Noncommutative Mathematics for Quantum Systems

Independence and Levy Processes in Quantum Probability ́ 103

includes also the free convolution. This shows that Levy processes ́
on dual semigroups are in one-to-one correspondence with
generating functionals, that is, hermitian, conditionally positive
linear functionals that vanish on the identity.

Theorem 1.9.5 Schoenberg correspondence: Let {φt}t≥ 0 be a
convolution semigroup of unital functionals with respect to the tensor,
boolean, monotone, or anti-monotone convolution on a dual semigroup
(B,∆,ε)and letψ:B →Cbe defined by

ψ(b) =lim

t↘ 0

1

t

(

φt(b)−ε(b)

)

for b∈B. Then the following statements are equivalent.

(i) φtis positive for all t≥ 0.

(ii) ψis hermitian and conditionally positive.

Conversely,φtcan be recovered from the generating functionalψ
as a kind of convolution exponential (defined by extendingψto the
symmetric tensor algebra overB). For details see [Sch95b, Theorem
3.3].
It is well known that freeness arises in the study of the behavior
of random matrices for large dimension, see, e.g., [Voi91, Bia97]
and the references therein. Ulrich has shown recently that free
Levy processes can also be obtained in this way. He has shown ́
that the limit in distribution of the d×d blocks of Brownian
motion in the unitary groupU(nd)asntends to infinity is a free
Levy processes on the dual group analog of the unitary group, ́ cf.
[Ulr14].

1.9.3 Time reversal

Denote byFtheflipmapF:B‰B →B‰B,F=mB‰B◦(i 2 ‰i 1 ),
wherei 1 ,i 2 :B → B‰Bare the inclusions ofBinto the first and
the second factors of the free productB‰B. The flip mapFacts on
i 1 (a 1 )i 2 (b 1 )···i 2 (bn)∈B‰Bwitha 1 ,... ,an,b 1 ,... ,bn∈Bas

F

(

i 1 (a 1 )i 2 (b 1 )···i 2 (bn)

)

=i 2 (a 1 )i 1 (b 1 )···i 1 (bn).

If j 1 : B → A 1 and j 2 : B → A 2 are two unital∗-algebra
homomorphisms, then we have(j 2 ‰j 1 )◦F= γA 1 ,A 2 ◦(j 1 ‰j 2 ).