Summary of Mathematics Doctoral Thesis: Some limit theorems in noncommutative probability

Objective of the research: The research objective of the thesis is to establish some limit theorems regarding the forms of law of large numbers for sequences and arrays of measurable operators under different conditions.

Supervisor: 1 Prof Dr Nguyen Van Quang

2 Dr Le Hong Son

Reviewer 1:

Reviewer 2:

Reviewer 3:

Thesis will be defended at university-level thesis

evaluating council at Vinh University

at , ,

Thesis can be found at:

- Vietnam National Library

- Nguyen Thuc Hao Library and Information Center - Vinh University

1 Rationale

1.1 The limit theorems have been interested in many mathematicians and have many tions in statistics, economics, medicine, and some other empirical sciences The limit theoremsregarding the forms of law of large numbers are studied for many different objects For exam-ple, the law of large numbers for single-valued random variables, set-valued random variables,fuzzy set-valued random variables; the law of large numbers in game theory, in noncommutativeprobability In particular, the limit theorems in noncommutative probability are attracting theattention of many authors and have yielded certain results

applica-1.2 The noncommutative integral theory was started studied in 1952-1953 by Segal Later,

it was continued studied by Kunze (1958), Stinespring (1959), Nelson (1974), Yeadon (1979),etc On the basis of noncommutative integral theory, the theory of noncommutative probabilityhas been studied by Batty (1979), Padmanabhan (1979), Luczak (1985), Jajte (1985) and iscontinuing to be of interest In noncommutative probability, there is no basic probability space,instead of studying random variables, we study operators on von Neumann algebra or measuredoperators Because multiplication of operators is not commutative and we cannot talk aboutthe max, min of operators, to study the problems of noncommutative probability theory, weare needed new tools, and techniques

1.3 The law of large numbers in noncommutative probability is studied in two main directions:bounded operators on von Neumann algebra with state and measured operators with tracialstate The difficulty in the first direction is the limited nature of the state, while in the second,the unbounded property of the measurable operators give rise to many complex problems.These characteristics contribute to the diversity of issues to be considered, studied of the limittheorems in noncommutative probability

1.4 As a result of many problems arising from quantum physics theory, problems of boundedoperators on von Neumann algebra or measurable operators have been extensively studied fromthe seventies of the last century and continue to be studied up to now Therefore, the study

of the limit theorems in noncommutative probability is not only theoretical sense but alsopractical sense

With the above reasons, we have chosen the topic for the thesis as follows:

“Some limit theorems in noncommutative probability”.

2 Objective of the research

The research objective of the thesis is to establish some limit theorems regarding theforms of law of large numbers for sequences and arrays of measurable operators under differentconditions

3 Subject of the research

The research subject of the thesis is the measurable operators and the law of large numbersfor the measurable operators with the tracial state in noncommutative probability

4 Scope of the research

The thesis focuses on studying the law of large numbers for measurable operators in ferent types of convergence such as: convergence bilaterally almost uniformly, convergence in

dif-LP, convergence in measure; the study expanded the integrable concepts into noncommutativeprobability space

5 Methodology of the research

We use a combination of the fundamental methods of probability theory in proving the law

of large numbers and the techniques of operator theory such as truncation method, subsequencesmethod, spectral representation technique of the operators

6 Contributions of the thesis

The results of the thesis contribute to enriching the research direction of the limit theorems

in noncommutative probability theory

The thesis can be used as a reference for students, masters students and PhD studentsbelong to the specialty: Theory of probability and mathematical Statistics

7 Overview and organization of the research

7.1 Overview of the research

In this thesis, we study limit theorems regarding the forms of law of large numbers forsequences and arrays of measurable operators

For strong law of large numbers, we first establish some strong law for sequences of itive measurable operators Using these results, we prove some strong laws of large numbersfor sequences of pairwise independent measurable operators that are identically distributed ornon-identically distributed Next, we prove the equivalence conditions of uniform integrability

pos-for a sequence of measurable operators Based on these results, we introduce some integrableconcepts for sequences of measurable operators in noncommutative probability Finally, thestrong law of large numbers for sequences of pairwise independent measurable operators andstrongly Ces`aro α-integrable is studied by us.

