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 ad
apt 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.
We compute the Chern-Connes character (a map from the $K$-theory of a C$^*$-algebra under the action of a Lie group to the cohomology of its Lie algebra) for the $L^2$-norm closure of the algebra of all classical zero-order pseudodifferential operato
rs on the sphere under the canonical action of ${rm SO}(3)$. We show that its image is $mathbb{R}$ if the trace is the integral of the principal symbol.
Given a separable unital C*-algebra A, let E denote the Banach-space completion of the A-valued Schwartz space on Rn with norm induced by the A-valued inner product $<f,g>=int f(x)^*g(x) dx$. The assignment of the pseudodifferential operator B=b(x,D)
with A-valued symbol b(x,xi) to each smooth function with bounded derivatives b defines an injective mapping O, from the set of all such symbols to the set of all operators with smooth orbit under the canonical action of the Heisenberg group on the algebra of all adjointable operators on the Hilbert module E. It is known that O is surjective if A is commutative. In this paper, we show that, if O is surjective for A, then it is also surjective for the algebra of k-by-k matrices with entries in A.
