Noncommutative Mathematics for Quantum Systems

94 Noncommutative Mathematics for Quantum Systems

This leads again to the formulae

φ 1 φ 2 (a 1 a 2 ···an) =

n

∏

i= 1

φei(ai),

φ 1 .φ 2 (a 1 a 2 ···an) = φ 1

(

∏

i:ei= 1

ai

)

∏

i:ei= 2

φ 2 (ai),

φ 1 /φ 2 (a 1 a 2 ···an) = ∏

i:ei= 1

φ 1 (ai)φ 2

(

∏

i:ei= 2

ai

)

,

fora 1 a 2 ···an∈ A 1 ‰A 2 ,ai ∈ A^0 ei,e 16 =e 26 =··· 6=en; however,

note that now the factorsaihave to be chosen fromA^0 ei. We use the
convention that the empty product is equal to the unit element.

These products can be defined in the same way for∗-algebraic
probability spaces, where the algebras are unital∗-algebras having
such a decompositionA=C 1 ⊕A 0 and the functionals are states.
To check thatφ 1 φ 2 ,φ 1 .φ 2 ,φ 1 /φ 2 are again states, ifφ 1 andφ 2
are states, one can verify that the following constructions give
their GNS representations. Let(π 1 ,H 1 ,ξ 1 )and(π 2 ,H 2 ,ξ 2 )denote
the GNS representations of (A 1 ,φ 1 ) and (A 2 ,φ 2 ). The GNS
representations of (A 1 ‰A 2 ,φ 1 φ 2 ), (A 1 ‰A 2 ,φ 1 .φ 2 ), and
(A 1 ‰A 2 ,φ 1 /φ 2 ) can all be defined on the Hilbert space
H = H 1 ⊗H 2 with the state vector ξ = ξ 1 ⊗ξ 2. The
representations are defined byπ( 1 ) =id and

π|A 0

1

= π 1 ⊗P 2 , π|A 0

2

= P 1 ⊗π 2 , for φ 1 φ 2 ,

π|A 0

1

= π 1 ⊗P 2 , π|A 0

2

= idH 2 ⊗π 2 , for φ 1 .φ 2 ,

π|A 0

1

= π 1 ⊗idH 2 , π|A 0

2

= P 1 ⊗π 2 , for φ 1 /φ 2 ,

whereP 1 ,P 2 denote the orthogonal projectionsP 1 :H 1 →Cξ 1 ,P 2 :
H 2 →Cξ 2. For the boolean case,ξ=ξ 1 ⊗ξ 2 ∈H 1 ⊗H 2 is not cyclic
forπ, only the subspaceCξ⊕H^01 ⊕H 20 can be generated fromξ.

1.8.4 Classification of the universal independences

We will consider products of the form
(
(A 1 ,φ 1 ),(A 2 ,φ 2 )

)

7→

(

(A 1 ‰A 2 ,φ 1 ·φ 2 ),iA 1 ,iA 2

)

on categories of algebraic probability spaces satisfying the
following conditions.