Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 87

How is this functor defined on the morphisms?
Show that the following relation holds between the free product
with identification of units ‰Alg and the free product without
identification of units‰NuAlg,

A 1 ̃

‰

NuAlg

A 2 ∼=A ̃ (^1) ‰
Alg
A ̃ 2
for allA 1 ,A 2 ∈ObNuAlg.
Note, furthermore, that the range of this functor consists of all
algebras that admit a decomposition of the formA =C 1 ⊕A 0 ,
where A 0 is a subalgebra. This is equivalent to having a
one-dimensional representation. The functor is not surjective, e.g.,
the algebra M 2 of 2×2-matrices can not be obtained as a
unitization of some other algebra.

1.8.1 Algebraic probability spaces

From the four categoriesAlg,NuAlg,∗-Alg, and∗-NuAlg, we form
the four categories AlgProb, NuAlgProb, ∗-AlgProb, and
∗-NuAlgProbofalgebraic probability spaceswith or without unit and
with or without involution. The objects in these categories of
algebraic probability spaces are pairs (A,φ) consisting of an
algebraAtaken from the corresponding category of algebras, and
a linear functionalφ:A→C. In the two categoriesAlgProband
∗-AlgProb constructed from unital algebras we impose the
condition φ( 1 ) = 1; in the two categories ∗-AlgProb and
∗-NuAlgProb built from involutive algebras we impose the
additional condition

φ(a∗a)≥ 0 for alla∈A.

The morphisms in the categoriesAlgProb,NuAlgProb,∗-AlgProb,
and∗-NuAlgProbare those morphisms from the categoriesAlg,
NuAlg, ∗-Alg, and ∗-NuAlg, respectively, which preserve
furthermore the functionals, that is,j: (A,φA) → (B,φB)has to
satisfy the condition

φB◦j=φA.

We can also consider the subcategories CommAlgProb,
CommNuAlgProb, ∗-CommAlgProb, and ∗-CommNuAlgProb of
commutativealgebraic probability spaces, obtained fromAlgProb,