Langlands reciprocity for C*-algebras

Abstract

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

to appear in Operator Theory: Advances and Applications

References in corpus (3)

• Comments on a note by M. Waldschmidt
• On a symmetry of complex and real multiplication
• On traces of Frobenius endomorphisms

Cited by in corpus (1)

• K-theory of rational quadratic forms