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.