38 Noncommutative Mathematics for Quantum Systems
Let nowLbe a generating functional. The sesqui-linear form
〈·,·〉L:B×B →Cdefined by
〈a,b〉L=L
((
a−ε(a) 1
)∗(
b−ε(b) 1
))
fora,b∈Bis positive, sinceLis conditionally positive. DividingB
by the null space
NL={a∈B|〈a,a〉L= 0 }
we obtain a pre-Hilbert spaceD=B/NLwith a positive definite
inner product〈·,·〉induced by〈·,·〉L. For the cocycleη:B →Dwe
take the canonical projection, this is clearly surjective and satisfies
Equation (1.5.3).
The∗-representationρis induced from the left multiplication on
Bon kerε, that is,
ρ(a)η
(
b−ε(b) 1
)
=η
(
a
(
b−ε(b) 1
))
orρ(a)η(b)=η(ab)−η(a)ε(b)
fora,b∈B. To show that this is well defined, we have to verify that
left multiplication by elements ofBleaves the null-space invariant.
Let thereforea,b∈B,b∈NL, then we have
∣∣
∣
∣∣
∣
(
a
(
b−ε(b) 1
))∣∣
∣
∣∣
∣
2
= L
((
ab−aε(b) 1
)∗(
ab−aε(b) 1
))
= L
((
b−ε(b) 1
)∗
a∗
(
ab−aε(b) 1
))
=
〈
b−ε(b) 1 ,a∗a
(
b−ε(b) 1
)〉
L
≤ ‖b−ε(b) 1 ‖^2
∣∣∣∣
a∗a
(
b−ε(b) 1
)∣∣∣∣ 2
=0,
with Schwarz’ inequality.
That the Schurmann triple ̈ (ρ,η,L) obtained in this way is unique
up to unitary equivalence follows similarly as for the usual GNS
construction.
Exercise 1.5.11 Let(Xt)t≥ 0 be a classical real-valued Levy process ́
with all moments finite (on some probability space(Ω,F,P)).
Define a Levy process on the free unital algebra ́ C[x]generated by
one symmetric elementx = x∗ with the coproduct and counit
determined by∆(x) =x⊗ 1 + 1 ⊗xandε(x) =0, such that
Φ
(
jst(xk)
)
=E
(
(Xt−Xs)k
)