作用素環論の最近の進展 (Recent Developments in Operator Algebras)
RIMS ４２０, ２０１６年９月１２日(月)  １４日(水)

Monday, 12 
Tuesday, 13 
Wednesday, 14 
09:40  10:30 
Welcome 


10:40  11:30 




13:00  13:50 


Program in pdf 
14:00  14:50 




15:10  16:00 


16:10  17:00 


We study the representation theory of Drinfeld doubles of $q$deformations. As applications, we prove central property (T) for general higher rank $q$deformations and the HoweMoore type theorem for $q$deformations. We also give a partial result to compute (co)homology of
the representation category of qdeformations.
We introduce Rokhlin properties for certain discrete group actions on C*correspondences as well as on Hilbert bimodules and analyze them. It turns out that the group actions on any C*correspondence $E$ with Rokhlin property induces group actions on the associated C*algebra $\mathcal{O}_E$ with the Rokhlin property and the group actions on any Hilbert bimodule with the Rokhlin property induces group actions on the linking algebra with the Rokhlin property. Permanence properties of several notions such as nuclear dimension and $\mathcal{D}$absorbing property with respect to crossed product of Hilbert $C^*$modules with groups, where group actions have Rokhlin property, are studied.
This is a joint work with Santanu Dey and Harsh Trivedi.
In this talk, we discuss on a variation of the Rokhlin property, the continuous Rokhlin property, of compact group actions on unital C*algebras. Here equivariant KKtheory and quantum group theory play key roles. Our main result is a classification of unital Kirchberg $G$algebras with the continuous Rokhlin property when G is a compact Lie group with torsionfree fundamental group.
We use an iterative procedure to construct maximal abelian *subalgebras (MASAs) that satisfy certain prescribed properties in a given separable $\mathrm{II}_1$ factor $M$. We prove this way that any $M$ contains uncountably many singular (respectively semiregular) MASAs $\{A_i\}_i$ with $A_i \nprec_M A_j$, $\forall i\neq j$. We then give an intrinsic characterization of $\mathrm{II}_1$ factors $M$ that have an sMASA, i.e. an abelian subalgeba $A\subset M$ such that $A \vee JAJ$ is maximal abelian in ${\cal B}(L^2M)$. Examples of sMASAs are the Cartan subalgebras. We use the above technique to prove that factors having an sMASA are closed to amplifications, inductive limits, finite index, and that their sMASA can be constructed either singular or semiregular.
Parallel sums of bounded positive operators play important roles in many places. A general theory for those of unbounded positive selfadjoint operators is explained together with some related topics.
Kirchberg and Wassermann introduced exactness for locally compact groups in order to study continuous bundles of C*reduced crossed products. For a discrete group, exactness is equivalent to topological amenability of the canonical action on its StoneCech compactification. In the talk, we will discuss their relation for general locally compact groups. Recently, Roe and Willett proved that a discrete metric space without property A always admits noncompact ghosts. As a consequence, we are able to answer an open question raised by AnantharamanDelaroche in 2002.
This is joint work with Jacek Brodzki and Christopher Cave.
We consider unitary equivalent classes of onedimensional quantum walks. We show that onedimensional quantum walks are unitary equivalent to quantum walks of Ambainis type and that translationinvariant onedimensional quantum walks are Szegedy walks.
Fraïssé theory is one of the classical topics of model theory within which countable structures with a kind of homogeneity are understood as special inductive limits of finitely generated structures. In this talk, I will explain how this theory is applied to the JiangSu algebra.
We would like to deal with the classification problem of BostConnes systems. By KMSclassification theorem of LacaLarsenNeshveyev, the Dedekind zeta function is an invariant of BostConnes systems. However, this invariant turned out to be an invariant of BostConnes C*algebras. In this talk, we will give an outline of the proof of this theorem.