Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 91

(i) The images of j 1 and j 2 commute, that is,
[
j 1 (a 1 ),j 2 (a 2 )

]

=0,

for all a 1 ∈A 1 , a 2 ∈A 2.

(ii) φsatisfies the factorization property

φ

(

j 1 (a 1 )j 2 (a 2 )

)

=φ

(

j 1 (a 1 )

)

φ

(

j 2 (a 2 )

)

,

for all a 1 ∈A 1 , a 2 ∈A 2.

Example 1.8.10 (The universal version of the tensor product)
We can also use the tensor product to define a product

φ 1 ⊗ ̃φ 2 :A 1 ‰A 2 →C

for two functionalsφi:Ai→C, simply by composing the tensor
productφ 1 ⊗φ 2 : A 1 ⊗A 2 → C⊗C ∼= Cwith the unique
morphismR:A 1 ‰A 2 →A 1 ⊗A 2 that makes the diagram

A 1

jA 1



iA 1

&&

A 1 ‰A 2 R //A 1 ⊗A 2

A 2

jA 2

OO

iA 2

88

commute. The product

(

(A 1 ,φ 1 ),(A 2 ,φ 2 )

)

7→

(

(A 1 ‰A 2 ,φ 1 ⊗ ̃φ 2 ),jA 1 ,jA 2

)

is called theuniversalversion of the tensor product of functionals.

Example 1.8.11 (Free independence)
We will now introduce another product with inclusions for the
category of algebraic probability spacesAlgProb. On the algebras
we take simply the free product of algebras with identifications of
units introduced in Example 1.8.2. This is the coproduct in the
category of algebras; therefore, we also have natural inclusions. It
only remains to define a unital linear functional on the free
product of the algebras.