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ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT $C^{\ast }$-MODULES

Part of: Selfadjoint operator algebras General theory of categories and functors

Published online by Cambridge University Press:  30 June 2016

Pierre Clare ,
Tyrone Crisp  and
Nigel Higson
Pierre Clare
Affiliation:
Department of Mathematics, Dartmouth College, HB 6188, Hanover, NH 03755, USA (clare@math.dartmouth.edu)
Tyrone Crisp
Affiliation:
Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany (tyronecrisp@mpim-bonn.mpg.de)
Nigel Higson
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA (higson@math.psu.edu)
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Abstract

Let $E$ be a (right) Hilbert module over a $C^{\ast }$-algebra $A$. If $E$ is equipped with a left action of a second $C^{\ast }$-algebra $B$, then tensor product with $E$ gives rise to a functor from the category of Hilbert $B$-modules to the category of Hilbert $A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clare et al. [Parabolic induction and restriction via $C^{\ast }$-algebras and Hilbert $C^{\ast }$-modules, Compos. Math.FirstView (2016), 1–33, 2].

Keywords

Hilbert C∗ -modulesadjoint functorsparabolic induction

MSC classification

Primary: 46L08: $C^*$-modules 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 2 , April 2018 , pp. 453 - 488
Copyright
© Cambridge University Press 2016 

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