Index theory for boundary value problems via continuous fields of -algebras |
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Authors: | Johannes Aastrup Ryszard Nest Elmar Schrohe |
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Institution: | aSFB 478 “Geometrische Strukturen”, Hittorfstrasse 27, 48149 Münster, Germany;bDepartment of Mathematics, Copenhagen University, Universitetsparken 5, 2100 Copenhagen, Denmark;cInstitut für Analysis, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany |
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Abstract: | 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 semi-groupoid generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field of C*-algebras over 0,1]. Its fiber in =0, , can be identified with the symbol algebra for Boutet de Monvel's calculus; for ≠0 the fibers are isomorphic to the algebra of compact operators. We therefore obtain a natural map . 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. |
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Keywords: | Index theory Boundary value problems Continuous fields of color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6WJJ-4WGHJT9-1&_mathId=mml11&_user=10&_cdi=6880&_rdoc=9&_acct=C000054348&_version=1&_userid=3837164&md5=f3bd925c35687953a796ee30bb1f0747" title="Click to view the MathML source" C*-algebras" target="_blank">alt="Click to view the MathML source">C*-algebras Groupoids |
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