2013 NIMS Hot Topics Workshops on

Invited Speakers :Luigi Accardi (Università di Roma Tor Vergata)
Nobuhiro Asai (Aichi University of Education)
Philiippe Biane (CNRS Universite Paris-Est)
Pradip Kumar Das (Indian Statistical Institute)
Fumio Hiai (Tohoku University)
Sejong Kim (Louisiana State University)
Yuh-Jia Lee (Natiional University of Kaohsiung)
Franz Lehner (Graz University of Technology)
Yongdo Lim (Sungkyunkwan University)
Wojciech Mlotkowski (University of Wroclaw)
Nobuaki Obata (Tohoku University)
Habib Ouerdiane (University of Tunis El manar)
Kimiaki Saito (Meijo University)
Takashi Sano (Yamagata University)
Tomoyuki Shirai (Kyushu University)
Adam Skalski (Polish Academy of Sciences & University of Warsaw)
Janusz Wysoczanski (University of Wroclaw)
Anna Wysoczanska-Kula (University of Wroclaw)
Hiroaki Yoshida (Ochanomizu University)

Local Organizing Committee :Un Cig Ji (Chungbuk National University)
Jaeseong Heo (Hanyang University)
Hyun Jae Yoo (Hankyong National University)
Hun Hee Lee (Seoul National University)

Supported by

National Institute for Mathematical Sciences
Chungbuk National University- Industry Academic Cooperation Foundation- Research Institute of Mathematical Finance2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Program
19 (Mon)

9:20~9:30

Opening

9:30~10:20

Lehner1

20 (Tue)

21 (Wed)22 (Thur)

23 (Fri)

Lehner3

Hiai1

Hiai3

9:30~10:10
Ouerdiane
10:10~10:50
Shirai

10:30~11:20

Lehner2

Biane2

Hiai2

Skalski2
11:10~11:50
Das

11:40~12:30

Biane1Biane3

Skalski1

Skalski3

11:50~12:20
Choi

12:30~13:30

Lunch

Lunch

Lunch

Lunch

Closing/
Lunch

13:30~14:20

Obata

AccardiLee

14:30~15:10

JanuszWysoczanski

Saito

Anna
Wysoczanska-Kula
Excursion

15:30~16:10

Kim

YoshidaMlotkowski

16:10~16:50

Sano

Lim

Asai

18:20~

DinnerBanquetDinner

2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Contents
Cumulants in noncommutative probability…………………………………………………………………… 7
Franz Lehner

Concavification of free entropy… …………………………………………………………………………… 8
Philippe Biane

Spectra of Manhattan Street Digraphs for Synchronization of Coupled Oscillators… ……………………… 9
Nobuaki Obata

FB-independence?… ……………………………………………………………………………………… 10
Janusz Wysoczanski

Relative Operator Entropy and Fidelity…………………………………………………………………… 11
Sejong Kim

Operator monotone functions and Kwong matrices… …………………………………………………… 12
Takashi Sano

On the multi-dimensional Favard theorem………………………………………………………………… 13
Luigi Accardi

Infinite dimensional analysis based on higher order derivatives of white noise…………………………… 14
Kimiaki Saitô

The free logarithmic Sobolev and transportation cost inequalities by time integration…………………… 15
Hiroaki Yoshida

Weighted least squares means on NPC space……………………………………………………………… 16
Yongdo Lim

Contraction theory in quantum information… …………………………………………………………… 17
Fumio Hiai

Haagerup property for locally compact classical and quantum groups… ………………………………… 18
Adam Skalski

Analysis of Complex brownian motion… ………………………………………………………………… 19
Yuh-Jia Lee

Construction of compact quantum groups related to functions on permutation…………………………… 20
Anna Wysoczanska-Kula

Distributions with binomial moments……………………………………………………………………… 21
Wojciech Mlotkowski

The Brenke-Chihara polynomials and q-deformed Bargmann measures… ……………………………… 22
Nobuhiro Asai

Unitarizing measures for the representation of Lie group and associated invariant difierential operators……… 23
Habib Ouerdiane

Zeros of a certain Gaussian power series… ……………………………………………………………… 24
Tomoyuki Shirai

An Experimental Scheme To Generate Non-Classical States In Interacting Fock Space… ……………… 25
Pradip Kumar Das

Convergence of Weighted Sums in Noncommutative Probability Space… ……………………………… 26
Byoung Jin Choi

2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Cumulants in noncommutative probability
Franz LehnerGraz University of Technology

