Index theory for boundary value problems viacontinuous fields of C~*-algebras

摘要

We prove an index theorem for boundary value problems in Boutet de Monvel's calculus on a compactmanifold X with boundary. The basic tool is the tangent semi-groupoid J– X generalizing the tangentgroupoid defined by Connes in the boundaryless case, and an associated continuous field C_r~*(J-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 deMonvel's calculus; for h≠0 the fibers are isomorphic to the algebra K of compact operators. We thereforeobtain a natural map K_0(C_r~*(T-X)) K_0(C_T~*X))→k) = Z. Using deformation theory we showthat this is the analytic index map. On the other hand, using ideas from noncommutative geometry, weconstruct the topological index map and prove that it coincides with the analytic index map.