Noncommutative Mathematics for Quantum Systems

48 Noncommutative Mathematics for Quantum Systems

is invertible, with inverse

(

f 1 (u(jks))

)
jk ∈ Mns(C),cf. [Wor87a,
Equation (5.24)].

Remark 1.6.3 The Haar state on compact quantum groups is a
trace if and only if the antipode is involutive, that is, we have
S^2 (a) =afor alla∈Pol(G). In this case we say that(A,∆)is of
Kac type. This is also equivalent to the following conditions,cf.
[Wor98, Theorem 1.5],

(i) fz=εfor allz∈C,

(ii)σt=id for allt∈R.

The antipodeS: Pol(G)→Pol(G)is a closable operator and its
closureSadmits a polar decomposition

S=R◦τi

2

, (1.6.7)

whereτi
2

is the analytic generator of a one-parameter group(τt)t∈R

of∗-automorphisms of the C∗-algebraAandR:A→Ais a linear
antimultiplicative norm preserving involution that commutes with
hermitian conjugation and with the semigroup(τt), that is,τt◦R=
R◦τtfor allt ∈R, see [Wor98, Theorem 1.6]. The operatorRis
called theunitary antipodeand the one-parameter group(τt)t∈Ris
called thescaling group.
Moreover,τandRare related to Woronowicz characters through
the following formulae

τt(a) = fit?a?f−it, (1.6.8)

R(a) = S(f 1

2

?a?f− 1

2

) (1.6.9)

fora∈Pol(G).

1.6.2 Translation invariant Markov semigroups

Our goal is to construct Markov semigroups on compact quantum
groups that reflect the structure of the quantum group. In this
section we show that it is exactly the translation invariant
Markovian semigroups that can be obtained from Levy processes ́
on the algebra of smooth functionsA =Pol(G)of the quantum
groupG= (A,∆).
For this purpose, we first prove that the Markov semigroup
(Tt)t≥ 0 of a Levy process on Pol ́ (G)has a unique extension to a