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,