Table Of Content' .Volume 1
C CRM
R MONOGRAPH
M SERIES
Centre de Recherches Mathématiques
UniverSité de Montreal . -
Free Random Variables
D. V.‘ Voiculescu
K. J. Dykema '
A. Nica ' '
American Mathematical Society
Volume 1
C RM
MONOGRAPH
SERIES
Centre de Recherches Mathématiques
Université de Montreal
Free Random Variables
A noncomrnutative probability approach
to free products with applications
to random matrices, operator algebras
and harmonic analysis on free groups
D. V. Voiculescu
K. J. Dykema
A. Nica
The Centre de Recherches Mathématiques (CRM) of
l'Université de Montreal was created in 1968 to promote
research in pure and applied mathematics and related
disciplines. Among its activities are special theme years,
summer schools, workshops, postdoctoral and publica-
tion programs. CRM is supported by l'Université de
Montreal, the Province of Quebec (FCAR), and the
Natural Sciences and Engineering Research Council of
Canada. It is afliliated with l'Institut des Sciences
Mathématiques de Montreal (ISM), whose constituent
members are Concordia University, McGill University,
l'Université de Montreal, 1'Université du Quebec 21
Montreal, and l'Ecole Polytechnique.
AmericanMathematical Society
Providence, RhodeIsland USA
The production of this volume was supported in part by the Chaire Andre Aisen-
stadt, the Fonds pour la Formation de Chercheurs et I’Aide a la Recherche (Fonds
FCAR),andtheNaturalSciencesandEngineering ResearchCouncilofCanada (NSERC).
The second author was supported in part by the John and Fannie Hertz Foundation.
1991 Mathematics Subject Classification. Primary 46L50; Secondary 46L10, 46L35, 47380,
60H25.
Library ofCongress Cataloging-in-Publication Data
Voiculescu, D. V. (Dan V.), 1949—
Free random variables: a noncommutative probabilityapproach to free products
with applicationsto random matrices, operatoralgebras, and harmonicanalysis on free
groups / D. V. Voiculescu, K. J. Dykema, A. Nica.
p. cm.
Includes bibliographical references.
ISBN 0-8218-6999-X (alk. paper)
1. Operatoralgebras. 2. Selfadjointoperators. 3. Free products (Grouptheory)
I. Dykema, K. J., 1967-. n. Nica, A., 1961— . m. Title.
QA326.V65 1992 92-37964
512’.55—dc20 cup
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Contents
Preface
Chapter 1. Free Products I—
1.1. Groups II—
1.2. Unital algebras r
I—
1.3. Hilbert spaces
Iw
1.4. Unital C'*—a1gebras o
o
1.5. Representations and states b
p
1.6. The commutant ofa free product lK
Chapter 2. Free Random Variables in Noncommutative
Probability Theory 11
2.1. Introduction 11
2.2. Noncommutative probability spaces 12
2.3. Random variables in noncommutative probability spaces 12
2.4. Independence 13
2.5. Freeness 14
2.6. Analogue ofthe Gaussian process over a Hilbert space 17
Chapter 3. Free Harmonic Analysis 21
3.1. Additive free convolution 21
3.2. The R—transform 22
3.3. Analytic function theory for the R—transform 23
3.4. Examples 26
3.5. The central limit theorem 28
3.6. Multiplicative free convolution 29
3.7. Infinite divisibility 32
3.8. Free products with amalgamation 37
Chapter 4. Random Matrices and Asymptotic Freeness 43
4.1. Complex Gaussian random matrices 44
4.2. Further results 50
4.3. Random unitary matrices 52
4.4. The non-Gaussian case 54
iii
iv CONTENTS
Chapter 5. Free Product Factors 55
5.1. Circular and semicircular systems 55
5.2. Direct applications 59
5.3. The fundamental group ofL(Foo) 62
5.4. Free products with injective factors 63
Postscript 67
References 69
Preface
I began putting free products into a noncommutative probabilistic framework,
some ten years ago, hoping this would lead to progress on the elusive I11 factors
of free groups. Recently, after discovering the connection,with random matrices,
this approach began to pay off. Moreover, the emerging “free probability theory”
has produced a unified way ofcomputing spectra of convolution operators on free
groups and asymptotic spectral densities of random matrices. A short book of
lecture notes should be useful as an introduction. The subject is still in progress,
so any attempt at completeness seemed out ofthe question. A postscript has been
added to serve as a guide to the original papers where recent advances were made.
