C*-algebra approach to the index theory of boundary value problems (original) (raw)

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 operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel [5] 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 Atiyah-Singer topological index map, applied to this class, then furnishes the index of the operator. Based on this result, Fedosov, Rempel-Schulze and Grubb have given index formulas in terms of the symbols. In this paper we survey how C * -algebra K-theory, as initiated in , can be used to give a proof of Boutet de Monvel's index theorem for boundary value problems, a task carried out in , and how the same techniques yield an index theorem for families of Boutet de Monvel operators, detailed in . The key ingredient of our approach is a precise description of the K-theory of the kernel and of the image of the boundary symbol.