74 Noncommutative Mathematics for Quantum Systems
Definition 1.7.30 Let XandY be two normal operators on a
Hilbert spaceH, not necessarily bounded. We say thatXandYare
boolean-independent, if the∗-algebras alg 0 (X) ={h(X):h∈Cb
(C),h( 0 ) = 0 }and alg 0 (Y) = {h(Y):h∈Cb(C),h( 0 ) = 0 }are
boolean-independent.
We will start by characterizing up to unitary transformations
the general form of two boolean-independent normal operators.
Given a measure space(M,M,μ), we shall denote byL^2 (M,μ) 0
the orthogonal complement of the constant function, that is,
L^2 (M,μ) 0 =
{
ψ∈L^2 (M,μ);
∫
M
ψdμ= 0
}
.
Proposition 1.7.31 Letμ,νbe two probability measures onCand
define normal operators Nx and Ny on Hμ,ν = C⊕L^2 (C,μ) 0 ⊕
L^2 (C,ν) 0 by
DomNx =
α
ψ 1
ψ 2
∈Hμ,ν;
∫
C
∣
∣x
(
ψ 1 (x) +α
)∣
∣^2 dμ(x)<∞
,
DomNy =
α
ψ 1
ψ 2
∈Hμ,ν;
∫
C
∣∣
y
(
ψ 2 (y) +α
)∣∣ 2
dν(y)<∞
,
Nx
α
ψ 1
ψ 2
=
∫
Cx
(
ψ 1 (x) +α
)
dμ(x)
x(ψ 1 +α)−
∫
Cx
(
ψ 1 (x) +α
)
dμ(x)
0
,
Ny
α
ψ 1
ψ 2
=
∫
Cy
(
ψ 2 (y) +α
)
dν(y)
0
y(ψ 2 +α)−
∫
Cy
(
ψ 2 (y) +α
)
dν(y)
.
Then Nxand Nyare boolean-independent w.r.t. the vectorω=
1
0
0
and we haveL(Nx,ω) =μ,L(Ny,ω) =ν.