Operator Algebras Associated to Groups and Noncommutative Convexity

从冯·诺伊曼关于量子物理学数学基础的工作开始，数学家们发现，将算子代数理论中出现的各种结构视为经典数学对象的非交换对立物是有益的。这种哲学是我的工作的基础，它跨越了算子代数领域内的广泛主题。我最近最重要的工作集中在两个领域：与群相关的算子代数的结构和非交换凸集的结构。我的建议的第一个主要组成部分涉及与群和动力系统相关的算子代数的结构。一方面，算子代数的群论构造提供了一个有价值的例子和问题来源。另一方面，越来越清楚的是，关于群和动力系统的解析结构的许多问题最自然地是在算子-代数框架内进行研究。在我的研究中考虑的基本问题是如何将群或动力系统的性质与相应的算子代数的性质联系起来。我的建议的第二个主要组成部分是关于非交换凸性的理论。1969年，在无限维空间中凸集结构的开创性工作的激励下，Arveson提出了一个非交换凸性理论，作为研究非交换数学产生的对象的框架。尽管这些想法具有巨大的潜力，但它们多年来一直没有得到发展，直到最近的突破，第一个是由Arveson本人提出的，第二个是由现任作者与k·r·戴维森合作提出的。我计划继续发展这些想法，这些想法已经在量子信息论、群论和半定优化等领域的问题上取得了巨大的成功。

Beginning with the work of von Neumann on the mathematical foundations of quantum physics, mathematicians have found it profitable to view various structures arising in the theory of operator algebras as noncommutative counterparts of classical mathematical objects. This philosophy underlies my work, which spans a wide range of topics within the field of operator algebras. My most significant recent work is concentrated in two areas: the structure of operator algebras associated to groups and the structure of noncommutative convex sets.The first major component of my proposal concerns the structure of operator algebras associated to groups and dynamical systems. On the one hand, group-theoretic constructions of operator algebras provide a valuable source of examples and questions. On the other hand, it is becoming increasingly clear that many problems about the analytic structure of groups and dynamical systems are most naturally studied within an operator-algebraic framework. The fundamental problem considered in my research is how to relate properties of the group or dynamical system to properties of the corresponding operator algebra.The second major component of my proposal concerns the theory of noncommutative convexity. In 1969, motivated by groundbreaking work on the structure of convex sets in infinite dimensional spaces, Arveson proposed a theory of noncommutative convexity as a framework for the study of objects arising from noncommutative mathematics. Despite the enormous potential of these ideas, they went undeveloped for many years until recent breakthroughs, the first by Arveson himself and the second by current author in joint work with K.R. Davidson. I plan to continue developing these ideas, which have already been applied with great success to problems in fields like quantum information theory, group theory and semidefinite optimization.