42 Noncommutative Mathematics for Quantum Systems
over the algebraic probability space (L∞(Ω,F,P),E), where E
denotes expectation w.r.t.P.
If the Levy process ́ (Xt)t≥ 0 has finite moments, then we can
restrict similarly to polynomials onRand we get a Levy process ́
overthe algebraic probability space(L∞−(Ω,F,P),E), with
L∞−(Ω,F,P) =⋂1 ≤p<∞Lp(Ω,F,P).Levy processes on finite semigroups ́
Exercise 1.5.16 Let (G,·,e) be a finite semigroup with unit
elemente. Then the complex-valued functionsF(G)onGform an
involutive bialgebra. The algebra structure and the involution are
given by pointwise multiplication and complex conjugation. The
coproduct and counit are defined by
∆(f)(g 1 ,g 2 ) = f(g 1 ·g 2 ) forg 1 ,g 2 ∈G,ε(f) = f(e),forf∈F(G).
Show that the classical Levy processes in ́ Gare in one-to-one
correspondence with Levy processes on the ́ ∗-bialgebraF(G).
1.6 Levy Processes on Compact Quantum Groups ́
and their Markov Semigroups
In the eighties of the last century Woronowicz introduced
‘compact matrix pseudogroups’, which are nowadays called
compact quantum groups,cf.[Wor87a, Wor87b, Wor88]. They can
be defined alternatively as C∗-bialgebras, that is, unital
C∗-algebras equipped with additional structure, see Definition
1.6.1 below, which satisfy the quantum cancellation properties, or
as involutive Hopf algebras satisfying additional conditions, see
[DK94]. For more background on compact quantum groups, see,
for example, [Wor98, MvD98, MT04, Tim08, NT13].
The additional structure makes compact quantum groups
correspond more closely to compact groups than arbirtrary
involutive bialgebras, see Subsection 1.6.1 below. This leads to a
richer theory for Levy processes on these structures, which we will ́
present in this Section.