88 Noncommutative Mathematics for Quantum Systems
NuAlgProb, ∗-AlgProb, and ∗-NuAlgProb by restricting to
commutative algebras.
It is the category∗-AlgProbthat has as objects the∗-algebraic
probility spaces with which we worked in Section 1.5, when we
developed the theory of Levy processes on involutive bialgebras. ́
The categories of algebraic probability spaces defined above do
not have coproducts. However, they have interesting products, as
we shall see in this section. Imposing several natural conditions,
such as functoriality and associativity, it becomes possible to
classify all such products, see Subsection 1.8.4. These so-called
universal products of algebraic probability spaces allow to define
the universal notions of independence, see Definition 1.9.1.
Definition 1.8.6 LetCbe any category equiped with a mapPthat
associates to any pair of objectsA,B∈Ob(C)a triple
P(A,B) = (D,jA,jB)consisting of an objectDand morphismsjA:A→DandjB:B→
D. Then we say that two morphismskA:A→CandkB:B→C
areindependent w.r.t. toPif there exists a morphismh:D→Csuch
thatkA=h◦jAandkB=h◦jB, that is, the diagram commutes.
A
jA
kA
D h //CBjBOOkB??Remark 1.8.7 If a category has a coproduct, then all pairs of
morphisms are independent w.r.t. to the coproduct, simply by the
universal property of the coproduct,cf.Definition 1.8.1.
1.8.2 Classical stochastic independence and the product of
probability spaces
Let us first consider the product of classical probability spaces and
its relation to classical independence from a category theoretical
point of view.
Two random variablesX 1 :(Ω,F,P) → (E 1 ,E 1 )andX 2 :(Ω,
F,P)→(E 2 ,E 2 ), defined on the same probability space(Ω,F,P)