Ng, Ka Shing. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 45-47). / Abstracts in English and Chinese. / Introduction --- p.v / Chapter 1 --- Preliminaries --- p.1 / Chapter 1.1 --- Noncommutative probability spaces and Free in- dependence --- p.1 / Chapter 1.2 --- C*-probability spaces --- p.4 / Chapter 1.3 --- Fock spaces --- p.5 / Chapter 1.4 --- Cauchy transform and R-transform of probability measures with bounded support --- p.7 / Chapter 1.5 --- Helton-Howe formula --- p.8 / Chapter 1.6 --- Stieltjes inversion formula --- p.9 / Chapter 1.7 --- Pick functions --- p.11 / Chapter 2 --- Free convolution and R-transform --- p.13 / Chapter 2.1 --- Additive free convolution and R-transform --- p.13 / Chapter 2.2 --- R-transform and algebraic Cauchy transform --- p.18 / Chapter 2.3 --- Properties of R-transform --- p.26 / Chapter 2.4 --- Properties of --- p.29 / Chapter 3 --- Examples of free convolution --- p.32 / Chapter 3.1 --- Measures with compact support --- p.32 / Chapter 3.2 --- Examples of free convolution --- p.33 / Chapter 4 --- Free Central Limit Theorem --- p.42 / Bibliography --- p.45

This dissertation consists of three more or less independent projects. In the first project, we find the microstates free entropy dimension of a large class of L1[0; 1]{ circular operators, in the presence of a generator of the diagonal subalgebra. In the second one, for each sequence {cn}n in l1(N), we de fine an operator A in the hyper finite II1-factor R. We prove that these operators are quasinilpotent and they generate the whole hyper finite II1-factor. We show that they have non-trivial, closed, invariant subspaces affiliated to the von Neumann algebra, and we provide enough evidence to suggest that these operators are interesting for the hyperinvariant subspace problem. We also present some of their properties. In particular, we show that the real and imaginary part of A are equally distributed, and we find a combinatorial formula as well as an analytical way to compute their moments. We present a combinatorial way of computing the moments of A*A. Finally, let fTkg1k =1 be a family of *-free identically distributed operators in a finite von Neumann algebra. In this paper, we prove a multiplicative version of the Free Central Limit Theorem. More precisely, let Bn = T*1T*2...T*nTn...T2T1 then Bn is a positive operator and B1=2n n converges in distribution to an operator A. We completely determine the probability distribution v of A from the distribution u of jTj2. This gives us a natural map G : M M with u G(u) = v. We study how this map behaves with respect to additive and multiplicative free convolution. As an interesting consequence of our results, we illustrate the relation between the probability distribution v and the distribution of the Lyapunov exponents for the sequence fTkg1k=1 introduced by Vladismir Kargin.

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No description available.

We introduce real second-order freeness in second-order noncommutative probability spaces. We demonstrate that under this definition, independent ensembles of the three real models of random matrices which we consider, namely real Ginibre matrices, Gaussian orthogonal matrices, and real Wishart matrices, are asymptotically second-order free. These ensembles do not satisfy the complex definition of second-order freeness satisfied by their complex analogues. This definition may be used to calculate the asymptotic fluctuations of products of matrices in terms of the fluctuations of each ensemble. We use a combinatorial approach to the matrix calculations similar to genus expansion, but in which nonorientable surfaces appear, demonstrating the commonality between the real ensembles and the distinction from their complex analogues, motivating this distinct definition. We generalize the description of graphs on surfaces in terms of the symmetric group to the nonorientable case. In the real case we find, in addition to the terms appearing in the complex case corresponding to annular spoke diagrams, an extra set of terms corresponding to annular spoke diagrams in which the two circles of the annulus are oppositely oriented, and in which the matrix transpose appears. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-09-09 11:07:37.414

