Noncommutative Mathematics for Quantum Systems

84 Noncommutative Mathematics for Quantum Systems

processes with tensor-independent increments. In order to define
and study Levy ́ processes whose increments are free,
monotonically independent, or boolean-independent, bialgebras
have to be replaced by dual groups, which we shall introduce in
Section 1.9.
These universal products of quantum probability spaces have
an interpretation in category theory; they are tensor products
equipped with canonical inclusions of the factors into the product.
Furthermore, it turns out that boolean, monotone, and
anti-monotone independence can be reduced to tensor
independence in a similar way as the bosonization of Fermi
independence [HP86] or the symmetrization of [Sch93, Section 3],
cf.[Fra02, Fra03c, Fra06b].
Let us recall the definition of a coproduct, it will be underlying
many constructions in this Section.

Definition 1.8.1 LetAandBbe two objects in some category. A
triple(A‰B,iA,iB)consisting of an objectA‰Band morphisms
iA:A→A‰BandiB:B→A‰Bis called acoproductofAandB,
if for any triple(D,kA,kB)consisting of an objectDand morphisms
kA : A → DandkB : B → Dthere exists a unique morphism
h:A‰B→Dsuch that the following diagram commutes

A

iA



kA

""

A‰B h //D

B

iB

OO

kB

<<

This means thathsatisfieskA=h◦iAandkB=h◦iB.

The free product of associative algebras is an example of a
coproduct, it will play an important role in our discussion.

Example 1.8.2 Denote by Algthe category whose objects are
unital associative algebras and whose morphisms are
unit-preserving algebra homomorphisms.
This category has a coproduct, it is also calledfree productof
algebraswithidentification of the units. Let us recall its defining
universal property. Let{Ak}k∈Ibe a family of unital algebras and