Combinatorics of free probability theory [Lecture notes]

129 Pages·2005·0.596 MB·English

by  Roland Speicher

Combinatorics of free probability
theory
Roland Speicher
Queen’s University, Kingston, Ontario
Abstract. The following manuscript was written for my lec-
turesatthespecialsemester’Freeprobabilitytheoryandoperator
spaces’, IHP, Paris, 1999. A complementary series of lectures was
given by A. Nica. It is planned to combine both manuscripts into
abookA. Nica and R. Speicher: Lectures on the Combinatorics of
Free Probability, which will be published by Cambridge University
Press
Contents
Part 1. Basic concepts 5
Chapter 1. Basic concepts of non-commutative probability theory 7
1.1. Non-commutative probability spaces and distributions 7
1.2. Haar unitaries and semicircular elements 10
Chapter 2. Free random variables 17
2.1. Definition and basic properties of freeness 17
2.2. The group algebra of the free product of groups 21
2.3. The full Fock space 23
2.4. Construction of free products 27
Part 2. Cumulants 35
Chapter 3. Free cumulants 37
3.1. Motivation: Free central limit theorem 37
3.2. Non-crossing partitions 46
3.3. Posets and M¨obius inversion 50
3.4. Free cumulants 52
Chapter 4. Fundamental properties of free cumulants 59
4.1. Cumulants with products as entries 59
4.2. Freeness and vanishing of mixed cumulants 66
Chapter 5. Sums and products of free variables 75
5.1. Additive free convolution 75
5.2. Description of the product of free variables 84
5.3. Compression by a free projection 87
5.4. Compression by a free family of matrix units 91
Chapter 6. R-diagonal elements 95
6.1. Definition and basic properties of R-diagonal elements 95
6.2. The anti-commutator of free variables 103
6.3. Powers of R-diagonal elements 105
Chapter 7. Free Fisher information 109
7.1. Definition and basic properties 109
3
4 CONTENTS
7.2. Minimization problems 112
Part 3. Appendix 123
Chapter 8. Infinitely divisible distributions 125
8.1. General free limit theorem 125
8.2. Freely infinitely divisible distributions 127
Part 1
Basic concepts

