The Lefschetz fixed point theorem follows easily from the identification of the Lefschetz number with the fixed point index. This identification is a consequence of the functoriality of the trace in symmetric monoidal categories. There are refinements of the Lefschetz number and the fixed point index that give a converse to the Lefschetz fixed point theorem. An important part of this theorem is...

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\subs...