For weak law of large numbers, we first study the convergence in L1 for sequences of surable operators, residually Ces`aro α-integrable and pairwise independent or m-dependent

mea-We then present the concepts: uniformly integrable in the Ces`aro sense, h-integrable with spect to the array of constant {ani} and h-integrable with exponent r of the array of measurableoperators Finally, we establish some mean convergence theorems and weak law large numbersfor arrays of measurable operators from the above concepts

re-7.2 The organization of the research

Besides the sections of usual notations, preface, general conclusions, and recommendations,list of the author’s articles related to the thesis and references, the thesis is organized into threechapters

Chapter 1 presents some preliminaries

Chapter 2 studies some limit theorems regarding the forms of strong law of large numbersfor sequences of measurable operators

Chapter 3 studies some limit theorems regarding the forms of weak law of large numbersfor sequences and arrays of measurable operators

CHAPTER 1 PRELIMINARIES

In this chapter, we introduce some basic knowledge of noncommutative probability theory

1.1 The operators on a Hilbert space

Definition 1.1.1 Let D be a subspace of H, the linear transformation T : D → H is calledpartially defined operator on H

A densely defined operator on H with domain D(T ) is a partially defined operator on Hwhose domain D(T ) is dense in H

A partially defined operator (or densely defined operator) may be bounded or unbounded

A partially densely defined operator on H is closed if its graph is closed in H × H

Theorem 1.1.2 If T is a self-adjoint, densely defined operator on H then there exist uniqueresolution E of the identity, on the Borel subsets of the real line, such that

Moreover, E is concentrated on σ(T ) ⊂ (−∞, +∞), in the sense that E σ(T )
= 1

Formula (1.1) is called spectral representation of operator T and usually written as:

1.2 Von Neumann algebra

Definition 1.2.1 A subalgebra A of L(H) is called von Neumann algebra if:

i) A is self-adjoint, i.e, if T ∈ A then T∗ ∈ A ;

ii) A contains the identity operator 1;

iii) A is weakly closed, i.e, if {Ti} ⊂ A is a net such that Ti → T in weak operator topology,then T ∈ A

Definition 1.2.2 Let A ⊂ L(H) be a von Neumann algebra and τ : A → C be linear tional Denote A+= {X ∈ A : X ≥ 0} Then

func-i) τ is called positive if τ (X) ≥ 0, ∀X ∈ A+

ii) τ is called faithful if τ (X) = 0 implies X = 0 for all X ∈ A+

iii) τ is called state if τ positive and τ (1) = 1

iv) State τ is called normal if, for all the net

{Xi} ⊂ A+, Xi↑ X (in strong operator topology), then τ (Xi) ↑ τ (X)

v) State τ is called tracial state if

decom-Denote A for the set of operator which affiliated to the von Neumann algebra A Ane

element of A is called a measurable operator.e

Definition 1.3.2 Let A ⊂ L(H) be a von Neumann algebra and τ be a faithful normal tracialstate on A.

Denote LP(A, τ ) (P ≥ 1) is a Banach space of all elements in A satisfyinge

||X||P = [τ (|X|P)]P1 < ∞

For notational consistency, A will be denoted by Le 0(A, τ ) Then we have natural inclusions:

A ≡ L∞(A, τ ) ⊂ LQ(A, τ ) ⊂ LP(A, τ ) ⊂ ⊂ L0(A, τ ) =A,e

for 1 ≤ P ≤ Q < ∞

1.4 Some kinds of convergence and independence

In this section, let A ⊂ L(H) be a von Neumann algebra, τ be a faithful normal tracialstate on A, and L0(A, τ ) be an algebra of measurable operator

Definition 1.4.1 The sequence {Xn, n ≥ 1}) ⊂ L0(A, τ ) is called converges in measure to

X ∈ L0(A, τ ), denoted by Xn −→ X, if any ε > 0, ττ 

Definition 1.4.3 The sequence {Xn, n ≥ 1} ⊂ L0(A, τ ) is called converges almost uniformly

to X ∈ L0(A, τ ), denoted by Xn −−→ X, if for each ε > 0, there exists a projection p ∈ A sucha.u.that

