.1.28)
is discontinuous at t = 0 : For t = 0 its value equals to 1, but for arbitrarily small
nonzero values of t ∈ R the values of (5.1.28) are found to be zero.
79
i.e. irreducibility of the action of A
in each H
. The weak closure of A
in L(H )
consists of all elements x ∈ L(H ) satisfying (5.1.22). The action of A
in H
is a
representation of this C
∗ -algebra. Such representations (all irreducible and faithful)
for two product vectors are unitarily equivalent iff these vectors are weakly equivalent. The center of the weak closure of A
in L(H ) is generated by the projectors
P
w
. Denote this weak closure by B
# and by Z
# its center: x ∈ Z
#
⊂ B
# iff [x, y] = 0
for all y ∈ B
# .
5.1.5 Proposition. The mapping
σ : G →
∗ -Aut A
, g → σ g ,
(5.1.24)
defined by (see (5.1.13))
σ g (x) := U (g)xU (g
−1
), ∀x ∈ A
, g ∈ G,
(5.1.25)
is a group homomorphism of G into the group
∗ - Aut A
of
∗ -automorphisms of the
C
∗ -algebra A
. For any normalized vector ∈ H define the vector state ω
on
A
by
ω
: x → ω
(x) := ((, x).
(5.1.26)
The functions
g → ω
(σ g (x))
(5.1.27)
for any x ∈ A
and any ∈ H are continuous functions from G to C.
Proof. The mapping A → U (g)AU (g
−1
) is a
∗ -automorphism of L(H), A ∈ L(H).
Since A
is generated by elements x := π j (A) ( j ∈ , A ∈ L(H)) defined in
(5.1.12) (i.e. A
is the norm-closure of finite linear combinations of finite products of such elements), the first statement follows from the definition (5.1.13) of U .
The functions (5.1.27) are continuous for all x = π j (A) and for all product states
ω
(i.e. states corresponding via (5.1.26) to product vectors of the form (5.1.11)).
The set of product vectors is total in H and any
∗ -automorphism of a C
∗ -algebra
is norm-continuous. These facts imply by standard considerations validity of the last
statement.
5.1.6 Note. Due to weak discontinuity of U , the second statement of 5.1.5 is not
valid if A
would be replaced by its weak closure B
# in L(H ). This can be seen
by setting :=
a from (5.1.19) with a j := 0 (for all j ∈ ) and with a choice
ϕ
0
∈ H such that it is not an eigenvector of the generator X ξ of U (G) for some
ξ ∈ g. Then, setting x := P
w
∈ B
# in (5.1.26), the function
t → ω
(σ exp(tξ) (P
w
))
(5.1.28)
is discontinuous at t = 0 : For t = 0 its value equals to 1, but for arbitrarily small
nonzero values of t ∈ R the values of (5.1.28) are found to be zero.