6.2 Spin Systems with Polynomial Local Hamiltonians Q N
133
Let B = B = conv(B) ⊂ B g be such that for any ξ ∈ g the following implication is
valid:
λ ∈ conv(sp(X ξ )) ⇒ ∃ F ∈ B : F(ξ) = λ.
(6.2.69)
The set B := supp E g , and also B := B g has the property (6.2.69).
Let F 0 ∈ g
∗ does not belong to B : F 0 /
∈ B. Then, according to
[157, Lemma (B.26)], there is an element of g
∗∗
= g, ξ 0 ∈ g, such that
inf{F(ξ 0 ) : F ∈ B} > F 0 (ξ 0 ).
(6.2.70)
But from (6.2.69) and from B ⊂ B g we see that {F(ξ) : F ∈ B} = conv(sp(X ξ ))
for all ξ ∈ g, hence F 0 (ξ 0 ) /
∈ conv(sp(X ξ 0 )), and this implies that F 0 /
∈ B g . We have
proved that B = B g , hence supp E g = B g . But
s G ≤ s π ⇒ E g ≤ E
π
g ⇒ supp E g ⊂ supp E
π
g ,
(6.2.71)
what with the help of (6.2.68) gives now the desired result.
6.2.18 Lemma. supp E g is convex.
Proof. The projection measure E g introduced in 6.2.2.(i) is built of its values E g (F),
(5.1.56), calculated on one point sets {F} ⊂ g
∗ . The measure E g is isomorphically
mapped onto the measure E
#
g := ρ G ◦ E g acting in the Hilbert subspace P G H of the
infinite (complete) tensor product space H , cf. 5.1.11. According to the definitions
in 5.1.7, 5.1.9 and 5.1.11, F ∈ supp E g means that there is a product-vector ∈ H :
:=
k∈
ϕ k , ϕ k ∈ H k := u k H, ϕ k = 1, for all k ∈ ,
(6.2.72)
such that the following relations are valid:
lim
N →∞
1
N
N
k=1
(ϕ k , π k (X ξ )ϕ k ) = F(ξ), for all ξ ∈ g.
(6.2.73)
Let F
( j)
∈ supp E g ( j = 1, 2) be determined according to (6.2.73) by the product
vectors
( j)
:= ⊗ k∈ ϕ
( j)
k ∈ D (g). We shall construct a product vector ∈ D (g),
for any rational number c : 0 < c =
r
s
< 1, such that the corresponding value of
F ∈ g, cf. (6.2.72) and (6.2.73), is
F = cF
(1)
+ (1 − c)F
(2)
.
(6.2.74)
This will prove the convexity of supp E g , since supp E g is a closed subset of g
∗ .