The conclusions of Chapter 1

In this chapter, we obtain some main results:

- A brief some of the basic concepts and properties of operators on a Hilbert space;

- Prove some properties of measurable operators;

- System some kinds of convergence in noncommutative probability and the relationshipbetween them;

- Present some independent concepts of sequences and arrays of measurable operators

Theorem 2.1.2 Let {Xn, n ≥ 1} ⊂ L0(A, τ ) be a sequence of positive operators such that there

is a sequence {Bn} of Borel subsets of R satisfying the following conditions:

in the special case, when f (n) = n, p = 2, an analogous result for successively independentsequence with the almost uniform convergence was given by Klimczak (2012).

Theorem 2.1.3 Let {Xn, n ≥ 1} ⊂ LP(A, τ ) (P ≥ 1) be a sequence of positive operators If(i) sup

1 In case f (n) = n, the following theorem is an extension of Theorem 1 from Cs¨org˝o et al.(1983) to noncommutative probability

Theorem 2.2.1 Let {Xn, n ≥ 1} ⊂ L0(A, τ ) be a sequence of pairwise independent, adjoint operators such that:

The next theorem is an extension of the main result of Klimczak (2012).

Theorem 2.2.2 Let {Xn, n ≥ 1} ⊂ L0(A, τ ) be a sequence of pairwise independent, adjoint operators Put G(x) = sup

self-n≥1

τ e[x,∞)(|Xn|)
, for x ≥ 0 If R0∞G(x)dx < ∞, then

1n

n

X

k=1

ck Xk− τ (Xk)
−−−→ 0 as n → ∞,b.a.u

for each bounded sequence {cn}

In the special case, when {Xn, n ≥ 1} ⊂ L1(A, τ ) is a sequence of pairwise independent,identically distributed, self-adjoint operators, we have

Corollary 2.2.3 [Theorem 2.1, Klimczak (2012)] Let {Xn, n ≥ 1} ⊂ L1(A, τ ) be a sequence

of pairwise independent, identically distributed, self-adjoint operators Then

1n

Theorem 2.2.4 Let gn : (0, ∞) → (0, ∞) be a sequence of nondecreasing functions satisfyingx

bounded If {Xn, n ≥ 1} ⊂ L0(A, τ ) is a sequence

of pairwise independent, self-adjoint operators such that

Sn− τ (Sn)

f (n)

b.a.u

−−−→ 0 as n → ∞

2.3 Some kinds of uniform integrability and laws of large numbers for sequences of measurable operators

In this section, we will first prove some equivalent properties of uniform integrability ditions for sequences of measured operators We will then construct some integrable concepts

con-in noncommutative probability and the relationship between them Fcon-inally, we will presentsome strong law of large numbers for sequences of pairwise independence measurable opera-tors that satisfy strongly Ces`aro α-integrable condition or strongly Ces`aro uniformly integrablecondition

Definition 2.3.1 A bounded subset K of L1(A, τ ) is said to be Uniformly Integrable (UI, inshort) if

We now prove the equivalent properties of the uniform integrability condition It is teresting that if X is a self-adjoint measurable operator then X can be permutable withany spectral projection eB(X) of itself, where B is a Borel subset of R, which implies that

in-eB(X)XeB(X) = XeB(X) This fact is helpful to establish the following theorem that is thefirst main result of this section

Theorem 2.3.2 Let {Xn, n ≥ 1} be a sequence of measurable operators The following areequivalent:

(i) {Xn, n ≥ 1} is uniformly integrable

(ii) there exists a convex function φ ∈ Φ such that

Proposition 2.3.4 (noncommutative criterion for Ces`aro uniform integrability)

Let {Xn, n ≥ 1} be a sequence of measurable operators The following properties areequivalent:

n

X

k=1

* This property is not used for (ii) ⇒ (i) It follows from Proposition 2.3.4 that if {Xn, n ≥ 1}

is SCUI then it is CUI

Definition 2.3.5 Let α ∈ (0, ∞) A sequence {Xn, n ≥ 1} of measurable operators is called(i) Ces`aro α-Integrable (CI(α), in short) if the following two conditions hold:

We now prove that for all α > 0, CUI =⇒ CI(α) and SCUI =⇒ SCI(α)

Lemma 2.3.6 Suppose that {Xn, n ≥ 1} is a sequence of measurable operators and let α be apositive real

(i) If {Xn, n ≥ 1} is CUI, then it is CI(α).

