Independence and L ́evy Processes in Quantum Probability 65Remark 1.7.14 It follows that the joint law of two monotonically
independent, normal operators is uniquely determined by their
marginal distributions, in the sense that the restriction of
Φ(·) = 〈Ω,·Ω〉 to alg(X,Y) = alg{h(X),h(Y);h ∈ Cb(C)}is
uniquely determined by L(X,Ω) and L(Y,Ω). But by
Lemma 1.7.8, also computations for unbounded functions ofX
andY, e.g., concerning the operatorsX+Yfor self-adjointXand
Y, or
√
XY√
Y for positive X and Y, reduce to the model
introduced in Proposition 1.7.12.
Additive monotone convolution onM 1 (R)
Definition 1.7.15 [Mur00] Let μ and ν be two probability
measures onRwith reciprocal Cauchy transformsFμandFν. Then
we define theadditive monotone convolutionλ=μ.νofμandνas
the unique probability measure on R with reciprocal Cauchy
transformFλ=Fμ◦Fν.
It follows from Subsection 1.7.1 that the additive monotone
convolution is well defined. Let us first recall some basic
properties of the additive monotone convolution.
Proposition 1.7.16 [Mur00] The additive monotone convolution is
associative and∗-weakly continuous in both arguments. It is affine in the
first argument and convolution from the right by a Dirac measure
corresponds to translation, that is,μ.δx = Tx−^1 μfor x ∈ R, where
Tx:R→Ris defined by Tx(t) =t+x.
This convolution is not commutative, that is, in generalμ.ν 6 =
ν.μ.
Letx∈Rand 0≤p≤1. Then one can compute, for example,
δx.(
pδ 1 + ( 1 −p)δ− 1)
=qδz 1 + ( 1 −q)δz 2where
z 1 =1
2(
x+√
x^2 + 4 ( 2 p− 1 )x+ 4)
,z 2 =1
2(
x−√
x^2 + 4 ( 2 p− 1 )x+ 4)
,q =x+ 4 p− 2 +√
x^2 + 4 ( 2 p− 1 )x+ 4
2√
x^2 + 4 ( 2 p− 1 )x+ 4.