We shall give a complex-analytic 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 non-commutative $L^p$ 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 Bass--Serre tree. Using an explicit identification with a Cuntz--Pimsner algebra we prove two kinds of "amenability" results for the C*-algebra. There are applications for approximation properties, embeddability, and KK-theory 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 Archey--Phillips (Z) and Phillips ($Z^d$, $d<\infty$) (which notably include non-Z-stable 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 half-braidings. We identify half-braidings with minimal central projections of the relative tube algebra and certain sectors related to the Longo--Rehren subfactors. We apply this general machinery to various fusion categories arising from alpha-induction applied to a modular tensor category and compute the relative Drinfeld commutants explicitly.

We complete the classification of Bost--Connes systems. We show that two Bost--Connes 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 Cornelissen--de Smit--Li--Marcolli--Smit, we conclude that two Bost--Connes 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 quasi-triangular Hopf algebra with the canonical element as the universal R-matrix, 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 R-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has the canonical element (the S-tensor) 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 3-manifolds up to colored moves. In this construction, a copy of the S-tensor is attached to each tetrahedron, and invariance under the colored Pachner (2,3) moves is shown by the pentagon relation of the S-tensor.

The main result of this talk is non-vanishing of the image of the index map from the G-equivariant K-homology of a proper G-compact G-manifold X to the K-theory of the C*-algebra of the group G. Under the assumption that a G-invariant Dirac type operator on X is detevted by the Kronecker pairing with a low-dimensional cohomology class, we prove that its G-index is non-zero. 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 K-groups 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 Gauss--Manin 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 2-cocycles. This setting allows us to integrate the Gauss--Manin 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).