K ‐theory of the Boutet de Monvel algebra with classical SG‐symbols on the half space

We compute the K ‐groups of the C * ‐algebra of bounded operators generated by the Boutet de Monvel operators with classical SG‐symbols of order (0,0) and type 0 on 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 ofR + n × R nandR n − 1 × 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 * ‐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.