Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 57

cf.[BB05] and the references therein.
In the following, if X is an operator with distribution
μ=L(X,Ω)w.r.t.Ω, then we will writeGX,FX,ψXorKXinstead
ofGL(X,Ω),FL(X,Ω),ψL(X,Ω), orKL(X,Ω)for the transforms of the
distribution ofX.

1.7.2 Free convolutions

ByAkwe denote the set of alternatingk-tuples of 1’s and 2’s, i.e.

Ak=

{

(ε 1 ,... ,εk)∈{1, 2}k;ε 16 =ε 26 =... 6 =εk

}

.

Definition 1.7.2 [Voi86] LetA 1 ,A 2 ⊆ B(H)be two∗-algebras of
bounded operators on a Hilbert space and assume 1 ∈ Ai,i =
1, 2. LetΩbe a unit vector inHand denote byΦthe vector state
associated with toΩ. We say thatA 1 andA 2 arefree, if we have

Φ(X 1 ···Xk) = 0

for allk≥1,ε∈Ak,X 1 ∈Aε 1 ,... ,Xk∈Aεksuch that

Φ(X 1 ) =···=Φ(Xk) =0.

Two normal operatorsX andY are called free, if the algebras
alg(X) ={h(X);h∈Cb(C)}and alg(Y) ={h(Y);h∈Cb(C)}they
generate are free.

Theorem 1.7.3 [Maa92, CG05, CG06] Letμandνbe two probability
measures on the real line, with reciprocal Cauchy transforms Fμand Fν.
Then there exist unique functions Z 1 ,Z 2 ∈Fsuch that

Fμ

(

Z 1 (z)

)

=Fν

(

Z 2 (z)

)

=Z 1 (z) +Z 2 (z)−z

for allz∈C+.
The functionF = Fμ◦Z 1 = Fν◦Z 2 also belongs toF and is
therefore the reciprocal Cauchy transform of some probability
measureλ. One defines theadditive free convolutionofμandνas
this unique probability measure and writesμ
ν = λ. This is
justified by the following theorem.

Theorem 1.7.4 [Maa92, BV93] Let X and Y be two self-adjoint
operators on some Hilbert space H that are free w.r.t. some unit vector
Ω∈H. IfΩis cyclic, that is, if

alg{h(X),h(Y);h∈Cb(R)}Ω=H.