Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 61

Remark 1.7.10

(a) Note that this notion depends on the order, that is, ifA 1 and

A 2 are monotonically independent, then this doesnotimply

thatA 2 andA 1 are monotonically independent. In fact, ifA 1

andA 2 are monotonically independent andA 2 andA 1 are

also monotonically independent, andΦ(·) = 〈Ω,·Ω〉does

not vanish on one of the algebras, then restrictions ofΦtoA 1

andA 2 have to be homomorphisms. To prove this for the

restriction to, e.g.,A 1 , take an elementY ∈ A 2 such that

Φ(Y) 6 =0, then

Φ(X 1 X 2 ) =

Φ(X 1 YX 2 )

Φ(Y)

=Φ(X 1 )Φ(X 2 )

for allX 1 ,X 2 ∈A 1.

(b) The sub-algebras are not required to be unital. IfA 1 contains

the identity operator 1 , then the restriction ofΦ(·) =〈Ω,·Ω〉

to A 2 has to be a homomorphism, since monotone

independence implies

〈Ω,XYΩ〉=〈Ω,X 1 YΩ〉=〈Ω,XΩ〉〈Ω,YΩ〉

forX,Y∈A 2.

(c) In the definition of monotone independence the condition

XYZ=〈Ω,YΩ〉XZ

for allX,Z ∈ A 1 ,Y∈ A 2 is often also imposed. If the state

vectorΩis cyclic for the algebra generated byA 1 andA 2 , then

this is automatically satisfied. LetX 1 ,X 3 ,... ,Z 1 ,Z 3 ,.. .∈ A 1

andY,X 2 ,X 4 ,... ,Z 2 ,Z 4 ,.. .∈A 2 , then

〈X 1 ···XnΩ,YZ 1 ···ZmΩ〉=〈Ω,X∗n···X∗ 1 YZ 1 ···ZmΩ〉

= 〈Ω,YΩ〉 ∏

keven

〈Ω,X∗kΩ〉 ∏

`even

〈Ω,Z`Ω〉〈X 1 X 3 ···Ω,Z 1 Z 3

···Ω

= 〈Ω,YΩ〉〈X 1 ···XnΩ,Z 1 ···ZmΩ〉,

for alln,m≥1, that is,X 1 ∗YZ 1 and〈Ω,YΩ〉X∗ 1 Z 1 coincide on

the subspace generated byA 1 andA 2 fromΩ.