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

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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. *****