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For a real reductive group $G$, we investigate the structure of the Casselmanalgebra $\mathcal{S}(G)$ and its similarities to the structure of the reducedgroup $C^*$-algebra $C_r^*(G)$. We demonstrate that the two algebras areassembled from very similar elementary components in a compatible way. Inparticular, we prove that the two algebras have the same $K$-theory whenrestricted to a finite set of $K$-types, which is a refinement of theConnes-Kasparov isomorphism.

Compatible Decomposition of the Casselman Algebra and the Reduced Group C*-algebra of a Real Reductive Group