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

研究计划是在算子代数和泛函分析领域。它处理了一类特殊的巴拿赫代数，以及C*代数和冯·诺伊曼代数。该提案部分致力于研究Lp算子代数中的几个开放问题。Lp算子代数是几年前才被引入的，它们的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，在某种意义上它们类似于C*-代数。为了使Lp算子代数表现良好，出现了一些可能的限制条件。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可服从作用和可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有关（表现良好的Lp算子代数，C*代数技术的使用等），它们将极大地推动Lp算子代数理论的发展。此外，我计划研究一些与q变形自由群因子有关的开放性问题。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，并证明了在q值很小的情况下，这些C*代数与Cuntz代数的紧化算子的扩展之间存在同构。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，但在von Neumann代数情况下q的区间比C*-代数的区间小得多。我打算研究是否可以使用Dykema和Nica使用的C*-代数技术来获得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。目的是提供对q变形自由群因子的更好理解，并探索C*代数技术是否可以用于改进在冯·诺伊曼代数设置中获得的结果。最后，我将使用冯·诺伊曼代数技术构造一个简单的可分离C*-代数，满足涉及代数的自同构及其k群的自同构的某些性质。使用冯·诺伊曼代数的思想和技术来解决C*代数中的问题是最近才出现的，但我已经在以前的工作中使用了这种方法。这种构造将产生具有期望性质的C*-代数的例子，这些性质是以前不知道的，并且可能为埃利奥特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.