Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 39

holds for allk∈Nand all 0≤s≤t.
Construct the Schurmann triple for Brownian motion and for a ̈
compound Poisson process (with finite moments).

For the classification of Gaussian and drift generating functionals
on an involutive bialgebraBwith counitε, we need the ideals

K = kerε,

K^2 = span{ab|a,b∈K},

K^3 = span{abc|a,b,c∈K}.

Proposition 1.5.12 Let L be a generating functional on B with
Sch ̈urmann triple(ρ,η,L). Then the following are equivalent.

(i) η= 0 ,
(ii) L|K 2 = 0 ,
(iii) L is anε-derivation, that is, L(ab) =ε(a)L(b) +L(a)ε(b)for all
a,b∈B,
(iv) The statesφtare multiplicative, that is,φt(ab) =φt(a)φt(b)for
all a,b∈Band t≥ 0.
If a generating functionalLsatisfies one of these conditions, then
we call it and the associated Levy process a ́ drift.

Proposition 1.5.13 Let L be a generating functional onB.
Then the following are equivalent.

(i) L|K 3 =0,

(ii) L(b∗b) = 0 for all b∈K^2 ,

(iii) L(abc) = L(ab)ε(c) +L(ac)ε(b) +L(bc)ε(a)−ε(ab)L(c)−
ε(ac)L(b)−ε(bc)L(a)for all a,b,c∈B,

(iv)ρ|K= 0 for the representationρin the surjective Sch ̈urmann triple
(ρ,η,L)associated to L by the GNS-type construction presented in
the proof of Theorem 1.5.10,

(v)ρ=ε 1 , for the representationρin the surjective Sch ̈urmann triple
(ρ,η,L)associated to L by the GNS-type construction presented in
the proof of Theorem 1.5.10,

(vi)η|K 2 = 0 for the cocycle ηin any Sch ̈urmann triple (ρ,η,L)
containing L,

(vii)η(ab) =ε(a)η(b) +η(a)ε(b)for all a,b∈ Band the cocycleηin
any Sch ̈urmann triple(ρ,η,L)containing L.