Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 93

φ 1 φ 2 (a 1 a 2 ···am) =

m

∏

k= 1

φek(ak),

φ 1 .φ 2 (a 1 a 2 ···am) = φ 1

(

→

∏

k:ek= 1

ak

)

∏

k:ek= 2

φ 2 (ak),

φ 1 /φ 2 (a 1 a 2 ···am) = ∏

k:ek= 1

φ 1 (ak)φ 2

( →

∏

k:ek= 2

ak

)

,

forφ 1 :A 1 →Candφ 2 :A 2 →Cand a typical elementa 1 a 2 ···am
∈ A 1 ‰A 2 ,ak∈ Aek,e 16 =e 26 =··· 6=em, that is, neighboringa’s
don’t belong to the same algebra. Note that we again have
canonical inclusionsAi → (A 1 ‰A 2 ),i = 1, 2, since the free
product without units is the coproduct in the category of not
necessarily unital algebras.
The monotone and anti-monotone product are not commutative,
but related by

φ 1 .φ 2 = (φ 2 /φ 1 )◦γA 1 ,A 2 ,

for all linear functionalsφ 1 :A 1 →C,φ 2 :A 2 →C, whereγA 1 ,A 2 :
A 1 ‰A 2 → A 2 ‰A 1 is the commutativity constraint (for the
commutativity constraint for the free product of unital algebras
see Equation (1.8.1)). The boolean product is commutative, that is,
it satisfies

φ 1 φ 2 = (φ 2 φ 1 )◦γA 1 ,A 2 ,

for all linear functionalsφ 1 :A 1 →C,φ 2 :A 2 →C.

Exercise 1.8.14 The boolean, the monotone, and the
anti-monotone products can also be defined for unital algebras, if
they are in the range of the unitization functor introduced in
Exercise 1.8.5.
Letφ 1 :A 1 →Candφ 2 :A 2 →Cbe two unital functionals on
algebrasA 1 ,A 2 , which can be decomposed asA 1 = C 1 ⊕A^01 ,
A 2 = C 1 ⊕ A^02. Then we define the boolean, monotone, or
anti-monotone product ofφ 1 andφ 2 as the unital extension of the
boolean, monotone, or anti-monotone product of their restrictions
φ 1 |A 0
1

andφ 2 |A 0

2

.