98 Noncommutative Mathematics for Quantum Systems
The third step, which was actually completed first in both cases,
see [Spe97] and [Mur02], is to find the conditions that the
universal constants have to satisfy, if the resulting product is
associative. It turns out that the universal coefficients form>5 are
already uniquely determined by the coefficients for 1 ≤m ≤5.
Detailed analysis of the non linear equations obtained for the
coefficients of order up to five then leads to the classifications
stated above.
Muraki [Mur13] has recently shown that a simpler proof is
possible if one imposes positivity.
1.9 Levy Processes on Dual Groups ́
We now want to study quantum stochastic processes whose
increments are free or independent in the sense of boolean,
monotone, or anti-monotone independence. The approach based
on bialgebras that we followed in the Section 1.5 and 1.6 works for
the tensor product and fails in the other cases because the
corresponding products are not defined on the tensor product, but
on the free product of the algebra. The algebraic structures that
have to replace bialgebras were introduced into quantum
probability by Voiculescu [Voi87, Voi90], who named them dual
groups. Closely related structures have also been studied in other
contexts under the name H-algebra [Zha91] or co-group
[Ber65, BH96, Fre98].
In this section we will introduce these algebras and present the
theory of their Levy processes. L ́ evy processes on dual groups ́
with boolean, monotonically, or anti-monotonically independent
increments can be reduced to Levy processes on involutive ́
bialgebra,cf.[Fra03a]. We do not know if this is also possible for
Levy processes on dual groups with free increments. ́
In the literature additive free L ́evy processes have been studied
most intensively, see, for example, [GSS92, Bia98, Ans02, Ans03]
[BNT02b, BNT02a, Vos13, Ulr14].
1.9.1 Dual groups
Denote byComAlgthe category of commutative unital algebras and
letB ∈ObComAlgbe a commutative bialgebra. Then the mapping
MorComAlg(B,·): ObComAlg3A7→MorComAlg(B,A)