Independence and L ́evy Processes in Quantum Probability 29See also [App14, Theorem 5.3.3] for more details and an outline
of the proof of this theorem. Michael Skeide has given a
C∗-algebraic proof of Hunt’s Theorem for compact Lie groups in
[Ske99].
Exercise 1.4.13 Show how we can recover the L ́evy-Khintchine
formula forRdfrom Hunt’s Theorem.
1.5 Levy Processes on Involutive Bialgebras ́
In this section we will give the definition of Levy processes on ́
involutive bialgebras, cf. Subsection 1.5.1, and develop their
general theory.
In Subsection 1.5.2, we will see that the marginal distributions
of a Levy process form a convolution semigroup of states. We will ́
associate a generating functional with a Levy process on an ́
involutive bialgebra, which characterizes uniquely its distribution,
like in classical probability. By a GNS-type construction we can get
a so-called Schurmann triple from the generating functional, see ̈
Subsection 1.5.3.
This Schurmann triple can be used to obtain a realization of the ̈
process on a symmetric Fock space. This realization can be found
as the unique solution of a quantum stochastic differential
equation. It establishes the one-to-one correspondence between
Levy processes, convolution semigroups of states, generating ́
functionals, and Schurmann triples. We will not present the ̈
representation theorem here, but refer to [Sch93, Chapter 2].
Finally, in Subsection 1.5.4, we look at several examples.
For more information on Levy processes on involutive ́
bialgebras, see also [Sch93], [Mey95, Chapter VII], [FS99], [Fra06b].
1.5.1 Definition of Levy processes on involutive bialgebras ́
In this Section and the following we will usually deal with
quantum probability spacesin the∗-algebraic sense, that is, pairs
(A,Φ) consisting of a unital∗-algebraAand a state (that is, a
normalized positive linear functional)ΦonA. Positivity in this
purely algebraic context simply meansΦ(a∗a)≥0 for alla∈ A. A
quantum random variable jover a quantum probability space(A,Φ)
on a∗-algebraBis a unit-preserving∗-algebra homomorphism
j : B → A. Aquantum stochastic processis an indexed family of