Study of recent topics on operator algebras

点击翻译按钮获取中文摘要

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