96 Noncommutative Mathematics for Quantum Systems
version⊗ ̃ of the tensor product defined in Example 1.8.10 and the free
product∗of states defined in Example 1.8.11.
In the non-unital case there exist more universal products.Theorem 1.8.16 (Muraki[Mur03]) There exist exactly five universal
products satisfying (Norm) on the category of non-unital algebraic
probability spacesNuAlgProb, namely the universal version⊗ ̃ of the
tensor product, the free product∗, the boolean product, the monotone
product., and the anti-monotone product/.
The monotone and the anti-monotone product are not
symmetric, that is,(A 1 ‰A 2 ,φ 1 .φ 2 )and(A 2 ‰A 1 ,φ 2 .φ 1 )are
not isomorphic in general. Actually, the anti-monotone product is
simply the mirror image of the monotone product,
(A (^1) ‰A 2 ,φ 1 .φ 2 )∼=(A (^2) ‰A 1 ,φ 2 /φ 1 )
for all(A 1 ,φ 1 ),(A 2 ,φ 2 ) in the category of non-unital algebraic
probability spaces. The other three products are symmetric.
Condition (Norm) is not essential. If one drops it and adds
symmetry, one finds in addition the degenerate product
(φ 1 • 0 φ 2 )(a 1 a 2 ···am) =
{
φe 1 (a 1 ) if m=1,
0 if m>1.
(1.8.3)
and families
φ 1 • qφ 2 =q
(
(q−^1 φ 1 )·(q−^1 φ 2 )
)
, (1.8.4)
parametrized by a complex numberq∈C{ 0 }, for each of the three
symmetric products,•∈{⊗ ̃,∗,}.
If one adds the condition that products of states are again states,
then one can also show that the constant has to be equal to one.
Exercise 1.8.17 Consider the category of non-unital∗-algebraic
probability spaces, whose objects are pairs(A,φ)consisting of a
not necessarily unital∗-algebraAand a stateφ:A →C. Here a
state is a linear functionalφ : A → Cwhose unital extension
φ ̃ :A ̃ ∼=C 1 ⊕A →C,λ 1 +a7→ φ ̃(λ 1 +a) =λ+φ(a), to the
unitization ofAis a state.
Assume we have products·:S(A 1 )×S(A 2 )→S(A 1 ‰A 2 )of
linear functionals on non-unital algebrasA 1 ,A 2 that satisfy