TY - GEN

T1 - C*-Algebra approach to the index theory of boundary value problems

AU - Melo, Severino

AU - Schrohe, Elmar

AU - Schick, Thomas

N1 - 17 pages, submitted to the "Rosenberg proceedings"

PY - 2012

Y1 - 2012

N2 - 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 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 previous work how C*-algebra K-theory 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 K-theory of the kernel and of the image of the boundary symbol.

AB - 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 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 previous work how C*-algebra K-theory 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 K-theory of the kernel and of the image of the boundary symbol.

KW - math.KT

KW - math.AP

KW - math.OA

KW - 19K56, 46L80, 58J32

U2 - 10.1090/conm/584/11587

DO - 10.1090/conm/584/11587

M3 - Conference contribution

SN - 0-8218-9144-8

T3 - Contemporary mathematics

BT - Analysis, geometry and quantum field theory

T2 - International Conference in Honor of Steve Rosenberg's 60th Birthday

Y2 - 26 September 2011 through 30 September 2011

ER -