Functional and Harmonic Analysis
at NSYSU, Oct. 13, 2016
Department of Applied Mathematics
National Sun Yat-sen University, Kaohsiung, Taiwan.
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
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 Ap(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.
Chia-Feng Yen 嚴嘉鳳, firstname.lastname@example.org, Tel: (886) 7-5252000 ext. 3849; or visit