Descent of Hilbert C*-modules

TL;DR

This paper explores how tensoring with a Hilbert C*-module induces a functor between module categories and characterizes its image using coalgebra coactions, with applications in representation theory and operator algebras.

Contribution

It introduces a framework linking tensor functors from Hilbert C*-modules to coalgebra coactions, extending understanding of module transformations in operator algebra contexts.

Findings

The image of the tensor functor can be described via coalgebra coactions under certain conditions.

Examples include parabolic induction, Hermitian connections, and Fourier algebras.

The framework applies to maximal C*-dilations of operator modules.

Abstract

Let F be a right Hilbert C*-module over a C*-algebra B, and suppose that F is equipped with a left action, by compact operators, of a second C*-algebra A. Tensor product with F gives a functor from Hilbert C*-modules over A to Hilbert C*-modules over B. We prove that under certain conditions (which are always satisfied if, for instance, A is nuclear), the image of this functor can be described in terms of coactions of a certain coalgebra canonically associated to F. We then discuss several examples that fit into this framework: parabolic induction of tempered group representations; Hermitian connections on Hilbert C*-modules; Fourier (co)algebras of compact groups; and the maximal C*-dilation of operator modules over non-self-adjoint operator algebras.