The organizationofthe materialgrew out oflectures Igave at GPOTS in 1987,
College de France in 1989, U.C. Berkeley in 1990 and the Aisenstadt Lectures
in Montreal in 1991. In particular the notes taken by Ken Dykema at my U.C.
Berkeley topics course in 1990 were a precursor ofthis book.
It was fortunate to find two free probability enthusiasts in the persons of my
students Ken Dykema and Andu Nica, with whose collaboration this book was
written. The present exposition also covers some results from their papers. In
another direction, Ken’s offer to type the manuscript on the computer was a great
advantage in speeding things up.
Most ofthe work on the book was done while I occupied the Aisenstadt Chair
at the CRM in Montreal during the ’91 operator algebra programme and this also
provided the possibility for Ken and Andu to spend the spring semester at the
CRM. I thank David Handelrnan, who organized the programme, for inviting me
toMontrealandFrancisClarkeforthehospitalityat the CRM. Itwasalsoapleasure
to have the opportunity ofmeeting Dr. André Aisenstadt, whose sponsorship made
this project possible.
Do free group Ill—factors deserve this special effort? Note that the only other
Ill—factor known to have a similar involvement with probability theory is the hy-
perfinite Ill—factor, which is related to the fermionic context of the canonical an-
ticommutation relations. The hyperfinite Ill—factor is, by far, the “best” among
all Ill—factors. By analogy, this could mean the free group factors are the “best”
among the “bad” (i.e. non—1") Ill—factors.
June ’92 Dan Voiculescu
CHAPTER 1
Free Products
1.1. Groups
The free product ofgroups is the universal object defined in the following way.
DEFINITION 1.1.1. Let (GQLE; be a family of groups. The group free product
ofthis family, denoted *LEIGL, is the unique (up to isomorphism) group G together
with homomorphisms 1m: G’L —> G such that given any group H and homomor-
phisms qfibz 0,, —> H there exists a unique homomorphism <1) 2 he; Q: G —> H
making the diagram in Figure 1.1 commute. (The homomorphisms, 2/)“ are injec-
tive.).
FIGURE 1.1
We may construct the free product by taking the set of reduced words,
G={glgz---gnlg]-EGLj\{e}withL1#L2#L3#--'¢LH}U{Z}
with multiplication defined as juxtaposition followed by reducing. Thus
(gi---gn)(9’1---g$n) = reduced word 0f91---gn9'1-"91n.
1.2. Unital algebras
The free product for unital algebras is a universal object defined in a manner
similar to that for groups.
DEFINITION 1.2.1. If (ALheI is a family of unital algebras, then their unital
algebra free product *LejAL is the unique unital algebra A together with unital
homomorphisms 'l/hi AL —> A such that given any unital algebra B and unital
homomorphisms Q: A —> B there exists a unique unital homomorphism (I) =
*LeIq: A —> B making the diagram in Figure 1.2 commute.
As a vector space, the free product *Le]AL is the quotient of the vector space
which has as basis the set
B={a1a2...anln€N,ajEALj,L17éL27é...7éLn}
1
2 1. FREE PRODUCTS
FIGURE 1.2
by the subspace generated by the relations ofthe form
(11 . .-04-1 (Aug-0) +[Lag-1)) aj+1 - ''an
= /\a1---aj_1a§0)aj+1---an +pa1--'aj_1a§1)aj+1---an (A7” E (C)
0.1- =1 2 a1---an=a1---aj_1aj+1---an.
Moreover we can check Without much difficulty that if A = (Cl 63 VL (as vector
spaces) then *LEIAL is isomorphic as a vector space to
C=C1€B®( ® v..®---®v..).
n21 (mess-37a.)
