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

研究方案是在算子代数和泛函分析领域。它研究了一类特殊的Banach代数，以及C*-代数和von Neumann代数。*该方案部分地致力于研究LP算子代数中的几个公开问题。LP算子代数在几年前才被引入，它们的理论虽然非常丰富和有趣，但仍在发展中。这一领域的主要问题之一是刻画表现良好的LP算子代数，因为它们类似于C*-代数。一些条件已经作为可能的限制条件出现，以便对LP算子代数施加良好行为。我打算探索其中的一些准则，并评估某些已知表现良好的LP算子代数是否满足这些条件。我还计划确定是否有可能将Kalantar和Kennedy关于群C*-代数的单性的结果推广到Lp群代数，并在Lp算子代数的背景下引入服从作用和服从等价关系的概念。我打算研究的具体问题与在LP算子代数理论中起主要作用的思想(行为良好的LP算子代数，C*-代数技巧的使用等)有关，它们将极大地推动LP算子代数理论的发展。*此外，我还计划研究一些与q-变形自由群因子有关的公开问题。Dykema和Nica在90年代研究了Q-变形自由群因子的C*-代数类似，证明了对于较小的Q值，这些C*-代数之间存在同构，并证明了Cuntz代数的紧算子的扩张。最近，也用自由概率方法证明了q-变形自由群因子的同构结果，但在von Neumann代数情形下得到的q的区间比C*-代数的区间小得多。我打算调查Dykema和Nica使用的C*-代数技巧是否可以用来获得q-变形自由群因子的同构结果，以及这种同构存在的区间是否可以扩大。其目的是为了更好地理解q-变形自由群因子，并探索C*-代数技术是否可以用来改进在von Neumann代数环境下得到的结果。*最后，我将利用von Neumann代数技巧来构造一个简单的可分C*-代数，它满足代数的自同构及其K-群的自同构的某些性质。使用von Neumann代数的思想和技巧来解决C*-代数中的问题是相当新的，但我已经在以前的工作中使用过这种方法。这种构造将产生具有所需性质的C*-代数的例子，这些性质在以前是未知的，并且可以为Elliott分类程序提供一些见解。*****

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. *****