90 Noncommutative Mathematics for Quantum Systems
1.8.3 Products of algebraic probability spaces
We will now define several products on the categories of algebraic
probability spaces. In Subsection 1.8.4 we shall see that these
products are the only products satisfying the conditions of
functoriality and associativity.
Example: Tensor independence in the category of algebraic probability
spaces
Let us first consider the usual tensor product on the categories
AlgProband∗-AlgProb. The product of (A,φ 1 ) andA 2 ,φ 2 ) is
(A 1 ⊗A 2 ,φ 1 ⊗φ 2 ), where the unital algebra structure ofA 1 ⊗A 2
is defined by
(^1) A 1 ⊗A 2 = (^1) A 1 ⊗ (^1) A 2 ,
(a 1 ⊗a 2 )(b 1 ⊗b 2 ) = a 1 b 1 ⊗a 2 b 2 ,
and the new functional is defined by
(φ 1 ⊗φ 2 )(a 1 ⊗a 2 ) =φ 1 (a 1 )φ 2 (a 2 ),
for alla 1 ,b 1 ∈A 1 ,a 2 ,b 2 ∈A 2.
Note that there exist canonical morphisms from the objects
(A,φ 1 )and(A 2 ,φ 2 )to the product(A 1 ⊗A 2 ,φ 1 ⊗φ 2 ), defined by
iA 1 (a 1 ) = a 1 ⊗ (^1) A 2 ,
iA 2 (a 2 ) = (^1) A 1 ⊗a 2 ,
fora 1 ∈A 1 ,a 2 ∈A 2.
If(A,φ 1 )andA 2 ,φ 2 )are∗-algebraic probability spaces, if the
underlying algebras have an involution and the functionals are
states, then an involution is defined onA 1 ⊗A 2 by(a 1 ⊗a 2 )∗ =
a∗ 1 ⊗a∗ 2 andφ 1 ⊗φ 2 is again a state.
The notion of independence associated to this product with
inclusions is the usual notion ofBoseortensor independenceused in
quantum probability, e.g., by Hudson and Parthasarathy, see also
Definition 1.5.1.
Proposition 1.8.9 Two quantum random variables j 1 : (B 1 ,ψ 1 ) →
(A,φ)and j 2 : (B 2 ,ψ 2 ) → (A,φ), defined on algebraic probability
spaces (B 1 ,ψ 1 ),(B 2 ,ψ 2 ) and with values in the same algebraic
probability space(A,φ)are independent if and only if the following two
conditions are satisfied.