作用素環論研究者シンポジウム
作用素環論の最近の進展 (Recent Developments in Operator Algebras)
RIMS １１１, ２０１７年９月０４日(月)  ０６日(水)

Monday, 04 
Tuesday, 05 
Wednesday, 06 
09:30  10:30 
Welcome 


10:45  11:45 




13:30  14:30 


Program in pdf 
14:45  15:45 


16:00  17:00 


We shall give a complexanalytic construction of the canonical trace on Takesaki's duals in modular theory,
which is then used to explore the Haagerup's key formula in his theory on noncommutative $L^p$ spaces.
For a given reduced amalgamated free product of C*algebras,
we introduce a C*algebra including it as a unital subalgebra.
This C*algebra is an analogue of (the crossed product of)
the compactification of the BassSerre tree.
Using an explicit identification with a CuntzPimsner algebra
we prove two kinds of "amenability" results for the C*algebra.
There are applications for approximation properties, embeddability,
and KKtheory for amalgamated free product C*algebras.
Matui has introduced the notion of almost finiteness for totally disconneccted groupoids, to study their homology groups and topological full groups. In this talk, I will explain that this notion is also useful to study the structure of groupoid C*algebras. As a main application, we show that for any minimal dynamical system of any abelian group with a totally disconnected free factor, the crossed product has stable rank one. This generalizes the recent works of ArcheyPhillips (Z) and Phillips ($Z^d$, $d<\infty$) (which notably include nonZstable crossed products). We also explain how our result applies to crossed products of general amenable groups.
We study the relative Drinfeld commutant of a fusion category in another
fusion category in terms of halfbraidings. We identify halfbraidings
with minimal central projections of the relative tube algebra and
certain sectors related to the LongoRehren subfactors. We apply this
general machinery to various fusion categories arising from
alphainduction applied to a modular tensor category and compute the
relative Drinfeld commutants explicitly.
We complete the classification of BostConnes systems. We show that two BostConnes C*algebras for number fields are isomorphic if and only if the original semigroups actions are conjugate. Together with recent reconstruction results in number theory by Cornelissende SmitLiMarcolliSmit, we conclude that two BostConnes C*algebras are isomorphic if and only if the original number fields are isomorphic. This is a joint work with Yosuke Kubota.
We will explain how to construct families of countable discrete groups using surgery techniques, and describe their intermediate rank properties. This talk is based on ongoing work with Sylvain Barre.
The Drinfeld double of a finite dimensional Hopf algebra is a quasitriangular Hopf algebra with the canonical element as the universal Rmatrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal Rmatrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum YangBaxter equation of the universal Rmatrix.
On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the Stensor) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using the Heisenberg double, and extend it to an invariant of equivalence classes of colored ideal triangulations of 3manifolds up to colored moves. In this construction, a copy of the Stensor is attached to each tetrahedron, and invariance under the colored Pachner (2,3) moves is shown by the pentagon relation of the Stensor.
The main result of this talk is nonvanishing of the image of the index map from the Gequivariant Khomology of a proper Gcompact Gmanifold X to the Ktheory of the C*algebra of the group G.
Under the assumption that a Ginvariant Dirac type operator on X is detevted by the Kronecker pairing with a lowdimensional cohomology class, we prove that its Gindex is nonzero. Neither discreteness of the locally compact group G nor freeness of the action of G on X are required. The case of free actions of discrete groups was considered earlier by B. Hanke and T. Schick.
A $\mathrm{II}_1$ factor $M$ is McDuff if it is isomorphic to $M \otimes R$ where $R$ is the hyperfinite $\mathrm{II}_1$ factor. In this talk, we will discuss the following well known open problem: if $M$ and $N$ are two $\mathrm{II}_1$ factors, is it true that $M \otimes N$ is McDuff if and only if $M$ is McDuff or $N$ is McDuff. We will give some partial results and present a solution to the similar question for equivalence relations.
Understanding the linear forms on Kgroups given by traces, and more generally by cyclic cocycles, is one of the central questions in operator algebra and noncommutative geometry. When there is a continuous family of algebra structures, the GaussManin connection on periodic cyclic (co)homology due to Getzler, Tsygan, and others in deformation quantization gives a powerful guiding principle on this problem, which is however difficult to substantiate in the operator algebraic (or "strict" deformation) setting. In this talk we consider algebras graded over discrete groups, and consider deformation of product structure by group 2cocycles. This setting allows us to integrate the GaussManin connection using the natural action of group cohomology, and it recovers previous computations for noncommutative tori and other twisted group algebras. Based partly on joint work with S. Chakraborty (Muenster).