We review our recent operator-algebraic techniques to produce interacting models of Quantum Field Theory. We also discuss how more complicated models can be constructed and how the modular theory plays a crucial role there.

We consider a C*-analogue of Connes's view, which says that bimodules over von Neumann algebras play a role of unitary representations of groups. Using this we prove a relative analogue of equivalence between nuclearity and CPAP. As an application, we show the K-nuclearity of any amalgamated free product of nuclear C*-algebras under a strengthened variant of `relative nuclearity'.

It is known that the notion of completely positive (CP) instrument describes physical processes of measurement. In quantum mechanics, there exists a one-to-one correspondence between CP instruments and statistical equivalence classes of ``measuring processes". We generalize this correspondence, with some modifications, to quantum systems of infinite degrees of freedom. We then discuss several topics on CP instruments from the viewpoint of quantum probability theory. The main part of this talk is based on a joint work with Masanao Ozawa.

We give a brief account of the recent development of the classification results of small index subfactors and related fusion categories. The Cuntz algebra method plays a crucial role.

We characterize the condition for two finite index endomorphisms on an AFD factor to be mutually approximately unitarily equivalent. The characterization is given by using the canonical extension of endomorphisms, which is introduced by Izumi. Our result is a generalization of the characterization of approximate innerness of endomorphisms of the AFD factors, obtained by Kawahiashi--Sutherland--Takesaki and Masuda--Tomatsu. Our proof, which does not depend on the types of factors, is based on recent development on the Rohlin property of flows on von Neumann algebras.

力学系がいつ無限次元立方体上のシフトに埋め込めるか？という質問について考える．この問題は４０年以上前から研究されているが，近年になって，「平均次元」という力学系の位相不変量と深くかかわることが分かってきた．この講演では，現在どこまで出来ているかを説明したい．このアブストラクトを書いている６月末時点でも研究は進展中であり，講演で何を話せるかは，これから２か月間の研究次第で変化するので，私にもまだ具体的にはわかりません．

We investigate certain families of functions of several noncommuting operators also knowns as free functions. In particular we intorduce operator monotone functions and operator means in several variables and we realize that operator monotonicty is characterized by operator concavity for free functions with domains that are unbounded from above in the positive definite order. We introduce the hypograph of such functions and conclude that a free function is operator concave if and only if its hypograph is a matrix convex set in the sense of Effros-Winkler. Then by applying the matricial Hahn-Banach theorem of Effros and Winkler we construct an (infinite dimensional) LMI such that its positivity domain coincides with the hypograph of our operator concave function, hence characterizing operator monotone functions as extremal solutions of LMIs.

We introduce a class of gapped Hamiltonians which allows an asymmetric ground state structure and consider its classification.

I will report a recent remarkable result of Tucker-Drob about structure of inner amenable groups satisfying the minimal condition on centralizers. It is applied to a characterization of linear groups having a stable probability-measure-preserving action.

In this talk we present our recent work on locally compact C*-simple groups. A locally compact group is called C*-simple if its reduced group C*-algebra is simple. In the past, only C*-simplicity of discrete groups was studied. However, a recent breakthrough in the work of Kalantar-Kennedy and Breuillard-Kalantar-Kennedy-Ozawa almost puts this subject to an end. This success motivates research on locally compact C*-simple groups. In our work, we show that every C*-simple group must be totally disconnected and present first examples of non-discrete C*-simple groups.

We introduce the class C_(AO) of von Neumann algebras that particularly contains free (quantum) group factors and free Araki-Woods factors. We show that any tensor product factor, which consists of finitely many factors in C_(AO), retains each tensor component (up to stable isomorphism). This generalizes Ozawa-Popa's pioneering work for free group factors and provides a new result for free Araki-Woods factors. In order to obtain this, we show that Connes's bicentralizer problem has a positive solution for all type III_1 factors in the class C_(AO). This is joint work with C. Houdayer.