Subfactors and Noncommutative Ergodic Theory

Nicoara研究的两个主要方向之一是子因子的不变量研究，主要是所谓的交换平方。这些是有限维C*代数包含的平方，自然出现在子因子的标准不变量中。交换平方也可用作子因子的构造数据，并通过这种方法得到了子因子最显式的例子。通过结合使用代数组合方法和解析方法，PI证明了交换平方的几个有限性结果，并发现了这些对象的素数（在孤立意义上）的一个很好的概念。得到的分离结果为构造非同构子因子的单参数族提供了方法。PI将继续研究这样的构造，特别是那些基于复Hadamard矩阵的交换平方的构造。平面代数技术将用于理解这些模型，以及由交换平方构造的其他子因子。Nicoara的另一个研究方向是从非交换遍历理论的角度研究冯·诺伊曼代数，特别是刚性在冯·诺伊曼代数中的应用。在20世纪30年代，约翰·冯·诺伊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，像海森堡测不准原理这样的基本自然定律是冯·诺伊曼抽象理论的自然结果。在80年代早期，Vaughan Jones引入了子因子理论，作为包含冯诺依曼代数的伽罗瓦理论。子因子理论迅速成为算子代数理论中最繁荣的分支之一，与结论、表示论、3流形、量子群、统计力学中的可积系统和共形场论有着许多深刻的联系。子因子可以被看作是一个类群对象，它编码了量子物理或数学情况下的广义对称性。为了解码这些信息，人们计算更高的相对交换子，这是一个包含有限维C*代数的系统，自然地与子因子相关。这个对象被称为标准不变量，具有非常丰富的代数组合结构，推广了有限生成群、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.