Noncommutative Mathematics for Quantum Systems

86 Noncommutative Mathematics for Quantum Systems

fora 1 ⊗···⊗an∈ Ae,b 1 ⊗···⊗bm∈ Aδ. Note that in the case
en=δ 1 the productan·b 1 is not necessarily inA^0 en, but is in general

a sum of a multiple of the unit ofAenand an element ofA^0 en. We
have to identifya 1 ⊗···an− 1 ⊗ 1 ⊗b 2 ⊗···bmwitha 1 ⊗···⊗an− 1 ·
b 2 ⊗···bm.
Since‰is the coproduct of a category, it is commutative and
associative in the sense that there exist natural isomorphisms

γA 1 ,A 2 : A 1 ‰A 2

∼=

→A 2 ‰A 1 , (1.8.1)

αA 1 ,A 2 ,A 3 : A 1 ‰

(

A 2 ‰A 3

) ∼=

→

(

A 1 ‰A 2

)

‰A 3

for all unital algebrasA 1 ,A 2 ,A 3. Leti: A → A 1 ‰A 2 and
i′: A → A 2 ‰A 1 ,` = 1, 2 be the canonical inclusions. The
commutativity constraintγA 1 ,A 2 :A 1 ‰A 2 → A 2 ‰A 1 maps an
element of A 1 ‰A 2 of the form i 1 (a 1 )i 2 (b 1 )···i 2 (bn) with
a 1 ,... ,an∈A 1 ,b 1 ,... ,bn∈A 2 to

γA 1 ,A 2

(

i 1 (a 1 )i 2 (b 1 )···i 2 (bn)

)

=i′ 1 (a 1 )i 2 ′(b 1 )···i 2 ′(bn)∈A 2 ‰A 1.

Exercise 1.8.3 We also consider non-unital algebras. Show that
the free product of algebras without identification of units is a
coproduct in the category NuAlg of non-unital (or rather not
necessarily unital) algebras. Give an explicit construction for the
free product of two non-unital algebras.

Exercise 1.8.4 If we want to have a notion of positivity, we have
to work with involutive algebras. Define the categories∗-Algand
∗-NuAlgof involutive unital or not necessarily unital algebras as
above, but requiring in addition that the morphisms respect also
the involution, that is, thatj(a∗) = j(a)∗for allain the domain
ofj. Show that these categories also have coproducts, which can be
constructed essentially as in the non-involutive case.

Exercise 1.8.5 Show that one can define a functor from the
category of non-unital algebrasNuAlgto the category of unital
algebrasAlg. For an algebraA∈ObNuAlg,A ̃, defineA ̃=C 1 ⊕A
as a vector space and equip this with the multiplication

(λ 1 +a)(λ′ 1 +a′) =λλ′ 1 +λ′a+λa′+aa′

forλ,λ′∈C,a,a′∈A. We will callA ̃theunitizationofA. Note that
A∼= 01 +A⊆A ̃is not only a subalgebra, but even an ideal inA ̃.