Index theory for boundary value problems via continuous fields of C^*-algebras

authored by

Johannes Aastrup, Ryszard Nest, Elmar Schrohe

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 C_r^* (T^- X) of C^*-algebras over [0, 1]. Its fiber in ℏ = 0, C_r^* (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 (C_r^* (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.

Organisation(s)

Institute of Analysis

External Organisation(s)

University of Copenhagen
University of Münster

Type

Article

Journal

Journal of functional analysis

Volume

257

Pages

2645-2692

No. of pages

48

ISSN

0022-1236

Publication date

15.10.2009

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Analysis

Electronic version(s)

https://doi.org/10.1016/j.jfa.2009.04.019 (Access: Embargoed)