Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 41

1.5.4 Examples

L ́evy processes on the circleTand on the real lineR
Consider the involutive bialgebra

B=span{eλ;λ∈R}

with the multiplication

eλ·eμ=eλ+μ, λ,μ∈R,

involutione∗λ=e−λforλ∈R, coproduct

∆(eλ) =eλ⊗eλ

and counitε(eλ) =1 for allλ,μ∈R. Consider also the subalgebras

Bα=span{ekα;k∈Z}

forα>0.
The basis elements eλ can be represented as exponential
functionseλ:R 3 x→ eiλx ∈C. As this representation ofBis
faithful, we can viewBas a subalgebra of the algebraR(R)of
representative functions onR.
The subalgebraBα is generated as a∗-algebra by one unitary
elementeα; it is therefore commutative and isomorphic to the
algebra of polynomials on the unit circleT.
Levy processes on ́ R(R),B, andBα can be constructed from
real-valued Levy processes. If ́ (Xt)t≥ 0 is a real-valued Levy ́
process, then we would like to define a Levy process on ́
f∈R(R)by

jst(f) =f(Xt−Xs)

for 0≤s≤tandf∈R(R), but since the expectation off(Xt−Xs)
might not be defined, we have to be more careful. We can do this by
restricting to appropriate subalgebras ofR(R). E.g., if we restrict
the homomorphisms(jst) 0 ≤s≤ttoBorBα, then

jst(eλ) =exp

(

iλ(Xt−Xs)

)

are bounded random variables defined on the same probability
space (Ω,F,P) as the Levy process ́ (Xt)t≥ 0 , so we can view
(jst) 0 ≤s≤tas a Levy process on the involutive bialgebras ́ BorBα