6.2 Spin Systems with Polynomial Local Hamiltonians Q N
125
is again a projection-valued mesure with the same support:
supp ˆ
ϕE g = supp E g .
(6.2.27)
6.2.9 Proposition. Let E g and ϕ be as above. Then the mapping
E g : f → E g ( f ) :=
f (F) E g (dF), f ∈ C(supp E g ),
(6.2.28)
introduced in 6.2.2 (ii) is a C
∗ -isomorphism of the commutative C
∗ -algebra of continuous complex valued functions C(supp E g ) on the compact subset supp E g of g
∗
(X ξ ’s are now bounded!) onto N
c .
The C
∗ -algebra N
c is generated by the finite set E g ( f ξ j ), j = 1, 2, . . . n of its
elements (ξ j ’s form a basis of g). The mapping
ϕ
∗
: f → ϕ
∗ f, with ϕ
∗ f (F) := f (ϕF),
(6.2.29)
restricted to f ∈ C(supp E g ) is a
∗ -automorphism of C(supp E g ). One has
ˆ
ϕ : E g ( f ) → ˆ
ϕ(E g ( f )) := ˆ
ϕE g ( f ) = E g (ϕ
∗ f ), f ∈ C(g
∗
),
(6.2.30)
and the mapping ˆ
ϕ in (6.2.30) is a
∗ -automorphism of N
c .
Proof. Since supp E g is compact (due to the compactness of spectra of all the
X ξ ’s), the function set C(supp E g ) is a C
∗ -algebra generated by polynomials in the
variables F j := F(ξ j ) = f ξ j (F) according to the classical Weierstrass theorem. The
∗ -morphism property of E g in (6.2.28) is a consequence of the standard functional
calculus of normal operators determined by a projection measure. One can show
that if f (F 0 ) = 0 for some F 0 ∈ supp E g and a continuous f , then E g ( f ) = 0, and
this implies that the mapping E g in (6.2.28) is the C
∗ -isomorphism of C(supp E g )
onto N
c .
The mapping ϕ
∗ is a norm preserving
∗ -morphism of C(supp E g ) into itself,
hence, it is a
∗ -automorphism.
The automorphism property of ˆ
ϕ in (6.2.30) is then a consequence of the
relation (6.2.27), since both the mappings E
−1
g : N
c
→ C(supp E g ) and ˆ
ϕE g :
C(supp E g ) → N
c are
∗ -isomorphisms, and we have:
ˆ
ϕ(E g ( f )) = ˆ
ϕE g ◦ E
−1
g (E g ( f )), f ∈ C(supp E g ).
(6.2.31)
The equality in (6.2.30) can be obtained from (6.2.26) and the integral representation
(6.2.28). This concludes the proof.