$K$-Theory of Boutet de Monvel algebras 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 $mathbb{R}_{+}^{n}$, as defined by Schrohe, Kapanadze and Schulze. In order to adapt the techniques used in Melo, Nest, Schick and Schrohes work on the K-theory of Boutet de Monvels algebra on compact manifolds, we regard the symbols as functions defined on the radial compactifications of $mathbb{R}_{+}^{n}timesmathbb{R}^{n}$ and $mathbb{R}^{n-1}timesmathbb{R}^{n-1}$. 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.