Independence and Levy Processes in Quantum Probability ́ 101As the composition of the three unital∗-algebra homomorphisms
∆:B → B‰B,j 1 ‰j 2 :B‰B → A‰A, andmA:A‰A → A,
this is obviously again a unital∗-algebra homomorphism. The
convolutionj 1 ?j 2 should not be thought of as an analog of the
convolution of probability distributions, but rather as an analog of
the composition of random variables by pointwise multiplication.
It is this operation that we will use to give a sense to the notion of
increments of a stochastic process.
Note that this convolution can not be defined for arbitrary linear
maps onBwith values in some algebra, as for bialgebras, but only
for unital∗-algebra homomorphisms.
1.9.2 Definition of Levy processes on dual groups ́
Definition 1.9.1 Letj 1 :B 1 →(A,Φ),... ,jn:Bn→ (A,Φ)be
quantum random variables over the same quantum probability
space (A,Φ) and denote their marginal distributions by
φi=Φ◦ji,i=1,... ,n. The quantum random variables(j 1 ,... ,jn)
are called tensor-independent (respectively, boolean-independent,
monotonically independent, anti-monotonically independent, or
free), if the state Φ◦mA◦(j 1 ‰··· ‰jn) on the free product
‰ni= 1 Bi is equal to the tensor product (boolean, monotone,
anti-monotone, or free product, respectively) ofφ 1 ,... ,φn.
Exercise 1.9.2 Derive Definition 1.9.1 from Definition 1.8.6 (for
n= 2).
Note that tensor, boolean, and free independence do not depend
on the order, but monotone and anti-monotone independence
do. An n-tuple (j 1 ,... ,jn) of quantum random variables is
monotonically independent, if and only if (jn,... ,j 1 ) is
anti-monotonically independent.
We are now ready to define tensor, boolean, monotone,
anti-monotone, and free Levy processes on dual semigroups. ́
Definition 1.9.3 [Sch95b] Let(B,∆,ε) be a dual semigroup. A
quantum stochastic process {jst} 0 ≤s≤t≤∞ on B over some
quantum probability space(A,Φ)is called atensor (resp. boolean,
monotone, anti-monotone, or free) L ́evy process on the dual semigroupB,
if the following four conditions are satisfied.