Noncommutative Mathematics for Quantum Systems

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.