Noncommutative Mathematics for Quantum Systems

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

)