์pfย_คาVWE
์pfย_ฬล฿ฬiW (Recent Developments in Operator Algebras)
RIMS PPP, QOPWNXOR๚()  OT๚(
)
Sept. 2018 
Monday, 03 
Tuesday, 04 
Wednesday, 05 
09:30  10:30 
Welcome 


10:45  11:45 




13:30  14:30 


Program in pdf 
14:45  15:45 


16:00  17:00 


Nonbacktracking operators have been studied for their own sake for a long time, and more recently
in order to (re)prove eigenvalues estimates in graph theory. We develop an operator valued version
of nonbacktracking operators and provide some new applications to the study of the norm of operators
arising naturally in free probability theory. This talk is based on past and ongoing work with
Charles Bordenave (CNRS).
The development of free probability theory (FPT for short) invented by Voiculescu expands
the scope of research of random matrices. The FPT is an invaluable tool for describing the
spectral distributions of many random matrices. However, little is known about its inverse
direction, that is, how to know a random matrix model from its spectral distribution.
Now, situations in many fields of research, such as digital communications and statistics,
can be modeled with random matrices. In this research, we introduce a new parameter estimation
method of random matrix models via the FPT, and apply it to statistics.
In this talk, we are interested in characters of inductive limits of compact quantum groups.
There is a lot of researches of characters of inductive limits when compact quantum groups are
ordinary compact groups. However, there was no research for quantum groups. We propose a natural
character theory for inductive limits of compact quantum groups and study characters following
the VershikKerov approach for case of groups. Moreover, we investigate when given compact quantum
groups coincide with quantum unitary groups. In this case, our work relate to Gorinfs work on
qcentral measures on qGeldandTsetlin graph. This relationship gives a representationtheoretic
interpretation of Gorinfs work.
In the recent classification theory of C*algebras, the absorption of the JiangSu algebra Z plays a central role. Actually, A. S. Toms and W. Winter conjectured that Zabsorption is equivalent to other regularity properties, strict comparison and finiteness of nuclear dimension, for standard nuclear C*algebras. Now, we know that this conjecture holds true under the assumption of unique tracial state. In this talk, we study a permanence property of Zabsorption for crossed products by countable amenable groups. Assuming strict comparison and uniqueness of tracial state, we show that the crossed product of unital separable simple nuclear C*algebra absorbs the JiangSu algebra again.
A Smale space is a hyperbolic dynamical system on a compact metric space with local product structure.
Smale spaces are regarded as higher dimensional analogue of twosided topological Markov shifts.
I will talk about several kinds of groupoids and their C*algebras constructed from Smale spaces.
We first introduce notions of asymptotic continuous orbit equivalence and asymptotic conjugacy in Smale
spaces and characterize them in terms of their asymptotic Ruelle C*algebras with their dual actions.
We next introduce a class of groupoid C*algebras which might be regarded as bilateral CuntzKrieger
algebras from Smale spaces. We call them the extended Ruelle algebras. The asymptotic Ruelle algebra is
realized as a fixed point algebra of the extended Ruelle algebra under certain circle action.
This talk is based on my preprint: Asymptotic continuous orbit equivalence of Smale spaces and Ruelle
algebras,
arXiv: 1703.07011v4, to appear in Canad. J. Math..
In 1951, Kadison gave the following result about isometries of operator algebras:
Every complex linear unital surjective isometry between two unital C*algebras is a Jordan *isomorphism.
On the other hand, the classical theorem due to Mazur and Ulam states that every surjective isometry
between two normed spaces is automatically affine. These results give rise to the following questions:
Which substructure of an operator algebra, as a metric space, determines the Jordan structure of the
operator algebra? Can we give a complete description of surjective isometries between such substructures?
In this talk, we survey recent developments in these lines of research.
The overall goal of my research is to formulate and prove an infinitedimensional version of
the equivariant index theorem for locally compact manifold with a proper and cocompact group action.
More precisely, I want to replace ``locally compact manifold and locally compact group'' with
``infinitedimensional manifold and loop group'', keeping the action ``properly and cocompactly''.
Although this project has not been completed, I have constructed several substitutes for several
core objects for the analytic side: $L^2$space, Dirac operator, group C*algebra, Kgroup valued
analytic index, a part of assembly map and several C*algebras related to this map.
In this talk, I will explain the progress so far.
We discuss definition of inner amenability of discrete measured groupoids, its basic properties, examples, nonexamples, and some questions. This is a joint work with Robin TuckerDrob.
We establish complete description theorems of intermediate operator algebras for crossed product
inclusions under a mild condition. As an application we give examples of inclusions with unexpected/new behavior.