A K-Theoretic Proof of Boutet de Monvel's Index Theorem for Boundary Value Problems
Severino T. Melo, Thomas Schick, Elmar Schrohe
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->Ki(C(X))->Ki(A/K)->K1-i(C0(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 K1(A/K)->K0(C0(T*X')) defined above. This relation was first established by Boutet de Monvel by different methods.
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