Subfactors and Noncommutative Ergodic Theory

其中两个主要方向的Nicoara的研究是研究不变量的子因素，主要是所谓的通勤广场。这些都是平方包含有限维C*-代数自然出现在标准不变的一个子因子。交换平方也可以用作子因子的构造数据，并且子因子的最显式示例已经通过这种方式获得。通过使用代数组合和分析方法的组合，PI证明了交换平方的几个有限性结果，并为这些对象找到了一个很好的素性概念（在隔离的意义上）。得到的隔离结果建议的方法，构建非同构子因子的单参数家庭。PI将继续研究这种构造，特别是那些基于复阿达玛矩阵的交换方。平面代数技术将被用来理解这些模型，以及从通勤广场构建的其他子因子。另一个方向的尼古拉的研究是研究冯诺依曼代数的角度来看，非交换遍历理论，特别是应用刚性的背景下，冯诺依曼代数。1930年，约翰·冯·诺依曼发现希尔伯特空间上的某些算子代数是理解量子物理系统对称性的自然框架。他的思想在量子力学中扮演着重要的角色，而自然界的基本定律，如海森堡测不准原理，则是冯·诺依曼的抽象理论的自然结果。在80年代初沃恩·琼斯介绍了理论的子因子，作为一个伽罗瓦理论的包含冯诺依曼代数。子因子理论很快成为算子代数理论中最繁荣的分支之一，与纽结理论、表示论、三维流形、量子群、统计力学中的可积系统和共形场论有着广泛的联系。 子因子可以被看作是一个类群的对象，它编码了量子物理或数学情况的广义对称性。为了解码这个信息，我们计算了高阶相对交换子，一个自然地与子因子相关联的有限维C*-代数的包含系统。这个对象，称为标准不变量，具有非常丰富的代数组合结构，推广了量子生成群，有限维Hopf C*-代数和其他大类量子群。

One of the two main directions of Nicoara's research is the study of invariants of subfactors, mainly the so called commuting squares. These are squares of inclusions of finite dimensional C*-algebras that arise naturally in the standard invariant of a subfactor. Commuting squares can also be used as construction data for subfactors, and the most explicit examples of subfactors have been obtained this way. By using a combination of algebraic-combinatorial and analytic methods, the PI proved several finiteness results for commuting squares and found a good notion of primeness (in the sense of isolation) for these objects. The isolation results obtained suggest methods of constructing one-parameter families of non-isomorphic subfactors. The PI will continue to investigate such constructions, especially those coming from commuting squares based on complex Hadamard matrices. Planar algebra techniques will be used to understand these models, as well as other subfactors constructed from commuting squares. The other direction of Nicoara's research is the study of von Neumann algebras from the point of view of non-commutative ergodic theory, especially applications of rigidity in the context of von Neumann algebras. In 1930's John von Neumann discovered that certain algebras of operators on a Hilbert space are the natural framework for understanding symmetries of quantum physical systems. His ideas play an important role in quantum mechanics, and fundamental laws of nature such as the Heisenberg uncertainty principle appear as a natural consequence of von Neumann's abstract theory. In the early 80's Vaughan Jones introduced the theory of subfactors, as a Galois theory for inclusions of von Neumann algebras. Subfactor theory quickly became one of the most flourishing branches of operator algebra theory, with a multitude of deep connections in knot theory, representation theory, 3-manifolds, quantum groups, integrable systems in statistical mechanics and conformal field theory. A subfactor can be viewed as a group-like object that encodes what one might call the generalized symmetries of a quantum physical or mathematical situation. To decode this information, one computes the higher relative commutants, a system of inclusions of finite dimensional C*-algebras naturally associated to the subfactor. This object, called the standard invariant, has an extraordinarily rich algebraic-combinatorial structure, generalizing finitely generated groups, finite dimensional Hopf C*-algebras and other large classes of quantum groups.