15 Physically Relevant Representations
103
of all operators in B(H) which commute with A
. Clearly, if π(A) is an irreducible
representation of a C
∗ -algebra A, then π(A)
= {λ 1, λ ∈ C} and π(A)
= B(H).
Theorem 15.1 (Von Neumann bicommutant). For a
∗ -subalgebra A of B(H), with
identity, the following three properties are equivalent
i) A = A
,
ii) A is weakly closed (briefly A = A
w ),
iii) A is strongly closed: A = A
s .
Proposition 15.2 In any irreducible representation π of the algebra A of observables, satisfying III i) and weak asymptotic abelianess, (14.6), the ground state is
the unique translationally invariant state.
Proof. In fact, if Ψ 0
describes another translationally invariant state, and therefore
can be taken orthogonal to Ψ 0 , ∀A ∈ A, (denoting by P 0 the projection on Ψ 0 ), we
have
(Ψ 0
, π(A) Ψ 0 ) = (Ψ 0
, π(A x ) Ψ 0 ) = (Ψ 0
, π(A x ) P 0 Ψ 0 ) =
(Ψ 0
, P 0 π(A x ) Ψ 0 ) + (Ψ 0
, [π(A x ), P 0 ] Ψ 0 ).
(15.1)
The first term on the right-hand side is zero because Ψ 0
is orthogonal to Ψ 0 . The
second term is independent of x and one can take the limit |x| → ∞. Now, by von
Neumann’s theorem and irreducibility
π(A)
s = (π(A)
s )
⊇ π(A)
= B(H),
so that P 0 belongs to π(A)
s and therefore, by the extension of asymptotic abelianess
(discussed after (14.6)), the second term vanishes in the limit |x| → ∞.
In conclusion, (Ψ 0
, π(A) Ψ 0 ) = 0 and, by the cyclicity of Ψ 0 , Ψ 0
= 0.
Under the same hypotheses, the above argument can be used to prove that
w − lim
|x|→∞
π(A x ) = (Ψ 0 , A Ψ 0 )1 ≡< A > 0 1,
(15.2)
(sometimes, in the following equations the subscript 0 in the brackets will be omitted
for simplicity). In fact, ∀B ∈ π(A) one has, by asymptotic abelianess
w− lim
|x|→∞
π(A x ) B Ψ 0 = w − lim
|x|→∞
B π(A x ) Ψ 0 =
B w − lim
|x|→∞
([π(A x ), P 0 ] Ψ 0 + P 0 π(A x ) Ψ 0 ).
By the extension of asymptotic abelianess, the first term on the right-hand side
vanishes and the second term is equal to B Ψ 0 < A > 0 . Thus, the above weak limit
exists and it equals the r.h.s. of (15.2).
103
of all operators in B(H) which commute with A
. Clearly, if π(A) is an irreducible
representation of a C
∗ -algebra A, then π(A)
= {λ 1, λ ∈ C} and π(A)
= B(H).
Theorem 15.1 (Von Neumann bicommutant). For a
∗ -subalgebra A of B(H), with
identity, the following three properties are equivalent
i) A = A
,
ii) A is weakly closed (briefly A = A
w ),
iii) A is strongly closed: A = A
s .
Proposition 15.2 In any irreducible representation π of the algebra A of observables, satisfying III i) and weak asymptotic abelianess, (14.6), the ground state is
the unique translationally invariant state.
Proof. In fact, if Ψ 0
describes another translationally invariant state, and therefore
can be taken orthogonal to Ψ 0 , ∀A ∈ A, (denoting by P 0 the projection on Ψ 0 ), we
have
(Ψ 0
, π(A) Ψ 0 ) = (Ψ 0
, π(A x ) Ψ 0 ) = (Ψ 0
, π(A x ) P 0 Ψ 0 ) =
(Ψ 0
, P 0 π(A x ) Ψ 0 ) + (Ψ 0
, [π(A x ), P 0 ] Ψ 0 ).
(15.1)
The first term on the right-hand side is zero because Ψ 0
is orthogonal to Ψ 0 . The
second term is independent of x and one can take the limit |x| → ∞. Now, by von
Neumann’s theorem and irreducibility
π(A)
s = (π(A)
s )
⊇ π(A)
= B(H),
so that P 0 belongs to π(A)
s and therefore, by the extension of asymptotic abelianess
(discussed after (14.6)), the second term vanishes in the limit |x| → ∞.
In conclusion, (Ψ 0
, π(A) Ψ 0 ) = 0 and, by the cyclicity of Ψ 0 , Ψ 0
= 0.
Under the same hypotheses, the above argument can be used to prove that
w − lim
|x|→∞
π(A x ) = (Ψ 0 , A Ψ 0 )1 ≡< A > 0 1,
(15.2)
(sometimes, in the following equations the subscript 0 in the brackets will be omitted
for simplicity). In fact, ∀B ∈ π(A) one has, by asymptotic abelianess
w− lim
|x|→∞
π(A x ) B Ψ 0 = w − lim
|x|→∞
B π(A x ) Ψ 0 =
B w − lim
|x|→∞
([π(A x ), P 0 ] Ψ 0 + P 0 π(A x ) Ψ 0 ).
By the extension of asymptotic abelianess, the first term on the right-hand side
vanishes and the second term is equal to B Ψ 0 < A > 0 . Thus, the above weak limit
exists and it equals the r.h.s. of (15.2).