Adelic C*-correspondences and parabolic induction

TL;DR

This paper develops a framework for constructing global C*-correspondences from local data, specifically capturing adelic parabolic induction, by using restricted tensor products of Hilbert C*-modules and correspondences.

Contribution

It introduces a novel method to build global C*-correspondences from local ones, advancing the understanding of adelic structures in operator algebras.

Findings

Constructs global C*-correspondences from local collections.

Provides a new perspective on adelic parabolic induction.

Establishes compatibility conditions for tensor products.

Abstract

In analogy with the construction of representations of adelic groups as restricted products of representations of local groups, we study restricted tensor products of Hilbert C*-modules and of C*-correspondences. The construction produces global C*-correspondences from compatible collections of local C*-correspondences. When applied to the collection of C*-correspondences capturing local parabolic induction, the construction produces a global C*-correspondence that captures adelic parabolic induction.