Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 53

Proof Letb ∈ Nh ={a ∈A;h(a∗a) = 0 }. We can assume that
T 6 =0, then we haveT( 1 ) 6 =0 and

0 ≤

h

(

T(b)∗T(b)

)

‖T( 1 )‖

≤h

(

T(b∗b)

)

= (h?h)

(

T(b∗b)

)

= (h⊗h)

(

∆◦T(b∗b)

)

=

(

h⊗(h◦T)

)(

∆(b∗b)

)

= h(b∗b)h

(

T( 1 )

)

=0,

where we used idempotence and left invariance of the Haar state.
This proves the first claim.
Lets,s′∈I,s 6 =s′, and 1≤j,k≤ns, 1≤p,q≤ns′. Using again
the fact that the Haar state is idempotent, we have

h

((

u(s

′)

pq

)∗

T ̃

(

u(jks)

))

= (h?h)

((

u(s

′)

pq

)∗

T ̃

(

u(jks)

))

=

ns′

∑

r= 1

(h⊗h)

(((

u(s

′)

pr

)∗

⊗

(

u(s

′)

rq

)∗)

∆

(

T ̃

(

u(jks)

)))

=

ns′

∑

r= 1

ns

∑

`= 1

(h⊗h)

((

u(s

′)

pr

)∗

⊗

(

u(s

′)

rq

)∗(

u(js`)⊗T ̃

(

u(`sk)

)))

=

ns

∑

`= 1

δss′

f 1 ((u(j ps))∗)

Ds

h

(

(u(s

′)

`q )

∗T ̃(u(s)

`k)

)

,

that is,h

((

u(s

′)

pq

)∗ ̃

T

(

u(jks)

))

=0 for alls,s′∈ I, withs 6 =s′, and all

1 ≤j,k≤ns, 1≤p,q≤ns′. Therefore,T ̃

(

u(jks)

)

∈Vs.

The following Theorem is the main result in this Section. In
classical probability, Levy processes with values, say, in ́ Rdor in a
Lie group, are exactly the Markov processes whose transition
probabilities are time- and space-homogeneous. Time
homogeneity means that the transition kernels form a semigroup,
and in the group-case space homogeneity has to be interpreted as
translation invariance. We can now show that the same
characterization also holds for Levy processes on compact ́
quantum groups.

Theorem 1.6.6 Let(A,∆)be a compact quantum group and(Tt)t≥ 0 a
quantum Markov semigroup on(A,∆).