L^p operator algebras that look like C*algebras and q-deformed free group factors

L^p 算子代数看起来像 C* 代数和 q 变形自由群因子

• 批准号: RGPIN-2019-06513
• 负责人: Viola, MariaGrazia
• 金额: $ 1.09万
• 依托单位: Lakehead University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=675578
• 关键词: operator; algebras; look; like; deformed

The research proposal is in the area of Operator Algebras and Functional Analysis. It deals with a special class of Banach algebras, as well as with C*-algebras and von Neumann algebras. ***The proposal is partly devoted to the investigation of several open questions in Lp operator algebras. Lp operator algebras were introduced only a few years ago, and their theory, although very rich and interesting, is still in development. One of the main problems in the area is to characterize Lp operator algebras that are well-behaved, in the sense that they resemble C*-algebras. A few conditions have emerged as possible restrictions to impose on an Lp operator algebra for it to be well-behaved. I intend to explore some of these criteria, and assess whether some classes of Lp operator algebras, which are known to be well-behaved, satisfy such conditions. I also plan to determine whether it is possible to generalize the results of Kalantar and Kennedy on the simplicity of group C*-algebras to Lp group algebras, and to introduce the notion of amenable actions and amenable equivalence relations in the context of Lp operator algebras. The specific problems that I intend to investigate are connected to ideas that play a major role in the theory of Lp operator algebras (well-behaved Lp operator algebras, use of C*-algebra techniques, etc.), and they will greatly advance the development of the theory of Lp operator algebras.***In addition I plan to investigate some open questions related to q-deformed free group factors. Dykema and Nica studied in the '90s the C*-algebra analogs of the q-deformed free group factor, and showed the existence, for small values of q, of an isomorphism between these C*-algebras and the extensions by the compact operators of the Cuntz algebras. Recently, an isomorphism result was also proved for q-deformed free group factors using free probability methods, but the interval obtained for q in the von Neumann algebra case is much smaller than the one provided for the C*-algebras. I intend to investigate if the C*-algebra techniques used by Dykema and Nica could be used to obtain an isomorphism result for q-deformed free group factors, and if the interval where such an isomorphism exists could be enlarged. The objective is to provide a better understanding of q-deformed free group factors, and explore whether C*-algebra techniques could be used to improve a result obtained in the von Neumann algebra setting. ***Lastly, I will use von Neumann algebra techniques to construct a simple separable C*-algebra satisfying a certain property involving the automorphisms of the algebra and the automorphisms of its K-groups. The use of von Neumann algebra ideas and techniques to address problems in C*-algebras is quite recent, but I have already used this approach in previous work. The construction would produce examples of C*-algebras with desired properties that were not known before, and may provide some insight into the Elliott classification program. *****

该研究计划属于算子代数和泛函分析领域。它涉及一类特殊的 Banach 代数，以及 C* 代数和冯诺依曼代数。 ***该提案部分致力于研究 Lp 算子代数中的几个开放问题。 Lp 算子代数几年前才被引入，其理论虽然非常丰富和有趣，但仍在发展中。该领域的主要问题之一是表征表现良好的 Lp 算子代数，因为它们类似于 C* 代数。一些条件已经出现，可以作为对 Lp 算子代数施加的可能限制，以使其表现良好。我打算探索其中一些标准，并评估某些已知表现良好的 Lp 算子代数类是否满足这些条件。我还计划确定是否可以将 Kalantar 和 Kennedy 关于 C* 群代数的简单性的结果推广到 Lp 群代数，并在 Lp 算子代数的背景下引入顺应作用和顺应等价关系的概念。我打算研究的具体问题与 Lp 算子代数理论中发挥重要作用的思想有关（良好的 Lp 算子代数、C* 代数技术的使用等），它们将极大地推进 Lp 算子代数理论的发展。 Lp 算子代数理论的发展。***此外，我计划研究一些与 q 变形自由群因子相关的开放问题。 Dykema 和 Nica 在 90 年代研究了 q 变形自由群因子的 C* 代数类似物，并表明，对于较小的 q 值，这些 C* 代数与紧算子的扩展之间存在同构Cuntz 代数。最近，利用自由概率方法也证明了q变形自由群因子的同构结果，但是在冯·诺依曼代数情况下为q获得的区间比为C*-代数提供的区间小得多。我打算研究Dykema和Nica使用的C*代数技术是否可以用于获得q变形自由群因子的同构结果，以及是否可以扩大这种同构存在的区间。目的是更好地理解 q 变形自由群因子，并探索 C* 代数技术是否可用于改进在冯诺依曼代数设置中获得的结果。 ***最后，我将使用冯·诺依曼代数技术构造一个简单的可分离 C* 代数，满足涉及代数自同构及其 K 群自同构的特定属性。使用冯诺依曼代数思想和技术来解决 C* 代数中的问题是最近才出现的，但我已经在之前的工作中使用过这种方法。该构造将产生具有以前未知的所需属性的 C* 代数示例，并且可以为艾略特分类程序提供一些见解。 *****