ABSTRACTS OF TALKS OF RIMS WORKSHOP：

MATHEMATICAL STUDIES ON

INDEPENDENCE AND DEPENDENCE STRUCTURE

　 −ALGEBRA MEETS PROBABILITY−

19-21 December 2011, RIMS, Kyoto University, Japan

Koji Aoyama（Chiba University）

Fixed point and ergodic theorems for hybrid mappings

Abstract: We first introduce the class of hybrid mappings in Hilbert spaces. This class contains the classes of nonexpansive mappings and nonspreading mappings in Hilbert spaces. Then we show a fixed point theorem and an ergodic theorem for such mappings.

Marek Bozejko（Wroclaw University）

Non-commutative Fock spaces with applications to constructions

of new models of non-commutative probability

Abstract: In my talk we will consider the following subjects:
1.Fock spaces with Yang-Baxter-Hecke (YBH) operators as deformation "parameter".
2.Monotone Fock spaces of Muraki as deformations of (YBH) type.
3.Monotone convolutions as special case of conditionally free convolutions.
5.q-CCR relations and anyonic probability for |q| =1.
6.Khinchine inequality for monotone probability .

Marie Choda (Osaka Kyoiku University)

A representation of unital completely positive maps

Abstract: We give a model of representations of unital completely positive maps via linearly independent finite operators, in order to approach a definition of entropy for them. The definition is the same type as the von Neumann entropy for states of matrix algebras, and satisfies the expected property for the notion of entropy in a natural relation with other type entropies.  

 Benoit Collins (University of Ottawa and RIMS)
Applications of free probability to quantum information theory

Abstract: in this talk I will explain how new results in random matrix theory help to understand the behaviour of typical quantum channels, and the applications to the problem of additivity of the minimum output entropy.

Ichiro Fujimoto, Hideo Miyata（Kanazawa Institute of Technology）

Quantization of information theory

Abstract: In scope of CP-convexity theory for C*-algebras (quantization of convexity, measure, entropy for completely positive maps), we investigate the operational structure of quantum interactions of entangled systems, and propose new information quantities which naturally generalize the classical information theory.

Takahiro Hasebe（Kyoto University）

Free independence and its generalization

Abstract: Independence is a basic concept in probability theory. However, if we consider independence on a noncommutative algebra, it is not unique. A well known example is free independence. We will look at free independence from a new point of view, which enables us to generalize free independence.

Mamoru Kaneko（University of Tsukuba）

Game Theoretical Decision Making: Logical Inference and Free-Will

Abstract: In this presentation, I will talk about the epistemic logic approach to decision making in a game situation. There, logical inference by a player is crucial, but has not been explicitly discussed. Also, the tendency in the literature is to focus on the behavioral outcome of the theory, while logical inference is treated implicitly. In this talk, we give, using small examples, what logical aspects are involved. I will talk about one logical system and some applications. In doing so, I argue that the free-will postulate is basic to game theory, contrasting to the tendency of determinism in the game theory literature.

Jun Kawabe (Shinshu University)

Metrizability of weak convergence of nonadditive measures

Abstract: We formalize the L\'{e}vy-Prokhorov metric and the Fortet-Mourier metric for nonadditive measures on a metric space and show that the L\'{e}vy topology on every uniformly equi-autocontinuous set of Radon nonadditive measures can be metrized by such metrics. This result is proved by the help of the uniformity for the L\'{e}vy convergence on a bounded　subset of Lipschitz functions. We also give applications to stochastic convergence of a sequence of measurable mappings on a nonadditive measure space.

Fumiaki Kohsaka (Oita University)

On fixed points of firmly nonexpansive-type mappings in Banach spaces

Abstract: We study the existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. Many problems in optimization and nonlinear analysis, such as convex minimization problems, variational inequality problems, saddle point problems, and equilibrium problems, can be formulated as the fixed point problem for a firmly nonexpansive-type mapping in Banach spaces.

Motoya Machida (Tennessee Technological University)

Positive definiteness and Frechet bounds in capacities

Abstract: Mass transportation problems have been developed as an important tool in probability theory, enabling us to show the existence of probability measures given marginals (Strassen theorem by Kellerer, 1984), or give a probabilistic interpretation of the Fortet-Mourier metric (Kantorovich-Rubinstein theorem by Dudley, 1989), to name a few applications. Roughly speaking, these problems are viewed as a measure theoretic formulation of prime and dual linear programming problems, and the optimal bounds in prime problems are often called Frechet bounds (Ruschendorf, 1991).
Choquet (1954) and independently Murofushi and Sugeno (1991) gave a probabilistic interpretation for nonadditive measures. Furthermore, in a certain topological setting Choquet characterized closed random sets in terms of completely alternating capacities defined over the family of compact subsets. The characterization is essentially an integral representation for positive definite functions on an idempotent Abelian semigroup (Choquet theorem by Berg, Christensen and Ressel, 1984).
In this talk we discuss an application of Frechet bounds for the Choquet theorem, and present their probabilistic interpretation when a capacity is not completely alternating.

