Independence and L ́evy Processes in Quantum Probability 95- Associativity:
(
(φ 1 ·φ 2 )·φ 3
)
◦αA 1 ,A 2 ,A 3 =φ 1 ·(φ 2 ·φ 3 ), (Assoc)for all (A 1 ,φ 1 ),(A 2 ,φ 2 ),(A 3 ,φ 3 ) in AlgProb, where
αA 1 ,A 2 ,A 3 denotes the canonical associativity morphism of the
free product of (unital or non-unital) algebras, see Equation
(1.8.1).- Functoriality: we want the product to be a functor; therefore,
the free productj 1 ‰j 2 of two morphisms should again be a
morphism. This is the case if and only if
(φ 1 ·φ 2 )◦(j 1 ‰j 2 ) = (φ 1 ◦j 1 )·(φ 2 ◦j 2 ) (Funct)
for all pairs of morphismsj 1 :(B 1 ,ψ 1 )→(A 1 ,φ 1 ),j 2 :(B 2 ,
ψ 2 )→(A 2 ,φ 2 ).- Inclusion: the inclusionsiAi : Ai → (A 1 ‰A 2 ),i = 1, 2,
should be random variables (that is, morphisms) in the
corresponding category. This is the case if and only if
(φ 1 ·φ 2 )◦iA 1 =φ 1 and (φ 1 ·φ 2 )◦iA 2 =φ 2 (Incl)for all pairs of objects(A 1 ,φ 1 ),(A 2 ,φ 2 ).- Normalization: in addition, we can also consider the condition
(φ 1 ·φ 2 )(a 1 a 2 ) =φe 1 (a 1 )φe 2 (a 2 ) (Norm)for all(e 1 ,e 2 )∈{
(1, 2),(2, 1)}
,a 1 ∈Ae 1 ,a 2 ∈Ae 2.Our Conditions (Assoc), (Incl), and (Funct) are exactly the axioms
(P2), (P3), and (P4) in Ben Ghorbal and Schurmann[BGS99], or the ̈
axioms (U2), the first part of (U4), and (U3) in Muraki[Mur03].
The Conditions (Assoc), (Incl), and (Funct) can be reformulated;
they express the fact that the product should be a tensor product
equipped with natural inclusions of the objects into their product,
cf. [Fra02, Fra06b, GL14, Ger15, Lac15]. We will call products
satisfying (Assoc), (Incl), and (Funct)tensor products with inclusions
or simplyuniversal products.
Theorem 1.8.15 (Muraki[Mur03], Ben Ghorbal, and Sch ̈urmann
[BG01, BGS99]) There exist exactly two universal products on the
category of algebraic probability spacesAlgProb, namely the universal