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,∆).