Noncommutative Mathematics for Quantum Systems

66 Noncommutative Mathematics for Quantum Systems

This example shows that convolution from the left by a Dirac mass
is in general not equal to a translation and that the additive
monotone convolution is not affine in the second argument.
Note that the continuity and the fact that the monotone
convolution is affine in the first argument imply the following
formula

μ.ν=

∫

R

δx.νdμ(x) (1.7.3)

for allμ,ν∈M 1 (R).
The following proposition is the key to treat the additive
monotone convolution for general probability measures onR.

Proposition 1.7.17 Letμandνbe two probability measures onRand
be denoted by Mx and My the self-adjoint operators on L^2 (R×R,
μ⊗ν)defined by multiplication with the coordinate functions. Denote
by P 2 the orthogonal projection onto the subspace of functions that do not
depend on the second coordinate, that is, P 2 : L^2 (R×R,μ⊗ν) 3
ψ7→

∫

Rψ(·,y)dν(y)∈L

(^2) (R×R,μ⊗ν). Then MxP 2 =P 2 Mxand
Myare self-adjoint and monotonically independent w.r.t. the constant
function, and the operator z−MxP 2 −Myhas a bounded inverse for all
z∈C\R, given by
(
(z−MxP 2 −My)−^1 ψ
)
(x,y)=
ψ(x,y)
z−y

• x
  ∫
  R
  ψ(x,y′)
  z−y′ dν(y
  ′)
  (z−y)( 1 −xGν(z))
  .
  (1.7.4)
  Proof MxP 2 and My are monotonically independent by
  Proposition 1.7.12.
  The first term on the right-hand-side of Equation (1.7.4) is
  obtained fromψby multiplication with a bounded function, the
  second by composition of multiplications with bounded functions
  and the projectionP 2. Equation (1.7.4) therefore clearly defines a
  bounded operator. To check that it is indeed the inverse ofz−Mx
  P 2 −My is straightforward. If we apply the operator(z−Mx
  P 2 −My)to the function
  
  ψ(x,y)
  z−y


• 
  

  x
  ∫
  R
  ψ(x,y′)
  z−y′ dν(y
  ′)
  (z−y)
  (
  1 −xGν(z)
  )
  
  ,