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 Howe-Moore type theorem for $q$-deformations. We also give a partial result to compute (co)homology of the representation category of q-deformations.

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 KK-theory 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 torsion-free 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 s-MASA, i.e. an abelian subalgeba $A\subset M$ such that $A \vee JAJ$ is maximal abelian in ${\cal B}(L^2M)$. Examples of s-MASAs are the Cartan subalgebras. We use the above technique to prove that factors having an s-MASA are closed to amplifications, inductive limits, finite index, and that their s-MASA 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 self-adjoint 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 Stone--Cech 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 non-compact ghosts. As a consequence, we are able to answer an open question raised by Anantharaman-Delaroche in 2002.
This is joint work with Jacek Brodzki and Christopher Cave.

We consider unitary equivalent classes of one-dimensional quantum walks. We show that one-dimensional quantum walks are unitary equivalent to quantum walks of Ambainis type and that translation-invariant one-dimensional 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 Jiang--Su algebra.

We would like to deal with the classification problem of Bost--Connes systems. By KMS-classification theorem of Laca--Larsen--Neshveyev, the Dedekind zeta function is an invariant of Bost--Connes systems. However, this invariant turned out to be an invariant of Bost--Connes C*-algebras. In this talk, we will give an outline of the proof of this theorem.