Abstract:
Cumulants (in the classical sense) were invented by Thiele at the end of the 19th century
and have had many applications in statistics.
A century later other kinds of cumulants have been introduced to cope with different
notions of noncommutative independence, like boolean and free cumulants and most recently
monotone cumulants.
We give an introduction and general combinatorial theory of cumulants, covering thenecessary lattice theory, set partitions, and some applications.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Concavification of free entropy
Philiippe Biane
CNRS Universite Paris-Est

Abstract:
I will review Voiculescu’e free entropy, and propose to modify slightly its definition in orderto obtain a concave quantity. This modification allows in particular to solve the additivity
problem for free entropy.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Spectra of Manhattan Street Digraphs for
Synchronization of Coupled Oscillators
Nobuaki Obata
Tohoku University

Abstract:
A Manhattan street network is a digraph (directed graph), where the underlying graph isthe n × m lattice and each edge is given direction in such a way that left and right directedhorizontal lines are placed alternately, and so are up and down directed vertical lines.
In general, spectrum of a digraph contains useful information for the analysis of (deterministic
or stochastic) dynamics on it.
We will report some results on spectral analysis for the preliminary case of 2 × n lattice andapply to the study of synchronization of coupled oscillators.
References[1] F.
‌ Comellas, C. Dalf'ó, M. A. Fiol and M. Mitjana: The spectra of Manhattan street
networks, Linear Algebra Appl. 429 (2008), 1823-1839.
[2] ‌N. Obata: Spectra of Manhattan products of directed paths Pn#P2, Interdisciplinary
Information Sciences 18 (2012), 43-54. DOI: 10.4036/iis.2012.43[3] N. Obata and J. Rodriguez: in preparation (2013)92013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

FB-independence?
Janusz Wysoczanski
University of Wroclaw

Abstract:
We present a construction of Hilbert space operators, indexed by partially ordered set, which
generalize free and boolean independences. More properties of this construction is to beshown, intended for new notion of independence.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Relative Operator Entropy and Fidelity
Sejong Kim
Louisiana State University

Abstract:
We consider operator versions of quantum Tsallis relative entropy that involve the weighted
matrix geometric mean and establish relationships with the fidelity of a pure state and an
arbitrary state. We define the symmetrized operator entropy based on the Tsallis relative
operator entropy and study its properties.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Operator monotone functions and Kwong matrices
Takashi Sano
Yamagata University)

Abstract:
In this talk, we study operator monotonefunctions on (0, 1) and show some difierences fromthose on (0, ∞) with consideration on Kwong matrices

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

On the multi{dimensional Favard theorem
Luigi Accardi
Università di Roma Tor Vergata

Abstract:
The problem of developing a satisfactory theory of multi{dimensional orthogonal polynomials
has been open for several decades.
The root of this problem is related to the fact that the multi{dimensional extensions ofFavard’s theorem existing, up to now, in the literature are notsatisfactory for several reasons.
In a recent paper with A. Barhoumi and A. Dhahri we have proved that the theoryof orthogonal polynomials in d variables (d2N) can be canonically identfied to the theoryof symmetric interacting Fock spaces over Cd. The proof exploits the properties of thequantum decomposition of vector valued random variables.
This result opens the way to the investigation of algebraic invariants of probability
measures on Rd.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Infinite dimensional analysis based onhigher order derivatives of white noise
Kimiaki Saitô*
Meijo University

Abstract:
In this talk we present an infinite dimensional analysis based on higher order derivativesof white noise and its recent developments. This analysis is associated with the exotic
Laplacians and the Gross-Volterra Laplacians with orthonormal bases consisting ofgeneralized functions.
The domain of the exotic Laplacian consists of higher powers of the derivative of white noisethat are not normal polynomials introduced by P. Lévy. The exotic Laplacian acting on
white noise distributions connects with the Lévy Laplacian acting on functionals of higher
order derivative of white noise by the second quantization of the adjoint operator of thehigher order differentiation.
This relationship comes from a generalization of the Accardi-Smolyanov theorem in thehigher order Cesàro mean and implies a stochastic analysis of an infinite dimensional
stochastic process generated by the exotic Laplacian.
We also discuss the relationship between the exotic Laplacians and the second quantizationof the Fourier transform. It implies an infinite dimensional stochastic process consisting fromthe product of independent Fractional Brownian motions associated with the space of theexotic Laplacian acting on the Fourier transform of exotic test functionals.
*This work was supported by JSPS Grant-in-Aid Scientific Research 24540149.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

The free logarithmic Sobolev andtransportation cost inequalities by time integration
Hiroaki Yoshida
Ochanomizu University

