Study of recent topics on operator algebras

本文的研究计划包括以下四个方面：1. Jones指标理论和Hopf代数Masuda对Popa的III_1型子因子分类给出了一个替代性的证明，并成功地对许多类群作用在子因子下进行了分类。Izumi和Kosaki分析了Kac代数的结构（即，Hopf代数），并对维数不超过31的Kac代数进行了完全分类。2. III型因子的自由积与群胚的自由积的混合Kosaki构造了可测等价关系的自由积的混合.得到的对象是群胚，并阐明了它们的基本性质。上田在自由概率的基础上研究了自由积和相关的算子代数。3.由图、Hilbert C^*-双模、子移位和复动力系统产生的C ^*-代数 ...更多信息 tani和松本研究了各种C^*-代数及其不变量的性质（如K-理论和熵）。Kajiwara Watatani处理的C^*-代数所产生的希尔伯特C^*-双模和复杂的动力系统，而松本处理的那些所产生的子移位和λ-图系统。Hamachi研究了符号动力系统的嵌入问题。Izumi在某些C^*-代数上得到了具有Rohlin性质的有限群作用的漂亮分类结果。和Kosaki对算子均值及其范数比较进行了系统的研究。这些结果与相关函数的正定性信息一起，将为我们带来许多新的算子平均范数不等式，从而进一步丰富了这方面的研究。内山在算子单调函数（在算子的比较中起着重要作用）方面获得了许多新的结果，这使他对古田不等式中的指标要求得到了一定的解释。少

The following four topics were mentioned in our research plan.1.Jones index theory and Hopf algebraMasuda obtained an alternative proof for Popa's classification of type III_1-subfactors, and also succeeded in classifying many classes of group actions in the subfactor setting. Izumi and Kosaki analyzed structure of Kac algebras (i.e., Hopf algebras equipped with ^*-structure) via index theoretical approach, and completely classified Kac algebras of dimension up to 31. As a related topic, a notion of the second coholomogy for a subfactor was introduced and studied.2.Amalgamated free product of type III factors and free product of groupoidsKosaki constructed amalgamed free products of measurable equivalence relations. The resulting objects are groupoids, and their basic properties were clarified. Ueda studied free products and related operator algebras based on free probability.3.C^*-algebras arising from graphs, Hilbert C^*-bimodules, subshifts and complex dynamical systemsKajiwara-Wata … More tani and Matsumoto studied properties of various C^*-algebras and their invariants (such as K-theory and entropies). Kajiwara-Watatani dealt with C^*-algebras arising from Hilbert C^*-bimodules and complex dynamical systems while Matsumoto dealt with those arising from subshifts and λ-graph systems. Hamachi investigated embedding problems for symbolic dynamical systems. Izumi obtained beautiful classification results on finite group actions with Rohlin property on certain C^*-algebras.4.Operator meansHiai (Tohoku Univ.) and Kosaki made systematic studies on operator means and their norm comparison. These together with information on positive definiteness of relevant functions will bring us many new norm inequalities for operator means, and hence research in this direction seems to be further enriched. Uchiyama obtained many new results on operator monotone functions (playing important roles in comparison of operators), which enabled him to obtain a certain interpretation for index requirements in the Furuta inequality. Less