C*-Structure and K-Theory of Boutet de Monvel's Algebra

We consider the norm closure $A$ of the algebra of all operators of order and class zero in Boutet de Monvel's calculus on a manifold $X$ with boundary $Y$. We first describe the image and the kernel of the continuous extension of the boundary principal symbol to $A$. If the $X$ is connected and $Y$ is not empty, we then show that the K-groups of $A$ are topologically determined. In case the manifold, its boundary and the tangent space of the interior have torsion-free K-theory, we prove that $K_i(A/K)$ is isomorphic to the direct sum of $K_i(C(X))$ and $K_{1-i}(C_0(TX'))$, for i=0,1, with $K$ denoting the compact ideal and $TX'$ the tangent bundle of the interior of $X$. Using Boutet de Monvel's index theorem, we also prove this result for i=1 without assuming the torsion-free hypothesis. We also give a composition sequence for $A$.