Noncommutative Mathematics for Quantum Systems

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].