Automorphic $C^{\ast }$-algebras of reductive groups

Let $G$ be a reductive group over a number field $F$. We define $C_{\mathrm{a}\mathrm{u}\mathrm{t}}^{\ast }(G(\mathbb{A}))$, the automorphic $C^{\ast }$-algebra of $G$, to be the $C^{\ast }$-algebraic image of the full automorphic representation of $G(\mathbb{A})$ on $L^2(G(F)\mathrm{\backslash }G(\mathbb{A}))$. Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective $\ast$-homomorphism $$C_{\mathrm{a}\mathrm{u}\mathrm{t}}^{\ast }(G(\mathbb{A}))⟶\underset{[M,\sigma ]}{⨁}{K}_{C_0(\widehat{A_M})}{({\mathrm{Ind}}_P^G{C}_0(\widehat{A_M},H_{M,\sigma }))}^{W(G,M,\sigma )},$$ where $[M,\sigma ]$ ranges over the associate classes of discrete Levi data. When $G$ is $\mathrm{GL}(n)$ or an inner form of it, we prove that this map is an isomorphism of $C^{\ast }$-algebras.