**Mini-Workshop on**

**Functional and Harmonic Analysis**

** at NSYSU, Oct. 13, 2016**

Department of
Applied Mathematics

National Sun Yat-sen University, Kaohsiung, Taiwan.

台灣•高雄•中山大學•應用數學系

Revised: Oct. 5, 2016.

Several functional and harmonic analysts are visiting National Sun Yat-sen
University, and offering a half-day workshop. All interested are welcome to join.

**Thursday,
Oct. 13, 2016. Venue: SC-4013, NSYSU**

**1.
**Quanhua
Xu 許全華 (Université de Franche-Comté and Harbin Institute
of Technology): *Maximal inequalities in
noncommutative analysis* (13:00-13:45pm)

Maximal
inequalities are of paramount importance in analysis. Here ``analysis" is
understood in a wide sense and includes harmonic analysis, probability theory
and ergodic theory. Consider, for instance, the following three fundamental
examples, each in one of the previous fields:

i.
**Hardy-Littlewood maximal function.**
Given define

where denotes an interval and
$|I|$ the length of .

ii.
**Maximal martingale function**.
Given a martingale on a probability space define

iii.
**Maximal ergodic function.**
Let be a contraction on for every . Form the
ergodic averages of

and
define

All three maximal functions satisfy the following
inequality: For ,

where is a constant depending only on . This
inequality fails for but we have a weak type substitute:

where denotes the measure of the subset where is bigger than . This
classical result is due to Hardy-Littlewood, Doob or Dunford-Schwartz according
to one of the three cases.

We will consider in this survey talk the analogues of
these classical inequalities (and some others) in the noncommutative analysis.
Then the usual -spaces are
replaced by noncommutative -spaces
associated to von Neumann algebras. The theory of noncommutative
martingale/ergodic inequalities was remarkable developed in the last 15 years.
Many classical results were successfully transferred to the noncommutative
setting. This theory has interesting interactions with operator spaces, quantum
stochastic analysis and
noncommutative harmonic analysis. We will discuss some of these noncommutative
results and explain certain substantial difficulties in proving them.

**2.
**Sen
Zhu 朱 森 (Jilin University): *Complex symmetric generators for operator
algebras* (13:55-14:40pm)** **

The first part of this talk will be a brief
introduction to complex symmetric operators. Recall that a bounded linear operator on a Hilbert space is said to be complex symmetric if admits a symmetric matrix representation
with respect to some orthonormal basis for .

The second part of the talk will focus on the
algebraic aspects of the theory of complex symmetric operators. More precisely, we shall discuss the
complex symmetric generator problem for operator algebras, that is, the problem
of determining which operator algebras can be generated by a single complex symmetric
operator. For type I von Neumann
algebras, properly infinite von Neumann algebras and a large class of finite
von Neumann algebras, we give a complete answer.

**3.
**Zipeng
Wang 王子鵬 (Shaanxi Normal University): *Singular Integral Operator On the Unit Disc* (14:50-15:35pm)** **

There is a rich theory for singular integral operators
on the whole Euclidean space, and there are also many generalizations to the
unit disc as a homogeneous space. But these theories usually are hard to use
when one faces concrete examples from the unit disc. So our focus is on basic, concrete
examples and we will discuss how to get rather precise results for these
concrete operators.

This is my series joint works with Guozheng Cheng at
Wenzhou, Xiang Fang at Chung-Li and Jiayang Yu at Chengdu..

**4.
**Hao-Wei Huang 黃皓瑋 (National Sun
Yat-sen University): *Harmonic Analysis
Approach to Bi-Free Probability Theory* (15:45-16:30pm)** **

In 2013, D. Voiculescu introduced the notion of
bi-free independence as a generalization of free independence in order to
simultaneously study the left and right random variables in a vector space.
This research field is later on called bi-free probability. In this talk, we
will provide an analytic approach to bi-free probability. Specifically, we will
begin with basic definitions and results, and then introduce bi-free limit
theorems, bi-free infinitely divisible distributions, and bi-free stable laws.

**5.
**Hang-Chin
Lai 賴漢卿 (National
Tsing Hua University) Multipliers and *A**p*(*G*)-Algebras
() (16:35-17:20pm)

Let be an infinite
noncompact locally compact abelian (LCA) group with a dual group in protrjagin
sense (). Consider the
space

We supply the norm of by

Then is a semisimple
commutative Banach algebra under convolution product with norm . In this talk,
we would propose the construction for multiplier problem of for and give a
suitable explanation for the solution process.

**Sponsors:**
Mathematics Research Promotion Center and National Sun Yat-sen University.

Contact:

Ngai-Ching
Wong黃毅青,
wong@math.nsysu.edu.tw, Tel: (886)
7-525-2000 ext. 3818,

Chia-Feng Yen 嚴嘉鳳, yencf@math.nsysu.edu.tw,
Tel: (886) 7-5252000 ext. 3849; or visit