Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 59

Note that here the free convolution of two discrete measures has
a density and no atoms. In many respects the free convolution has
a stronger regularizing effect than classical convolution, see also
[BV98, BB04, BB05, Bel08].
Let us now look at the analogous results for the multiplicative
free convolutions of probability measures on the unit circle and the
positive half-line,cf.[BV92, CG05, CG06].

Theorem 1.7.6

(i) Letμandνbe two probability measures on the unit circle with

transforms K∫ μ and Kν and whose first moments do not vanish,

Txdμ(x)^6 =^0 ,

∫

Txdν(x)^6 =^0. Then there exist unique functions

Z 1 ,Z 2 ∈Ssuch that

Kμ

(

Z 1 (z)

)

=Kν

(

Z 2 (z)

)

=

Z 1 (z)Z 2 (z)

z

for all z∈D\{ 0 }. The multiplicative free convolutionλ=μν

is defined as the unique probability measureλ with transform

Kλ=Kμ◦Z 1 =Kν◦Z 2.

(ii) Let U and V be two unitary operators on some Hilbert space H that

are free w.r.t. some unit vectorΩ∈H. Then the products UV and

VU are also unitary and their distributions w.r.t. toΩare equal to

the free convolution of the distributions of U and V w.r.t.Ω, that is,

L(UV,Ω) =L(VU,Ω) =L(U,Ω)L(V,Ω).

Theorem 1.7.7

(i) Letμandνbe two probability measures on the positive half-line

such thatμ 6 =δ 0 ,ν 6 =δ 0 and denote their transforms by Kμand

Kν. Then there exist unique functions Z 1 ,Z 2 ∈Psuch that

Kμ

(

Z 1 (z)

)

=Kν

(

Z 2 (z)

)

=

Z 1 (z)Z 2 (z)

z

for all z∈C\R+. The multiplicative free convolutionλ=μν

is defined as the unique probability measureλwith transform Kλ=

Kμ◦Z 1 =Kν◦Z 2.

(ii) Let X and Y be two positive operators on some Hilbert space H that

are free w.r.t. some unit vectorΩ∈H. Assume, furthermore, that

Ωis cyclic, that is, that