Abstract:
We consider the time evolution of compactly supported probability measures on R by thefree Fokker-Planck equation with the potential. Then we find that the formulas of the time
derivation of the 2-Wasserstein distance with the Brenier map and of the dissipation of thefree entropy imply the free functional inequalities, involving free LSI and free TCI, by time
integration in case of a strictly convex potential.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Weighted least squares means on NPC space
Yongdo Lim
Sungkyunkwan University

Abstract:
Complete nonpositively curved metric spaces also called Hadamard spaces or simply NPC
spaces have many fundamental properties which make them an attractive geometric object
to study. These spaces somewhat resemble Hilbert spaces which provide an interesting new
setting to study and carry out averaging of convex functions. Recently the so called leastsquares mean definedas the unique solution of the minimization problem.

Pnwhere ai2M, wi>0,
i=1 w = 1 and M is an NPC space, received broader attention. We
introduce a new weighted random walk converging to the weighted least squares mean.
i

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Contraction theory in quantum information
Fumio Hiai
Tohoku University

Abstract:
The first part of my lecture is a survey talk on Hilbert’s projective metric and the irkoffHopf contraction theorem in Banach spaces. The subjects are very old but do not seem very
familiar. After introducing basic concepts and properties, I explain the Birkoff-Hopf theorem
and its application to generalization of Perron-Frobenius theorem. Thomson’s norm and basenorm in Banach spaces are also discussed.
In the second part, my lecture turns to a contraction theory in matrix algebras in connection
with quantum information. Contraction coefficients are introduced for a stochastic map (i.e.,a trace-preserving linear map Φ : M →M between matrix algebras such that the adjoint Φ*
is a Schwarz map) in terms of several quantum
information quantities such as quasi-entropies, monotone metrics, χ2-divergences, etc. I show
new as well as known results concerning relations among those contraction coefficients. in theclassical case where stochastic maps reduce to column-stochastic matrices, the theory waswell developed by Cohen et al., Linear Algebra Appl. 179(1993) and by Choi-Ruskai-Seneta,
Linear Algebra Appl. 208/209 (1994), however quite a few problems are still open in thequantum case.
In the third part, I talk about the reversibility (or sufficiency) of stochastic maps. Here, astochastic map Φ : M →M is said to be reversible for a set Cof states (or density matrices)
if there is a stochastic map Ψ : M →M such that Ψ(Φ(ρ)) = ρfor all ρ ∈ C. A number ofcharacterizations for reversibility were obtained by Hiai-Mosonyi-Petz-Bénny, Rev. Math.
Phys. 23 (2011) and by Jenčová, Rev. Math. Phys. 24 (2012). Quantitative version of thereversibility problem for Φ on Cis also discussed by introducing certain distinguishability
coefficients of Φ, the notion opposite to contraction coefficients.
d

d

d̃

d̃

d̃

d

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Haagerup property for locally compact classicaland quantum groups
Adam Skalski
Polish Academy of Sciences & University of Warsaw

Abstract:
The Haagerup property is one of the most intensively studied recently properties of locallycompact groups, exhibiting both geometric and analytic character. I will discuss its various
equivalent characterisations for a given group G, in terms of the existence of some particulara) representation of G;b) positive definite functions on G;c) conditionally negative definite function on G;d) affine action of G on a Hilbert space,adding a characterisation in terms of ‘typical’ representations of G. I will then pass to thecontext of locally compact quantum groups, outlining respective definitions and explaining
difficulties and subtleties one encounters in the quantum context. Several examples ofclassical and quantum groups with the Haagerup property will be introduced.
In the last part I will prove that a free product of discrete quantum groups with theHaagerup property has the Haagerup property and explain some connections with ‘free
products' of quantum Levy processes.
(partly based on joint work with Matt Daws, Pierre Fima and Stuart White).

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Analysis of Complex brownian motion
Yuh-Jia Lee*
Natiional University of Kaohsiung)

Abstract:
A theory of generalized functions based on the complex Brownian motion fZ (t) : t 2 Rg,
for which each Z(t) is N(0; jtj) is established on the probability space (S c0 ; B(S c0 ); (dz)),
where S 0 is the dual of the Schwartz space S, S c0 , the complexification of S ,0 identified as
the product space S 0 × S ,0 B(S c0 ) the Borel field of S 0 × S0 and v(dz) denotes the productmeasure μ1(dx) μ1(dy). Using the representation of the complex Brownian motion

whereand employing the technique of white noise calculus initiated by Hida we analyze functionals
of complex Brownian motion. To define generalized complex Brownian functionals, we adopt
the space of CKS entire functionals as test functions. As applications, the stochastic integral
with respect to a complex Brownian motion are defined and studied. The It^o formula forcomplex Brownian functionals is obtained and it is shown that the evaluation of stochastic
integral with respect to a complex Brownian motion follows the rule of Stratonovich integral.
Key words and phrases. complex Brownian motion, white noise analysis.
* Joint work with Kuang-Ghieh Yen and supported by the National Science Council of Taiwan.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Construction of compact quantum groups related tofunctions on permutation
Anna Wysoczanska-Kula
University of Wroclaw

