$K$-Theory of Boutet de Monvel algebras with classical SG-symbols on the half space

Abstract

We compute the $K$-groups of the $C^{\ast }$-algebra of bounded operators generated by the Boutet de Monvel operators with classical SG-symbols of order (0,0) and type 0 on ${\mathbb{R}}_{+}^n$, as defined by Schrohe, Kapanadze and Schulze. In order to adapt the techniques used in Melo, Nest, Schick and Schrohe's work on the K-theory of Boutet de Monvel's algebra on compact manifolds, we regard the symbols as functions defined on the radial compactifications of ${\mathbb{R}}_{+}^n\times {\mathbb{R}}^n$ and ${\mathbb{R}}^{n-1}\times {\mathbb{R}}^{n-1}$. This allows us to give useful descriptions of the kernel and the image of the continuous extension of the boundary principal symbol map, which defines a $C^{\ast }$-algebra homomorphism. We are then able to compute the $K$-groups of the algebra using the standard K-theory six-term cyclic exact sequence associated to that homomorphism.

References in corpus (2)

• Families index for Boutet de Monvel operators
• Non-commutative residue of projections in Boutet de Monvel's calculus