Independence and L ́evy Processes in Quantum Probability 49strongly continuous Markov semigroup on both its reduced and
its universal C∗-algebra. We then show that the characterization of
Levy processes in topological groups as the Markov processes that ́
are invariant under time and space translations extends to
compact quantum groups.
If(jst) 0 ≤s≤tis a Levy process on an ́ ∗-algebra Pol(G)with the
convolution semigroup of states (φt)t≥ 0 on Pol(G) and the
Markov semigroup(Tt)t≥ 0 on Pol(G), then, by universality, each
φtextends to a continuous functional on the universal C∗-algebra
Cu(G)generated by Pol(G)(see also [BMT01, Theorem 3.3] for a
detailed exposition). In the literature this C∗-algebra is also
denoted byAu. Then the formulaTt= (id⊗φt)◦∆makes sense
onCu(G)(where∆:Cu(G)→Cu(G)⊗Cu(G)denotes the unique
unital∗-homomorphism that extends ∆ : Pol(G) → Pol(G)⊗
Pol(G)) and one easily shows (in the same way as in the
Proposition below) that (Tt)t becomes a strongly continuous
Markov semigroup of contractions onCu(G). This means that each
Tt(t≥0) is a unital, completely positive contraction and(Tt)tis a
strongly continuous semigroup onCu(G).
For us, however, it will be more natural to consider the reduced
C∗-algebra generated by Pol(G). This is the C∗-algebraAr=Cr(G)
obtained by taking the norm closure of the GNS representation of
Pol(G)with respect to the Haar state h. The Haar state his by
construction faithful onCr(G). The coproduct on Pol(G)extends
to a unique unital∗-homomorphism∆:Cr(G)→Cr(G)⊗Cr(G),
which makes the pair(Cr(G),∆)a compact quantum group. The
following result shows that, even thoughφt: Pol(G)→Ccan be
unbounded with respect to the reduced C∗-norm and therefore not
extend toCr(G),(Tt)talways extends to a strongly continuous
Markov semigroup onCr(G).
States on anyC∗-algebraic versionC(G)ofGdefine continuous
convolution operators on the reduced versionCr(G),cf. [Bra12,
Lemma 3.4]. We will need a similar result for convolution
semigroups of states on Pol(G).
Theorem 1.6.4 Each L ́evy process(jst) 0 ≤s≤ton the Hopf∗-algebra
Pol(G)gives rise to a unique strongly continuous Markov semigroup
(Tt)t≥ 0 on Cr(G), the reduced C∗-algebra generated byPol(G).
This follows from the fact that the coproduct∆: Pol(G)→Pol
(G)⊗Pol(G)extends to a∗-homomorphismCr(G) → Cr(G)⊗