(ii) If {Xn, n ≥ 1} is SCUI, then it is SCI(α)

Definition 2.3.7 Let α ∈ (0, ∞) A sequence {Xn, n ≥ 1} of measurable operators is called(i) Residually Ces`aro α-Integrable (RCI(α), in short) if the following two conditions hold:sup

Theorem 2.3.8 Let {Xn, n ≥ 1} ⊂ L0(A, τ ) be a sequence of pairwise independent, adjoint operators If the sequence {Xn, n ≥ 1} satisfies the condition SCI(α) for some α ∈(0,1

self-2), then

Sn− τ (Sn)n

b.a.u

−−−→ 0 as n → ∞

The conclusions of Chapter 2

In this chapter, we obtain some main results:

- Establish some strong law of large nubers for sequences of positive measurable operators;

- Prove some strong laws of large numbers for pairwise independent non-identically tributed measurable operators and for pairwise independent identically distributed measurableoperators;

dis Establish some equivalent conditions of uniformly integrability for a sequences of meadis surable operators;

mea Construct some integrable concepts in noncommutative probability and the relationshipbetween them;

- Prove some strong laws for sequences of pairwise independent measurable operators andsatisfies condition strongly Ces`aro α-integrable

CHAPTER 3

SOME LIMIT THEOREMS REGARDING THE FORMS OF WEEK LAWS OF LARGE NUMBERS FOR SEQUENCES AND

ARRAYS OF MEASURABLE OPERATORS.

In this chapter, we study some limit theorems regarding the forms of week laws of largenumbers for sequences and arrays of measurable operators

3.1 The weak laws of large numbers for sequences of measurable erators

op-In this section, we will establish some convergence theorems in L1 We know that theconvergence in L1 implies the convergence in measure Therefore, from the theorems in thissection, we can obtain the results regarding the forms of the weak law of large numbers forsequences of measurable operators

Theorem 3.1.1 Let {Xn, n ≥ 1} ⊂ L0(A, τ ) be a sequence of pairwise independent, adjoint operators If the sequence {Xn, n ≥ 1} satisfies the condition RCI(α) for some α ∈(0, 1), then

self-Sn− τ (Sn)

n → 0 in L1 as n → ∞

By applying Theorem 3.1.1 we obtain the following corollary immediately:

Corollary 3.1.2 [Theorem 4.1(b), Lindsay and Pata (1997)] Let {Xn, n ≥ 1} ⊂ L0(A, τ )

be a sequence of pairwise independent, self-adjoint operators For each n ≥ 1, put Sen =

Sn− τ (Sen)

n → 0 in L1 as n → ∞

As in clasiccal situation, we say that a sequence {Xn, n ≥ 1} of measurable operators arepairwise m-dependent, if Xi and Xj are independent whenever |i − j| > m.

Theorem 3.1.3 Let {Xn, n ≥ 1} ⊂ L0(A, τ ) be a sequence of pairwise m-dependent, adjoint operators If the sequence {Xn, n ≥ 1} satisfies the condition RCI(α) for some α ∈(0, 1), then

We now provide some lemmas which will be helpful in obtaining main results

The proof of the following Lemma 3.2.1 is the same as that of Lemma 2.1 in Wu and Guan(2011) and is omitted

Lemma 3.2.1 Let {Xni, un ≤ i ≤ vn, n ≥ 1} be an array of measurable operators and

r > 0 Let moreover {h(n), n ≥ 1} be an increasing sequence of positive constants withh(n) ↑ ∞ as n → ∞ Let {ani, un ≤ i ≤ vn, n ≥ 1} be an array of constans satisfyingh(n) supun≤i≤vn|ani|r → 0 as n → ∞ Suppose the following conditions hold:

Taking ani = k−1/rn for un ≤ i ≤ vn and n ≥ 1 in Lemma 3.2.1, we can get the followinglemma