Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 55

by Lindsay and Skalski (cf.[LS11, Proposition 3.2]). This covers,
for example the case of universal locally compact quantum groups
and, as special case, discrete quantum groups and coamenable
compact quantum groups. Their proof is more technical than the
algebraic argument we presented here.

1.7 Independences and Convolutions in

Noncommutative Probability

Until now we only considered tensor independence, which is the
natural generalization of the notion of stochastic independence
used in classical probability and also corresponds to the notion of
independent observables used in quantum mechanics. However,
in quantum probability there exist also other notions of
independence. In this section we shall study the most prominent
examples: freeness, monotone independence, and boolean
independence, and the corresponding convolutions for probability
measures onR,R+, andTderived from them.
The material in this Section is taken from [Fra09].
In Subsection 1.7.1 we recall the necessary pre-requisites about
Cauchy-Stieltjes transforms of probability measures on the real line,
the unit circle, or the positive half-line.
In Subsection 1.7.2 we summarize the definition of freeness and
the formulae for computing free convolutions. See also, for
example, [VDN92, Voi00, BNT05, NS06], and the references
therein.
Then we state a short Lemma that we use in the following
Sections, see Lemma 1.7.8.
In Subsections 1.7.4 and 1.7.5 we study the monotone and
boolean convolutions for probability measures on the real line, the
positive half-line, and the unit circle.

1.7.1 Nevanlinna theory and Cauchy–Stieltjes transforms

Denote byC+={z∈C; Imz> 0 }andC−={z∈C; Imz< 0 }
the upper and lower half plane. Forμa probability measure onR
andz∈C+, we define itsCauchy–Stieltjes transform Gμby

Gμ(z) =

∫

R

1

z−x

dμ(x)