Independence and Levy Processes in Quantum Probability ́ 105{jopst} 0 ≤s≤t≤Thas anti-monotonically independent increments, and
vice versa.
Since tensor and boolean independence and freeness do not
depend on the order, {jst} 0 ≤s≤t≤T has tensor (boolean, free,
respectively) independent increments, if and only if
{jopst} 0 ≤s≤t≤T has tensor (boolean, free, respectively)
independent increments.
1.10 Open Problems
We close this lecture with a list of interesting research topics and
open questions.
(i) We defined Gaussian generating functionals in Proposition
1.5.13. Let L be a generating functional. We say that a
Gaussian generating functionalLGis a Gaussian component
of L, if L−LG is conditionally positive, and that LG is
maximal ifLG−L′is conditionally positive for all Gaussian
componentsL′ofL. If it exists, then the maximal Gaussian
component is unique up to a hermitian left invariant
derivation, that is, a drift, see [Sch90a, Proposition 3.8].
Schurmann has shown that generating functionals on ̈
commutative involutive bialgebras or on the so-called
‘noncommutative coefficient algebra of the unitary group’
admit maximal Gaussian components, see [Sch90a] [Sch93,
Section 5], but in general this question is still open.^1
(ii) In Equation (1.6.11) we showed that any translation invariant
Markov semigroup on a compact quantum groupGis equal
to the Markov semigroup of a Levy process on its ́ ∗-Hopf
algebra, up to elements fromNh, the null space of the Haar
state. Is it possible to removeNhfrom Equation (1.6.11)?
(iii) There are other notions of quantum groups, whose structure
can not be captured in involutive bialgebras as in
Woronowicz’ theory of compact quantum groups, and
therefore Schurmann’s theory of L ̈ evy processes can not be ́
applied directly. It would be interesting to develop a theory
of L ́evy processes for these quantum groups, too. There are in(^1) After these notes where written, there was some progress on this problem. Franz,
Gerhold, and Thom showed that there exist generating functionals on the group algebra of
fundamental groups of oriented surfaces of genusk≥2 that do not admit such a ‘Levy- ́
Khintchine’ decomposition into a maximal Gaussian part and a ‘rest’,cf.[FGT15].