C*Algebra approach to the index theory of boundary value problems
Abstract
Boutet de Monvel's calculus provides a pseudodifferential framework which encompasses the classical differential boundary value problems. In an extension of the concept of Lopatinski and Shapiro, it associates to each operator two symbols: a pseudodifferential principal symbol, which is a bundle homomorphism, and an operatorvalued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms. The crucial observation is that elliptic operators can be mapped to compactly supported $K$theory classes on the cotangent bundle over the interior of the manifold. The AtiyahSinger topological index map, applied to this class, then furnishes the index of the operator. Based on this result, Fedosov, RempelSchulze and Grubb have given index formulas in terms of the symbols. In this paper we survey previous work how C*algebra Ktheory can be used to give a proof of Boutet de Monvel's index theorem for boundary value problems, and how the same techniques yield an index theorem for families of Boutet de Monvel operators. The key ingredient of our approach is a precise description of the Ktheory of the kernel and of the image of the boundary symbol.
 Publication:

arXiv eprints
 Pub Date:
 March 2012
 arXiv:
 arXiv:1203.5649
 Bibcode:
 2012arXiv1203.5649M
 Keywords:

 Mathematics  KTheory and Homology;
 Mathematics  Analysis of PDEs;
 Mathematics  Operator Algebras;
 19K56;
 46L80;
 58J32
 EPrint:
 17 pages, submitted to the "Rosenberg proceedings"