106 Noncommutative Mathematics for Quantum Systems
particular Van Daele’s (∗-)algebraic quantum groups
[VD98, VD03] and Kusterman and Vaes’ locally compact
quantum groups [KV99, KV00].
The case of Poisson-type Levy processes on locally compact ́
quantum groups has been worked out in [LS11], and some
further initial work in this direction has been done in [LS12].
(iv) A construction that reduces the monotone, anti-monotone,
and the boolean product of quantum probability spaces to
the tensor product was introduced in [Fra03b]. This
construction allows to construct monotone, anti-monotone,
and boolean Levy processes on dual groups from L ́ evy ́
processes on∗-bialgebras. A construction of free products
from tensor products was given by Lenczewski [Len98];
however, it is not clear how to apply it to the theory of free
Levy processes. Does there exist a ‘reduction’ from freeness ́
to tensor independence, or vice versa, similar to the
construction in [Fra03b]?
(v) One question remains open in the classification of universal
products of algebraic probability spaces: Is the degenerate
universal product defined in Equation (1.8.3) the only
universal product in the case where both constantsc 1 andc 2
in Equations (1.8.5) and (1.8.6) are equal to zero? See [Lac15,
Page 7].
(vi) It would be interesting to find new examples of dual groups
and to study their properties. Voiculescu has defined dual
group analogs of the classical series of simple Lie groups,cf.
[Voi87], but only the dual group analog of the unitary groups
has been studied in the literature, see [GvW89, McC92]. In
quantum probability only the examples of the tensor algebra,
the free group algebra, and the dual group analog of the
unitary group have been studied, see, e.g,
[Sch95b, BG01, Vos13].
(vii) Ulrich [Ulr14] has shown that the limit in distribution of the
d×dblocks of Brownian motion in the unitary groupU(nd)
asntends to infinity is a free Levy process on the dual group ́
analog of the unitary group,cf.[Ulr14]. In [CU15] this result
was generalized to a larger class of Levy processes. It would ́
be interesting to study if such a result also holds for
orthogonal and symplectic groups and if it can be applied to
get new insights into the behavior of large random matrices.