  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.