Naofumi Muraki (Iwate Prefectural University)

On a q-interpolation betweeen free independence and classical independence

Abstract: In this talk “independence” means a universal calculation rule for mixed moments of noncommutative random variables. We construct a one-parameter family of independence with parameter q which interpolates between classical independence and free independence of Voiculescu.

Hiroshi Nagaoka（The University of Electro-Communications）

On a large deviation problem concerning quantum hypothesis testing

Abstract: We discuss a large deviation problem which arises in the asymptotic theory of quantum hypothesis testing. The problem is compared with its classical counterpart and is studied from an information geometrical viewpoint.

Yasuo Narukawa (Tokyo Institute of Technology)

Fuzzy measure and integral on multi sets

Abstract: Fuzzy measures on multisets are studied. We show that a class of multi sets can be represented as a subset of positive integers. We define the comonotonicity for multisets. and  show that a fuzzy measure on multisets with some comonotonicity condition can be represented by generalized fuzzy integral.

Nobuaki Obata（Tohoku University）

Quantum Probabilistic Spectral Analysis of Graphs

Abstract: The graph spectrum is a key concept for the analysis of large/growing complex networks. We review several new aspects for spectral analysis of graphs in line with quantum probability theory and report some recent achievements on Manhattan products of digraphs.

Izumi Ojima (Kyoto University)

How to Unify Interactions? − Independence and Dependence in Physics

Abstract: In the quadrality scheme based on Micro-Macro duality, spacetime is shown to be an epigenetic empirical notion of a posteri nature, arising from emergence processes from microscopic motions, in sharp contrast to the standard picture adopted in modern physics. Here spacetime points are indices to parametrize independent sectors whose inside structures consist of microscopic motions without gravity (i.e., "free-falling systems") and their inter-dependence relations are described by the gravitational field. Along this line of thought, we can find a new meaning in the "unification of four interactions", totally different from its common understanding.

Kazuya Okamura (Kyoto University)

The Quantum Relative Entropy and Statistical Inference

　−Hypothesis Testing and Model Selection−

Abstract: We will reveal that the quantum relative entropy has the same probabilistic and statistical meaning as the relative entropy in classical probability has, which could not be done by past studies. Hypothesis testing and model selection will be discussed here.

Hayato Saigo（Nagahama Institute of Bio-science and Technology）

Arcsine law and ``Quantum-Classical correspondence''

Abstract: Arcsin law is a well-known probability distribution in classical probability. Muraki discovered that it appears in a ``central limit theorem'' in non-commutative probability: We will review (and speculate) some aspects of  ``quantum-classical correspondence '' , focusing on Arcsin law  and the notion of ``independence'' in noncommutative probability.

Jiun-Chau Wang (University of Saskatchewan)

Strict Limit Types for Monotone Convolution

Abstract: We will discuss the recent progress in limit theorems for monotone convolution, including the Levy type characterization for strictly stable laws, the law of large numbers, and the central limit theorem. These results are obtained by complex analytic methods without reference to the combinatorics of monotone convolution.

Kenjiro Yanagi, Satoshi Kajiwara（Yamaguchi University）

　　　 Generalized uncertainty relation associated with a monotone or anti-monotone

pair skew information

Abstract: We give a trace inequality related to the uncertainty relation based on the monotone or anti-monotone pair skew information which is one of generalizations of result given by Ko-Yoo(JMAA, 2011). And it includes the result for generalized Wigner-Yanase-Dyson skew information as a particular case given by Yanagi (LAA, 2010).

Hiroaki Yoshida (Ochanomizu University)

An integral representation of the relative free entropy

Abstract: Using the logarithmic energy with the potential function, the free analogue of the relative entropy between two compactly supported probability measures on the real line was introduced by Biane and Speicher. In this talk, we shall give an integral representation of the free relative entropy associated with semicircular gradients.

Kei Zembayashi (Koen Girls' School）

Converegence theorems for equilibrium problems in Banach spaces

Abstract: Numerous problems in physics, optimization, and economics reduce to find a solution of the equilibrium problem. We will discuss the convergence theorems for solving the equilibrium problem in Banach spaces.