Abstract:
In 1989 S. L. Woronowicz suggested a method which allowed to construct the twisted
SU(n) compact quantum groups (CQG). The method relies in general on a choice of a nondegenerate function on n-tuples. However, in many cases the related CQG becomes trivial (i.e.
the C*-algebra is commutative). During the talk, we shall remind the method and shortly
review the known examples of non-trivial CQG’s comming from this construction. Then weshall try to classify the CQG’s related to functions on permutations for n=3.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Distributions with binomial moments
Wojciech Mlotkowski
University of Wroclaw

Abstract:
We prove that if p ≥ 1 and –1 ≤ r ≤ p–1 then the binomial sequence ( )is positive definite
and therefore is the moment sequence of a probability measure V(p, r). The support of V(p; r)
is contained in the interval [0; pp(p –1)1 –p]. If p>1 isa rational number and –1 <r ≤ p –1 thenV(p; r) is absolutely continuous and itsdensity function Vp;rcan be expressed in terms of the
Meijer G-function. In particular cases Vp;r is an elementary function. Finally we prove thatthe binomial sequence ( ) is positive definite if and only if either p ≥ 1 –1 ≤ r ≤ p –1 or p ≤ 0,
p –1 ≤ r ≤ 0. The measures corresponding to the latter case are reections of the former ones.
We also show connections with the distributions having Fuss-Catalan moments ().
This is joint work with Karol A. Penson.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

The Brenke-Chihara polynomials andq-deformed Bargmann measures.
Nobuhiro Asai
Aichi University of Education

Abstract:
I shall discuss the Brenke-Chihara polynomials generated from the Boas-Buck typegenerating functions (IDAQP, 2013).
Moreover, some connections with q-deformed Bargmann measures will be presented.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Unitarizing measures for the representation of
Lie group and associated invariant difierential operators
Habib Ouerdiane
University of Tunis El manar

Consider a Lie group with a unitary representation into a space of holomorphic functions
defined on a domain D of Cn and in L2(μ), the measure μ being the unitarizing measure of therepresentation.
On finite dimensional examples, we show that this unitarizing measure is also the invariant
measure for some difierential operators on D.
In this talk we calculate these operators in the following elementary cases:A) The commutative groups (R;+) and (R – 0; ×).
B) The three dimensional Heisenberg group.
C) The ane group on the real line.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Zeros of a certain Gaussian power series
Tomoyuki Shirai
Kyushu University

Abstract:
In this talk, we discuss the zeros of a certain real Gaussian process in the unit disk, which isa random power series defined as the limit of the so-called Kac random polynomials.
We give a pfaffian expression of correlation functions of the zeros of this Gaussian power
series.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

An Experimental Scheme To Generate
Non-Classical States In Interacting Fock Space
Pradip Kumar Das
Indian Statistical Institute

Abstract:
An experimentally realizable scheme is considered for manipulating quantum states using
a general superposition of products of interacting annihilation and creation operators.
Application of such an operation on states with classical features introduces strong
nonclassicality. This provides the possibility of engineering quantum states with nonclassicalfeatures.

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2013 NIMS Hot Topics Workshops on The 8th Jikji workshop : Infinite Dimensional Analysis and Quantum Probability

Convergence of Weighted Sums
in Noncommutative Probability Space
Byoung Jin Choi
Chungbuk National University

Abstract
We study various convergences of weighted sums of random variables in a non-commutative
probability space of which the weights are in a von Neumann algebra.
More precisely, we study the weak law of large numbers, convergence rates and exponential
convergence rates which give upper bounds of large deviation principle for weighted sums
in noncommutative probability space. As applications, we first study convergence rates ofweighted sums of random variables in the noncommu-tative Lorentz space, and secondly westudy convergence rates of weighted sums of probability measures with respect to the free
additive convolution. This is a joint work with my supervisor Un Cig Ji.
Mathematics Subject Classifications (2000): 46L53.
Keywords: noncommutative probability space, weighted sum, weak law of large numbers, convergence rates,
exponential convergence rates, large deviation, free convolution.

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