52 Noncommutative Mathematics for Quantum Systems
completely positive,Ttis completely positive and contractive. Let
us now check that(Tt)tforms a strongly continuous semigroup on
Cr(G).
For a givena∈Cr(G)we choose by densityb∈Pol(G)such
that‖a−b‖r <e. Recall that forb ∈ Pol(G),Tt(b) = φt?b =
(id⊗φt)◦∆(b), where(φt)tis the convolution semigroup of states
on Pol(G)(cf.Subsection 1.5.1). Then
‖Tt(a)−a‖r ≤ ‖Tt(a)−Tt(b)‖r+‖Tt(b)−b‖r+‖b−a‖r
≤ 2 ‖a−b‖r+‖(φt?b)−b‖r
≤ 2 e+∑‖b( 1 )φt(b( 2 ))−b( 1 )ε(b( 2 ))‖r
= 2 e+∑|φt(b( 2 ))−ε(b( 2 ))|‖b( 1 )‖r.
Since limt→ 0 +φt(b) =ε(b)for anyb∈Pol(G)and the sum is finite,
we conclude that
lim
t→ 0 +
‖Tt(a)−a‖r=0 for eacha∈Cr(G).
The next results give the characterization of Markov semigroups
that are related to L ́evy processes on compact quantum groups.
Lemma 1.6.5 LetG = (A,∆)a compact quantum group and T :
A → Aa completely bounded 2-positive linear map.
If T is translation invariant, that is, satisfies
∆◦T= (id⊗T)◦∆
then T leaves the null space of the Haar state invariant and induces a
translation invariant mapT ̃:Cr(G)→Cr(G).
Furthermore, we haveT ̃(Vs)⊆Vsfor all s∈ Iand thereforeT leaves ̃
the∗-Hopf algebraA=Pol(G)invariant.
2-Positivity means that the mapT 2 :M 2 (A)→M 2 (A)defined by
T 2
(
(ajk) 1 ≤j,k≤ 2
)
=
(
T(ajk)
)
1 ≤j,k≤ 2
preserves positivity; this implies that
T(b)∗T(b)≤‖T( 1 )‖T(b∗b)
for allb ∈ A,cf. [Pau02, Exercise 3.4]. Complete boundedness
guarantees that id⊗Tis well defined,cf.[Pau02].