ADJOINT FUNCTORS between CATEGORIES of HILBERT C∗-MODULES

Pierre Clare, Tyrone Crisp, Nigel Higson

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

Let be a (right) Hilbert module over C∗-algebra . If is equipped with a left action of a second C∗-algebra , then tensor product with gives rise to a functor from the category of Hilbert -modules to the category of Hilbert -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∗-algebras and Hilbert -modules, Compos. Math. FirstView (2016), 1-33, 2].

Original languageEnglish (US)
Pages (from-to)453-488
Number of pages36
JournalJournal of the Institute of Mathematics of Jussieu
Volume17
Issue number2
DOIs
StatePublished - Apr 1 2018

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Adjunction
Functor
Hilbert Modules
C*-algebra
Proof by induction
Restriction
Category Theory
Reductive Group
Representation Theory
Sort
Tensor Product
Hilbert space

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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abstract = "Let be a (right) Hilbert module over C∗-algebra . If is equipped with a left action of a second C∗-algebra , then tensor product with gives rise to a functor from the category of Hilbert -modules to the category of Hilbert -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∗-algebras and Hilbert -modules, Compos. Math. FirstView (2016), 1-33, 2].",
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ADJOINT FUNCTORS between CATEGORIES of HILBERT C∗-MODULES. / Clare, Pierre; Crisp, Tyrone; Higson, Nigel.

In: Journal of the Institute of Mathematics of Jussieu, Vol. 17, No. 2, 01.04.2018, p. 453-488.

Research output: Contribution to journalArticle

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