Page 106 - Special Topic Session (STS) - Volume 3

STS518 Steen T.

A survey of recent progress in free infinite
divisibility
Steen Thorbjørnsen
Depart. of Math., University of Aarhus, Ny Munkegade 118, 8000 Aarhus C,
Denmark

Abstract
The aim of this paper is to provide a brief introduction to the theory of the
additive convolution associated to the notion of free independence and the
derived concept of free infinite divisibility. We shall emphasize many parallels
and some differences to the classical theory of infinitely divisible probability
measures. We aim further to provide an overview of some of the recent
developments in this field with an ample amount of references.

1. Introduction
With the work of Speicher [19], Ben Ghorbal and Schu¨rmann [5] and
Muraki [16] it was clarified that there are only five notions of “probabilistic
independence” which satisfy some naturally required conditions (associativity,
universality, extension and normalization). These five notions of independence
are: Classical (or tensor) independence, Free independence, Boolean
independence, Monotone independence and Anti-monotone independence.
Two classical random variables X and Y cannot satisfy any of the four last
notions of independence, unless either X or Y is a constant. In fact,
disregarding constant random variables, the four last notions of independence
will all entail that the product XY is distinct from Y X. Thus these last four
notions of independence are (in essence) only realizable in the framework of
non-commutative (or quantum) probability, where the “random variables” x
and y are realized as Hermitian (possibly unbounded) operators on an infinite
dimensional Hilbert space H, and the expectation functional can be realized as
a vector state corresponding to a (fixed) unit vector on ℋ:

[] = 〈, 〉,

where 〈. , . 〉 denotes the inner product on ℋ. In case x is a continuous (i.e.
bounded) Hermitian operator on ℋ , there exists a unique compactly
supported probability measure µ on ℝ, satisfying that

[()] = ∫ () ()

ℝ
for any polynomial : ℝ → ℝ and with () defined in the obvious way. This
measure µ is referred to as the (spectral) distribution of (with respect to the

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