Noncommutative Mathematics for Quantum Systems

Independence and L ́evy Processes in Quantum Probability 35

(b) Let(φt)t≥ 0 be a convolution semigroup on some coalgebraC. Then
the limit

L(b) =lim

t↘ 0

1

t

(

φt(b)−ε(b)

)

exists for all b∈ C. Furthermore, we haveφt =exp?tL for all

t≥ 0.

The proof of this lemma relies on the fundamental theorem of
coalgebras [DNR01, Theorem 1.4.7], which states that for any
element c of a coalgebra C there exists a finite-dimensional
subcoalgebraCc ofCthat containsc. This allows to reduce the
study of convolution semigroups of functionals on coalgebras to
the study of semigroups in finite-dimensional algebras, see
[ASW88, Sch93].

Proposition 1.5.8 (Schoenberg correspondence) Let B be an
involutive bialgebra, (φt)t≥ 0 a convolution semigroup of linear
functionals onBand

L=lim

t↘ 0

1

t

(

φt−ε

)

.

Then the following are equivalent.

(i) (φt)t≥ 0 is a convolution semigroup of states.

(ii) L : B → C satisfies L( 1 ) = 0 , and it is hermitian and

conditionally positive, i.e.,

L(b∗) =L(b)

for all b∈B, and

L(b∗b)≥ 0

for all b∈Bwithε(b) = 0.

Proof We prove only the (easy) direction (i)⇒(ii), the converse
follows from the representation theorem, which can be found in
[Sch93, Chapter 2].
The first property follows by differentiatingφt( 1 ) =1 w.r.t.t.