Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 33

In particular, it follows that the antipodeSof an involutive Hopf
algebra is invertible. This is not true for Hopf algebras in general.

Exercise 1.5.4 Show that if (kt)t≥ 0 is any quantum stochastic
process on an involutive Hopf algebra, then the quantum
stochastic process defined by

jst=mA◦

(

(ks◦S)⊗kt

)

◦∆,

for 0≤s≤t, satisfies the increment property (1) in Definition 1.5.3.
A one-parameter stochastic process(kt)t≥ 0 on a Hopf∗-algebraH
is called aL ́evy process on H, if itsincrement process(jst) 0 ≤s≤twith
jst=mA◦

(
(ks◦S)⊗kt)◦∆is a Levy process on ́ Hin the sense of
Definition 1.5.3.
Let(jst) 0 ≤s≤tbe a Levy process on some involutive bialgebra. We ́
will denote the marginal distributions of(jst) 0 ≤s≤tbyφt−s=Φ◦
jst. Owing to the stationarity of the increments this is well defined.

Lemma 1.5.5 The marginal distributions(φt)t≥ 0 of a L ́evy process on
an involutive bialgebraBform acontinuous convolution semigroup
of statesonB, that is, they satisfy

(i) φ 0 =ε,φs?φt=φs+tfor all s,t≥ 0 , andlimt↘ 0 φt(b) =ε(b)

for all b∈B, and

(ii) φt( 1 ) = 1 , andφt(b∗b)≥ 0 for all t≥ 0 and all b∈B.

Proof φt=Φ◦j 0 tis clearly a state, sincej 0 tis a∗-homomorphism
andΦa state.
From the first condition in Definition 1.5.3 we get

φ 0 =Φ◦j 00 =Φ( 1 )ε=ε,

and

φs+t(b) =Φ

(

j0,s+t(b)

)

=Φ

(

∑j 0 s(b( 1 ))js,s+t(b( 2 ))

)

,

for b ∈ B, ∆(b) = ∑b( 1 )⊗b( 2 ). Using the independence of
increments, we can factorize this and get

φs+t(b) = ∑Φ

(

j 0 s(b( 1 ))

)

Φ

(

js,s+t(b( 2 ))

)

=∑φs(b( 1 ))φt(b( 2 ))

= φs⊗φt

(

∆(b)

)

=φs?φt(b)

for all∈B.