36 Noncommutative Mathematics for Quantum Systems
Letb∈ B,ε(b) =0. If allφtare states, then we haveφt(b∗b)≥ 0
for allt≥0 and therefore
L(b∗b) =lim
t↘ 0
1
t
(
φt(b∗b)−ε(b∗b)
)
=lim
t↘ 0
φt(b∗b)
t
≥0.
Similarly,Lis hermitian, since allφtare hermitian.
We will call a linear functional satisfying condition (ii) of the
preceding Proposition agenerating functional. Lemma 1.5.7 and
Proposition 1.5.8 show that Levy processes can also be ́
characterized by their generating functionalL= ddt
∣
∣∣
t= 0
φt.
1.5.3 The Schurmann triple of a L ̈ evy process ́
LetDbe a pre-Hilbert space. Then we denote byL(D)the set of all
linear operators onDthat have an adjoint defined everywhere on
D, that is,
L(D) =
{
X:D→Dlinear
∣∣
∣∣there existsX
∗:D→Dlinear s.t.
〈u,Xv〉=〈X∗u,v〉for allu,v∈D
}
.
L(D)is clearly a unital∗-algebra.
Definition 1.5.9 LetBbe a unital∗-algebra equipped with a unital
hermitian characterε:B →C(that is,ε( 1 ) =1,ε(b∗) =ε(b), and
ε(ab) =ε(a)ε(b)for alla,b∈ B). ASch ̈urmann triple on(B,ε)is a
triple(ρ,η,L)consisting of
- a unital∗-representationρ:B →L(D)ofBon some pre-Hilbert
spaceD, - aρ-ε-1-cocycleη:B →D, that is, a linear mapη:B →Dsuch
that
η(ab) =ρ(a)η(b) +η(a)ε(b) (1.5.2)
for alla,b∈B, and
- a hermitian linear functionalL : B → Cthat has the bilinear
mapB×B 3(a,b)7→ −〈η(a∗),η(b)〉as aε-ε-2-coboundary, that
is, that satisfies
−〈η(a∗),η(b)〉=∂L(a,b) =ε(a)L(b)−L(ab) +L(a)ε(b) (1.5.3)
for alla,b∈B.