Noncommutative Mathematics for Quantum Systems

78 Noncommutative Mathematics for Quantum Systems

This proves the first formula. The second formula follows by
symmetry.
Letf,g∈Cb(C),f( 0 ) =0, and note that
∣
∣
∣∣

∣

∣

∣∣f(X)g(Y)Ω−

∫

C

gdνf(X)Ω

∣

∣

∣∣

∣

∣

∣∣

2

= 〈Ω,g(Y)∗|f(X)|^2 g(Y)Ω〉−

∫

C

gdν〈Ω,g(Y)∗|f(X)|^2 Ω〉

−

∫

C

gdν〈Ω,|f(X)|^2 g(Y)Ω〉+

(∫

C

gdν

) 2

〈Ω,|f(X)|^2 Ω〉

= 0,

that is, f(X)g(Y)Ω =

∫

∫ Cgdνf(X)Ω. Similarly f(Y)g(X)Ω =
Cgdμf(Y)Ωand thus

alg{h(X),h(Y):h∈Cb(C)}Ω = span{Ω,f(X)Ω,f(Y)Ω;f∈Cb(C)}

= W

(

C⊕L^2 (C,μ) 0 ⊕L^2 (C,ν) 0

)

.

IfΩis cyclic, thenWis surjective and therefore unitary.

Remark 1.7.33 As in the monotone case,cf.Remark 1.7.14, this
theorem shows that the joint law of bounded functions ofXandY
is uniquely determined byL(X,Ω)andL(Y,Ω). Furthermore, the
characterization and computation of the law of unbounded

functions ofXandYlike, e.g.,X+Yor

√

XY

√
Y, is also reduced
to the model introduced in Proposition 1.7.31.

Additive boolean convolution onM 1 (R)
Definition 1.7.34 [SW97] Let μ and ν be two probability
measures onRwith reciprocal Cauchy transformsFμandFν. Then
we define theadditive boolean convolutionλ=μ]νofμandνas
the unique probability measureλon Rwith reciprocal Cauchy
transform given by

Fλ(z) =Fμ(z) +Fν(z)−z

forz∈C+.
That the additive boolean convolution is well defined follows
from Subsection 1.7.1. It is commutative and associative,∗-weakly
continuous, but not affine,cf.[SW97].