In this thesis, we study the operator algebraic structure of various classes of unimodular free quantum groups, including thefree orthogonal quantum groups $O_n^+$, free unitary quantum groups $U_n^+$, and trace-preserving quantum automorphism groups associated to finite dimensional C$^\ast$-algebras. The first objective of this thesis to establish certain approximation properties for the reduced operator algebras associated to the quantum groups $\G = O_n^+$ and $U_n^+$, ($n \ge 2$). Here we prove that the reduced von Neumann algebras $L^\infty(\G)$ have the Haagerup approximation property, the reduced C$^\ast$-algebras $C_r(\G)$ have Grothendieck's metric approximation property, and that the quantum convolution algebras $L^1(\G)$ admit multiplier-bounded approximate identities. We then go on to study trace-preserving quantum automorphism groups $\G$ of finite dimensional C$^\ast$-algebras $(B, \psi)$, where $\psi$ is the canonical trace on $B$ induced by the regular representation of $B$. Here, we extend several known results for free orthogonal and free unitary quantum groups to the setting of quantum automorphism groups. We prove that the discrete dual quantum groups $\hG$ have the property of rapid decay, the von Neumann algebras $L^\infty(\G)$ have the Haagerup approximation property, and that $L^\infty(\G)$ is (in most cases) a full type II$_1$-factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness of trace for the reduced C$^\ast$-algebras $C_r(\G)$, and the existence of multiplier-bounded approximate identities for the convolution algebras $L^1(\G)$. We also show that when $B$ is a full matrix algebra, $L^\infty(\G)$ is an index $2$ subfactor of $L^\infty(O_n^+)$, and thus solid and prime. Finally, we investigate strong Haagerup inequalities in the context of quantum symmetries arising from actions of free quantum groups on non-commutative random variables. We prove a generalization of the strong Haagerup inequality for $\ast$-free R-diagonal families due to Kemp and Speicher, and apply this result to study strong Haagerup inequalites for the free unitary quantum groups. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-07-31 12:45:57.767

Thesis (M.S. in Defense Analysis)--Naval Postgraduate School, Dec. 2004. / Thesis Advisor(s): Gordon McCormick, Saverio Manago. Includes bibliographical references (p. 53). Also available online.

Semigroups of completely positive maps arise naturally both in noncommutative stochastic processes and in the dynamics of open quantum systems. Since its inception in the 1970's, the study of completely positive semigroups has included among its central topics the dilation of a completely positive semigroup to an endomorphism semigroup. In quantum dynamics, this amounts to embedding a given open system inside some closed system, while in noncommutative probability, it corresponds to the construction of a Markov process from its transition probabilities. In addition to the existence of dilations, one is interested in what properties of the original semigroup (unitality, various kinds of continuity) are preserved. Several authors have proved the existence of dilations, but in general, the dilation achieved has been non-unital; that is, the unit of the original algebra is embedded as a proper projection in the dilation algebra. A unique approach due to Jean-Luc Sauvageot overcomes this problem, but leaves unclear the continuity of the dilation semigroup. The major purpose of this thesis, therefore, is to further develop Sauvageot's theory in order to prove the existence of continuous unital dilations. This existence is proved in Theorem 6.4.9, the central result of the thesis. The dilation depends on a modification of free probability theory, and in particular on a combinatorial property akin to free independence. This property is implicit in some Sauvageot's original calculations, but a secondary goal of this thesis is to present it as its own object of study, which we do in chapter 3.

フランス国リヨン高等師範学校との共同学位プログラムによる学位 / 京都大学 / 新制・課程博士 / 博士(理学) / 甲第23449号 / 理博第4743号 / 新制||理||1680(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 COLLINS Benoit Vincent Pierre, 教授 泉 正己, 教授 日野 正訓 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This talk is a survey specifically oriented towards the needs and interests of a computationally inclined audience. We include the important mathematics (free probability) that permit the characterization of a large class of random matrices. We discuss how computational software is transforming this theory into practice by highlighting its use in the context of a stochastic eigen-inference application. / Singapore-MIT Alliance (SMA)