Noncommutative Mathematics for Quantum Systems

72 Noncommutative Mathematics for Quantum Systems

Therefore,

G√XY√X(z) =

∫

R+×R+

(

1 +g 1 (x)

z−y

+h 1 (x)

)

dμ⊗ν(x,y)

=

∫

R+

Gν(z)

( 1 −x)zGν(z) +x

dμ(x)

=

Gν(z)

zGν(z)− 1

Gμ

(

zGν(z)

zGν(z)− 1

)

. (1.7.7)

Using the relation

Gμ(z) =

1

z

(

ψμ

(

1

z

)

+ 1

)

to replace the Cauchy transforms by theψ-transforms, this becomes

ψ√XY√X

(

1

z

)

=ψμ

(

ψν(1/z)

ψν(1/z) + 1

)

,

or finally

K√XY√X(z) =Kμ

(

Kν(z)

)

=Kμmν(z).

Multiplicative monotone convolution onM 1 (T)
Definition 1.7.24 [Ber05a] Let μ and ν be two probability
measures on the unit circleTwith transformsKμandKν. Then the
multiplicative monotone convolutionofμ and νis defined as the
unique probability measure λ = μmν on T with transform
Kλ=Kμ◦Kν.

It follows from Subsection 1.7.1 that the multiplicative monotone
convolution onM 1 (T)is well defined.
Let us first recall some basic properties of the multiplicative
monotone convolution.

Proposition 1.7.25 The multiplicative monotone convolution on the
unit circleTis 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 rotation, that is,μmδeiθ = R−θ^1 μfor

θ∈[0, 2π[, where Rθ:T→Tis defined by Rθ(t) =eiθt

This convolution is not commutative, that is, in generalμmν 6 =
νmμ. As in the additive case, it is not affine in the second argument,