套代数中的指标理论

Hilbert空间Fredholm算子的指标理论是算子理论和算子代数的重要组成部分，并在现代数学的很多领域(非交换几何、大范围分析、K-理论、微分方程、函数论等)有着广泛和重要的应用。近年来，非自伴算子代数理论不断的发展和深入，它的应用也日益广泛。自然地，非自伴算子代数中的指标理论需要得到深入的研究。在本课题中，我们将研究典型的非自伴算子代数- - 套代数的指标理论。我们拟从"指标是否是套代数中Fredholm算子的完全同伦不变量"这一核心问题入手，研究套代数中的指标理论的相关问题并探求套代数内部更为深入的拓扑与几何结构，期望能够建立套代数与拓扑、几何间更密切的联系。

The index theory of Fredholm operators in Hilbert space is an important part of operator theory and operator algebras, and nowadays plays a key role in many areas of modern mathematics (noncommutative geometry, large-scale analysis, K-theory, differential equations, function theory, etc.). Recently,the theory of non-selfadjoint operator algebras has undergone a vigorous development,and its applications are becoming more and more widespread. Naturally, the index theory in non-selfadjoint operator algebras need to be studied deeply. In this project, we will study the index theory in nest algebra, which is a kind of typical non-selfadjoint operator algebras. We will focus on the key problem "Whether the Fredholm index constitutes a complete systems of homotopy invariants for all Fredholm operators in a nest algebra" and study some related problems. We will also explore some fine topological and geometrical structures of nest algebras, and expect to establish a close connection between topology, geometry and nest algebras.