Independence and L ́evy Processes in Quantum Probability 43We will show that Levy processes on compact quantum groups ́
can be characterized as time- and space-homogeneous Markov
process, just like in classical probability, see Theorem 1.6.6. This
observation is the starting point of [CFK14], where other
symmetry properties, like GNS- or KMS-symmetry or invariance
under the adjoint action, of the Markov semigroups associated to
Levy processes on compact quantum groups are studied. It is ́
possible to associate Dirichlet forms, derivations, and Dirac
operators to these Markov semigroups. We refer the interested
reader to the paper [CFK14] for more information. See also the
article [CS15] by Caspers and Skalski for an application of
Dirichlet forms on compact quantum groups to the Haagerup
approximation property.
In this Section, we will work both with algebraic and topologic
dual spaces and tensor products. We use the following
conventions. For an algebraA, if no norm or topology is specified,
A′ will denote the algebraic dual ofA, that is, the space of all
linear functionals fromAtoC. For a C∗-algebraA, byA′we will
mean the dual space consisting of all linear continuous functionals
fromAtoC. The symbol⊗will denote the spatial tensor product
when it is used for C∗-algebras and the algebraic tensor product
otherwise. We will never need algebraic tensor products of
C∗-algebras. See, e.g., [Ped89] and the lecture ‘Quantum
dynamical systems from the point of view of non commutative
mathematics’ by Adam Skalski for the spatial tensor product and
other facts about C∗-algebras.
1.6.1 Compact quantum groups
The notion of compact quantum groups has been introduced in
[Wor87a]. Here we adopt the definition from [Wor98] (Definition
1.1 of that paper).
Definition 1.6.1 AC∗-bialgebra(a compact quantum semigroup)
is a pair(A,∆), whereAis a unital C∗-algebra,∆:A→A⊗Ais a
unital,∗-homomorphic map which is coassociative, that is,
(∆⊗idA)◦∆= (idA⊗∆)◦∆.If thequantum cancellation properties
Lin(( 1 ⊗A)∆(A)) =Lin((A⊗ 1 )∆(A)) =A⊗A,