Details
| Original language | English |
|---|---|
| Pages (from-to) | 2645-2692 |
| Number of pages | 48 |
| Journal | Journal of functional analysis |
| Volume | 257 |
| Issue number | 8 |
| Publication status | Published - 15 Oct 2009 |
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 T- X generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field Cr* (T- X) of C*-algebras over [0, 1]. Its fiber in ℏ = 0, Cr* (T- X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for ℏ ≠ 0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K0 (Cr* (T- X)) = K0 (C0 (T* X)) → K0 (K) = Z. 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.
Keywords
- Boundary value problems, Continuous fields of C-algebras, Groupoids, Index theory
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
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In: Journal of functional analysis, Vol. 257, No. 8, 15.10.2009, p. 2645-2692.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Index theory for boundary value problems via continuous fields of C*-algebras
AU - Aastrup, Johannes
AU - Nest, Ryszard
AU - Schrohe, Elmar
N1 - Funding Information: J. Aastrup and E. Schrohe gratefully acknowledge the support of Deutsche Forschungsge-meinschaft (DFG). Copyright: Copyright 2009 Elsevier B.V., All rights reserved.
PY - 2009/10/15
Y1 - 2009/10/15
N2 - 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 T- X generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field Cr* (T- X) of C*-algebras over [0, 1]. Its fiber in ℏ = 0, Cr* (T- X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for ℏ ≠ 0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K0 (Cr* (T- X)) = K0 (C0 (T* X)) → K0 (K) = Z. 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.
AB - 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 T- X generalizing the tangent groupoid defined by Connes in the boundaryless case, and an associated continuous field Cr* (T- X) of C*-algebras over [0, 1]. Its fiber in ℏ = 0, Cr* (T- X), can be identified with the symbol algebra for Boutet de Monvel's calculus; for ℏ ≠ 0 the fibers are isomorphic to the algebra K of compact operators. We therefore obtain a natural map K0 (Cr* (T- X)) = K0 (C0 (T* X)) → K0 (K) = Z. 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.
KW - Boundary value problems
KW - Continuous fields of C-algebras
KW - Groupoids
KW - Index theory
UR - http://www.scopus.com/inward/record.url?scp=68949184855&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2009.04.019
DO - 10.1016/j.jfa.2009.04.019
M3 - Article
AN - SCOPUS:68949184855
VL - 257
SP - 2645
EP - 2692
JO - Journal of functional analysis
JF - Journal of functional analysis
SN - 0022-1236
IS - 8
ER -