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A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems

Severino T. Melo·Thomas Schick·Elmar Schrohe
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Abstract

We study the C*-closure A of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a compact connected manifold X with non-empty boundary. We find short exact sequences in K-theory 0->K_i(C(X))->K_i(A/K)->K_{1-i}(C_0(T*X'))->0, i= 0,1, which split, where K denotes the compact ideal and T*X' the cotangent bundle of the interior of X. Using only simple K-theoretic arguments and the Atiyah-Singer Index Theorem, we show that the Fredholm index of an elliptic element in A is given as the composition of the topological index with mapping K_1(A/K)->K_0(C_0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.

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Severino T. Melo, Thomas Schick, Elmar Schrohe. 2005-10-11. A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems. https://doi.org/10.1515/crelle.2006.083

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