C ~*-代数连续域上边值问题的指标理论
Index Theory for Boundary Value Problems via Continuous Fields of C*-algebras
摘要：
证明了紧流形X上Boutet de Monvel微积分中边值问题的一个指标定理。基本工具是切半群$\cT^-X$，它推广了Connes在无边界情形下定义的切群胚，以及[0，1]上C*-代数的一个相关连续域C* r（\cT^- X）.它在h=0中的纤维C*_r（T^-X）可以用Boutet de Monvel演算的符号代数来表示;当h \not =0时，纤维同构于紧算子的代数K。因此我们得到一个自然映射K_0（C*_r（T^-X））=K_0（C_0（T*X））-> K_0（K）=Z。利用变形理论，我们表明，这是解析指数映射。另一方面，利用非交换几何的思想，构造了拓扑指数映射，并证明了它与解析指数映射是一致的。
英文摘要：
We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compact manifold X with boundary. The basic tool is the tangent semigroupoid $\cT^-X$ generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field C*_r(\cT^-X) of C*-algebras over [0,1]. Its fiber in h=0, C*_r(T^-X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for h\not=0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K_0(C*_r(T^-X))=K_0(C_0(T*X)) -> K_0(K)=Z. Using deformation theory we show that this is the analytic index map. On the other hand, using ideas from noncommutative geometry, we construct the topological index map and prove that it coincides with the analytic index map.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