If each of the AL has an involution, then so does A = he]AL in the obvi—
ous manner, and moreoVer, A is the universal object as in Definition 1.2.1 in the
category ofunital *—algebras and unital *—homomorphisms.
The following example relates free products ofgroups and algebras.
EXAMPLE 1.2.2. (GROUP ALGEBRAS). If G is a group, then its group algebra
C[G] is the *—algebra with basis G, multiplication given by group multiplication
and involution by g* = 9—1.
PROPOSITION 1.2.3. Let (GL)L61 be groups. Then (C[*L€IGL] : *LEI(C[GL].
PROOF. If G and H are groups and A and B are *—algebras let Hom(G,H)
be homomorphisms from G into H, Homalg(A,B) be the unital *—homomorphisms
from A into B and U(B) be the group ofunitary elements of B. Then
HHom(GL,H) : Hom( :1GHH),
LEI
HHomalg(AL,B) 2 Homa;g(LEIA,B)
LEI
and
Homalg(C[GL],B) : HOm(GL,U(B)).
Thus
Homalg(LEIC[Gb], B) 2 HHomalg(ClGLla B)
LEI
: HHom(GL,U(B)) 2 Hom< :10“U(B)) : Homalg(C[ 210.] ,B),
LEI
and the above equation is functorial in B. III
1.4. UNITAL C*—ALGEBRAS 3
1.3. Hilbert spaces
Let C be the category of Hilbert spaces with distinguished unit vector and
contractionswhichpreservethesevectors, i.e. objectsare (fl-C,E) withg 6 5H, “gH = 1
and morphisms are linear maps T: U-Cl —> 1H2 such that “T“ g 1 and T51 = 52.
Note that if T is a morphism then T(fI-C1 9 C51) Q 9C2 9 C52, (where 1H 9 {Hg is
the orthocomplement offl-Co in 1%). Indeed, simply show that T*§2 = £1.
DEFINITION 1.3.1. Let (9-6,,§,),61 be a family of Hilbert spaces with distin-
guished unit vectors. Then their Hilbert space free product *,61(9{,,§,) is (9&5),
givenby
.
sc=cgea®( EB 9{,,---®®0{L,,,)
"21 (b1¢t2¢"'#bn)
where at, is the orthocomplement of5, in II'C,.
It should be remarked that (ii-(,5) is not a universal object in the way that
group and algebra free products are, i.e. it is not the category theoretic direct sum.
EXAMPLE 1.3.2. Let l2(G,) have basis {69 |g E G’}. Then *,61(l2(G,),6eL) :
(12(*,61G,),6e), where e, is the identity element of G, and e the identity element
ofG.
EXAMPLE 1.3.3. (FULL FOCK SPACE). Let fl-C be a'Hilbert space. Itsfull Fock
space is
7m) = «:169 EB9cm
n>1
Then fl-C v—> (TCH), 1) is a functor from the category ofHilbert spaces and contrac—
tions to the category, (9. Indeed, for X: 9C1 —> 1H2 a contraction, the associated
morphism form (T(IH1),1) to (T(9-C2),1) is T(X) = 1 63 GB”), X®".
Moreover, we have (T($LE]9t), 1) = *,EI(T(J-C,),1). To see this, note that
®n
($9‘CL) = $ HL1®'.I®g-Cbn)
LEI L1,...,LHEI
sum over n and regroup.
1.4. Unital C*—algebras
The free product for unital C'*—algebras is the universal object defined in a
manner exactly analogous to that for unital algebras.
DEFINITION 1.4.1. If(A,),eI isafamilyofunitalC*—algebras, thentheir unital
C*—algebra free product *LEIA, is the unique unital C*—algebra A together with
unital *—homomorphisms 1b,: A, —>.A such that given any unital C*—algebra B and
unital*—homomorphisms (15,: A, —> Bthereexistsauniqueunital *—homomorphism
II> = *LeI(7,5,: A —> B making the diagram in Figure 1.3 commute.
A, A
FIGURE 1.3
