Approaching some open problems in algebraic combinatorics and in C*-algebras theory using von Neumann algebras.

本文的研究方向是算子代数，可以看作是对C*-代数和von Neumann代数的研究。这两种代数都是在20年代末和30年代初引入的。 该建议部分地致力于研究非同构于其反代数的C~*-代数上的一些公开问题。这项研究的基本原理是，要构造一个具有某些所需性质的C*-代数的例子，可以在von Neumann代数环境中工作，而不是在C*-代数环境中工作。我打算研究的具体问题涉及在C*-代数理论中起主要作用的概念，即核性和纯无限C*-代数。通过这个提议，我打算通过给出一些以前未知的C*-代数的例子，来为Elliott的C*-代数的分类程序提供一些见解。此外，我还计划研究一些与自由量子正交群和量子置换群有关的公开问题。自由量子群是最近的一个研究领域，至今仍是相当神秘的天体。由于自由量子群及其von Neumann代数在数学物理中有重要的应用，本工作的目的是更好地理解与自由量子群相关的von Neumann代数的性质和结构。最后，我计划用子因子理论证明代数组合数学中的一个猜想。近几年来，人们对拟对称函数越来越感兴趣。拟对称函数是指在多项式环上对称群的某一作用下不变的函数。这个作用诱导了Temperley-Lieb代数在多项式环上的忠实作用。该猜想声称Temperley-Lieb代数的作用存在唯一的推广到Fuss-Catalan代数的忠实作用。解决这个猜想将是研究在Fuss-Catalan代数作用下不变的函数的起点。希望这项研究能像对拟对称函数的研究一样有趣和丰富。

The research proposal is in the area of Operator Algebras, which can be viewed as the study of C*-algebras and von Neumann algebras. Both kinds of algebras were introduced between the late '20s and the early '30s. The proposal is partly devoted to the investigation of some open questions on C*-algebras non-isomorphic to their opposite algebra. The underlying philosophy of the proposed research is that to construct an example of a C*-algebra with some desired property, one can work in a von Neumann algebra setting instead of a C*-algebra setting. The specific problems that I intend to investigate deal with concepts that play a major role in the theory of C*-algebras, namely nuclearity and purely infinite C*-algebras. With this proposal I intend to provide some insights into Elliott's classification program for C*-algebras, by producing examples of C*-algebras which were not known before. In addition, I plan to investigate some open problems related to free quantum orthogonal groups and quantum permutation groups. Free quantum group are a recent area of research, and are still quite mysterious objects. The objective of this work is to gain a better understanding of the properties and structure of the von Neumann algebras associated to free quantume groups, since free quantum groups and their von Neumann algebras have important applications to mathematical physics. Lastly, I plan to prove a conjecture in algebraic combinatorics using subfactor theory. There has been an increased interest in recent years in quasi-symmetric functions. Quasi-symmetric functions are functions which are invariants under a certain action of the symmetric group on the ring of polynomials. This action induces a faithful action of the Temperley- Lieb algebra on the ring of polynomials. The conjecture claims that there is a unique extensions of the action of the Temperley-Lieb algebra to a faithful action of the Fuss-Catalan algebra. Solving this conjecture will be the starting point of the study of functions which are invariant under the action of the Fuss-Catalan algebra. Hopefully this study will be as interesting and as rich in results as the study of quasi-symmetric functions has been.