CHAPTER 1
Basic concepts of non-commutative probability
theory
1.1. Non-commutative probability spaces and distributions
Definition 1.1.1. 1) A non-commutative probability space
(A,ϕ) consists of a unital algebra A and a linear functional
ϕ : A → C; ϕ(1) = 1.
2) If in addition A is a ∗-algebra and ϕ fulfills ϕ(a∗) = ϕ(a) for all
a ∈ A, then we call (A,ϕ) a ∗-probability space.
3) If A is a unital C∗-algebra and ϕ a state, then we call (A,ϕ) a C∗-
probability space.
4) If A is a von Neumann algebra and ϕ a normal state, then we call
(A,ϕ) a W∗-probability space.
5) A non-commutative probability space (A,ϕ) is said to be tracial
if ϕ is a trace, i.e. it has the property that ϕ(ab) = ϕ(ba), for every
a,b ∈ A.
6) Elements a ∈ A are called non-commutative random variables
in (A,ϕ).
7) Let a ,...,a be random variables in some probability space (A,ϕ).
1 n
If we denote by ChX ,...,X i the algebra generated freely by n non-
1 n
commuting indeterminates X ,...,X , then the joint distribution
1 n
µ of a ,...,a is given by the linear functional
a1,...,an 1 n
(1) µ : ChX ,...,X i → C
a1,...,an 1 n
which is determined by
µ (X ...X ) := ϕ(a ...a )
a1,...,an i(1) i(k) i(1) i(k)
for all k ∈ N and all 1 ≤ i(1),...,i(k) ≤ n.
8) Let a ,...,a be a family of random variables in some ∗-probability
1 n
space (A,ϕ). If we denote by ChX ,X∗,...,X ,X∗i the algebra gener-
1 1 n n
ated freely by 2n non-commuting indeterminates X ,X∗,...,X ,X∗,
1 1 n n
then the joint ∗-distribution µ of a ,...,a is given by the
a1,a∗1,...,an,a∗n 1 n
linear functional
(2) µ : ChX ,X∗,...,X ,X∗i → C
a1,a∗1,...,an,a∗n 1 1 n n
7
8 1. BASIC CONCEPTS
which is determined by
µ (Xr(1)...Xr(k)) := ϕ(ar(1)...ar(k))
a1,a∗1,...,an,a∗n i(1) i(k) i(1) i(k)
for all k ∈ N, all 1 ≤ i(1),...,i(k) ≤ n and all choices of
r(1),...,r(k) ∈ {1,∗}.
Examples 1.1.2. 1) Classical (=commutative) probability spaces
fit into this frame as follows: Let (Ω,Q,P) be a probability space in
the classical sense, i.e., Ω a set, Q a σ-field of measurable subsets
of Ω and P a probability measure, then we take as A some suitable
algebra of complex-valued functions on Ω – like A = L∞(Ω,P) or
A = L∞−(Ω,P) := T Lp(Ω,P) – and
p≤∞
Z
ϕ(X) = X(ω)dP(ω)
for random variables X : Ω → C.
R
Note that the fact P(Ω) = 1 corresponds to ϕ(1) = 1dP(ω) =
P(Ω) = 1.
2) Typical non-commutative random variables are given as operators
on Hilbert spaces: Let H be a Hilbert space and A = B(H) (or more
generallyaC∗-orvonNeumannalgebra). Thenthefundamentalstates
areoftheformϕ(a) = hη,aηi,whereη ∈ Hisavectorofnorm1. (Note
that kηk = 1 corresponds to ϕ(1) = hη,1ηi = 1.)
All states on a C∗-algebra can be written in this form, namely one has
the following GNS-construction: Let A be a C∗-algebra and ϕ : A → C
a state. Then there exists a representation π : A → B(H) on some
Hilbert space H and a unit vector η ∈ H such that: ϕ(a) = hη,π(a)ηi.
Remarks 1.1.3. 1)Ingeneral, thedistributionofnon-commutative
random variables is just the collection of all possible joint moments
of these variables. In the case of one normal variable, however, this
collection of moments can be identified with a probability measure on
C according to the following theorem: Let (A,ϕ) be a C∗-probability
space and a ∈ A a normal random variable, i.e., aa∗ = a∗a. Then there
exists a uniquely determined probability measure µ on the spectrum
σ(a) ⊂ C, such that we have:
Z
ϕ(p(a,a∗)) = p(z,z¯)dµ(z)
σ(a)
for all polynomials p in two commuting variables, i.e. in particular
Z
ϕ(ana∗m) = znz¯mdµ(z)
σ(a)
1.1. NON-COMMUTATIVE PROBABILITY SPACES AND DISTRIBUTIONS 9
for all n,m ∈ N.
2) We will usually identify for normal operators a the distribution µ
a,a∗
with the probability measure µ.
In particular, for a self-adjoint and bounded operator x, x = x∗ ∈
B(H), its distribution µ is a probability measure on R with compact
x
support.
3) Note that above correspondence between moments and probability
measures relies (via Stone-Weierstrass) on the boundedness of the op-
erator, i.e., the compactness of the spectrum. One can also assign a
probability measure to unbounded operators, but this is not necessarily
uniquely determined by the moments.
4) The distribution of a random variable contains also metric and al-
gebraic information, if the considered state is faithful.
Definition 1.1.4. A state ϕ on a ∗-algebra is called faithful if
ϕ(aa∗) = 0 implies a = 0.
Proposition 1.1.5. Let A be a C∗-algebra and ϕ : A → C a
faithful state. Consider a self-adjoint x = x∗ ∈ A. Then we have
(3) σ(x) = suppµ ,
x
and thus also
(4) kxk = max{|z| | z ∈ suppµ }
x
and
(5) kxk = lim(cid:0)ϕ(xp)(cid:1)1/p.
p→∞
Proposition 1.1.6. Let (A ,ϕ ) and (A ,ϕ ) be ∗-probability
1 1 2 2
spaces such that ϕ are ϕ are faithful. Furthermore, let a ,...,a in
1 2 1 n
(A ,ϕ ) and b ,...,b in (A ,ϕ ) be random variables with the same
1 1 1 n 2 2
∗-distribution, i.e., µ = µ .
a1,a∗1,...,an,a∗n b1,b∗1,...,bn,b∗n
1) If A is generated as a ∗-algebra by a ,...,a and A is generated
1 1 n 2
as a ∗-algebra by b ,...,b , then 1 7→ 1, a 7→ b (i = 1,...,n) can be
1 n i i
∼
extended to a ∗-algebra isomorphism A = A .
1 2
2) If (A ,ϕ ) and (A ,ϕ ) are C∗-probability spaces such that A is as
1 1 2 2 1
C∗-algebra generated by a ,...,a and A is as C∗-algebra generated
1 n 2
∼
by b ,...,b , then we have A = A via a 7→ b .
1 n 1 2 i i
3) If (A ,ϕ ) and (A ,ϕ ) are W∗-probability spaces such that A is
1 1 2 2 1
as von Neumann algebra generated by a ,...,a and A is as von
1 n 2
∼
Neumann algebra generated by b ,...,b , then we have A = A via
1 n 1 2
a 7→ b .
i i
10 1. BASIC CONCEPTS
1.2. Haar unitaries and semicircular elements
Example 1.2.1. Let H be a Hilbert space with dim(H) = ∞ and
orthonormal basis (e )∞ . We define the two-sided shift u by (k ∈
i i=−∞
Z)
(6) ue = e and thus u∗e = e .
k k+1 k k−1
We have uu∗ = u∗u = 1, i.e., u is unitary. Consider now the state
ϕ(a) = he ,ae i. The distribution µ := µ is determined by (n ≥ 0)
0 0 u,u∗
ϕ(un) = he ,une i = he ,e i = δ
0 0 0 n n0
and
ϕ(u∗n) = he ,u∗ne i = he ,e i = δ .
0 0 0 −n n0
Since u is normal, we can identify µ with a probability measure on
σ(u) = T := {z ∈ C | |z| = 1}. This measure is the normalized Haar
measure on T, i.e., dµ(z) = 1 dz. It is uniquely determined by
2π
Z
(7) zkdµ(z) = δ (k ∈ Z).
k0
T
Definition 1.2.2. A unitary element u in a ∗-probability space
(A,ϕ) is called Haar unitary, if
(8) ϕ(uk) = δ for all k ∈ Z.
k0
Exercise 1.2.3. Let u be a Haar unitary in a ∗-probability space
(A,ϕ). Verify that the distribution of u+u∗ is the arcsine law on the
interval [−2,2], i.e.
(
√1 dt, −2 ≤ t ≤ 2
(9) dµ (t) = π 4−t2
u+u∗
0 otherwise.
Example 1.2.4. Let H be a Hilbert space with dim(H) = ∞ and
basis (e )∞ . The one-sided shift l is defined by
i i=0
(10) le = e (n ≥ 0).
n n+1
Its adjoint l∗ is thus given by
(11) l∗e = e (n ≥ 1)
n n−1
(12) l∗e = 0.
0
We have the relations
(13) l∗l = 1 and ll∗ = 1−|e ihe |,
0 0