Independence and L ́evy Processes in Quantum Probability 85

‰k∈IAk their free product, with canonical inclusions
{ik:Ak→‰k∈IAk}k∈I. IfBis any unital algebra, equipped with
unital algebra homomorphisms{i′k:Ak→B}k∈I, then there exists
a unique unital algebra homomorphismh:‰k∈IAk→Bsuch that

h◦ik=i′k, for all k∈I.

It follows from the universal property that for any pair of unital
algebra homomorphismsj 1 :A 1 → B 1 ,j 2 :A 2 → B 2 , there exists a
unique unital algebra homomorphismj 1 ‰j 2 :A 1 ‰A 2 →B 1 ‰B 2
such that the diagram commutes

A 1

iA 1

zz

j (^1) //
B 1
iB 1
$$
A 1 ‰A 2 j 1 ‰j 2 //B 1 ‰B 2
A 2
iA 2
dd
j 2
//B 2
iB 2
::
The free product‰k∈IAkcan be constructed as a sum of tensor
products of theAk, where neighboring elements in the product
belong to different algebras. For simplicity, we illustrate this only
for the case of the free product of two algebras. Let
A=
⋃
n∈N
{e∈{1, 2}n|e 16 =e 26 =···6=en}
and decomposeAi=C 1 ⊕A^0 i,i=1, 2, into a direct sum of vector
spaces. As a coproductA 1 ‰A 2 is unique up to isomorphism;
therefore, the construction does not depend on the choice of the
decompositions.
ThenA 1 ‰A 2 can be constructed as
A (^1) ‰A 2 =
⊕
e∈A
Ae,
whereA∅ =C,Ae = A^0 e 1 ⊗···⊗A^0 enfore= (e 1 ,... ,en). The
multiplication inA 1 ‰A 2 is inductively defined by
(a 1 ⊗···⊗an)·(b 1 ⊗···⊗bm)

{
a 1 ⊗···⊗(an·b 1 )⊗···⊗bm ifen=δ 1 ,
a 1 ⊗···⊗an⊗b 1 ⊗···⊗bm ifen 6 =δ 1 ,