Model theory and operator algebras

模型理论和算子代数

• 批准号: RGPIN-2018-05445
• 负责人: Hart, Bradd
• 金额: $ 1.68万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=653119
• 关键词: Model; theory; operator; algebras

Model theory is a branch of mathematical logic which has had success in providing novel attacks on problems in other branches of mathematics. Generally speaking, to study a mathematical object model theoretically, one considers not the object in question but the theory of the object in a suitably chosen language. By studying all the models of this theory, it is sometimes possible to draw conclusions about the original structure. Recently, with the development of continuous model theory, applications to certain branches of analysis which were previously out of reach for model theory have become more tractable. In this proposal, I suggest a systematic model theoretic study of operator algebras based on recent success of work of myself and co-authors in settling questions related to ultrapowers of separable algebras. There seems to be the intriguing possibility that model theory can provide assistance with problems related to questions about von Neumann algebras and a critical classes of C*-algebras such as nuclear C*-algebras. The short term objective is to widen the understanding of model theoretic methods in application areas but in the long run, with a better understanding of the techniques from both fields, it will be possible to prove significant results on the borders of two of Canada's strong research communities.

模型论是数学逻辑的一个分支，它已经成功地为其他数学分支中的问题提供了新的攻击。一般来说，要从理论上研究数学对象模型，人们考虑的不是所讨论的对象，而是用适当选择的语言研究对象的理论。通过研究这一理论的所有模型，有时可以得出关于原始结构的结论。近年来，随着连续模型理论的发展，一些以前模型理论无法触及的分析分支的应用变得更加容易。在这个提议中，我建议基于我和合作者最近在解决可分离代数的超幂相关问题上的成功工作，对算子代数进行系统的模型理论研究。似乎有一种有趣的可能性，即模型理论可以帮助解决与冯·诺伊曼代数和一类关键的C*代数（如核C*代数）有关的问题。短期目标是扩大对模型理论方法在应用领域的理解，但从长远来看，更好地理解这两个领域的技术，将有可能在加拿大两个强大的研究社区的边界上证明重要的结果。