Langlands Reciprocity for C ∗-Algebras

We introduce a $C^*$-algebra $\mathscr{A}_V$ of variety $V$ over the number field $K$ and $\mathscr{A}_G$ reductive group $G$ ring adeles $K$. Using Pimsner's Theorem we construct an embedding $\mathscr{A}_V\hookrightarrow \mathscr{A}_G$, where is $G$-coherent variety, e.g. Shimura $G$. The analog Langlands reciprocity for $C^*$-algebras. It follows from $K$-theory inclusion $\mathscr{A}_V\subset\mathscr{A}_G$ that Hasse-Weil $L$-function product automorphic $L$-functions corresponding to irreducible representations