### 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 language | English (US) |
---|---|

Pages (from-to) | 453-488 |

Number of pages | 36 |

Journal | Journal of the Institute of Mathematics of Jussieu |

Volume | 17 |

Issue number | 2 |

DOIs | |

State | Published - Apr 1 2018 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Journal of the Institute of Mathematics of Jussieu*,

*17*(2), 453-488. https://doi.org/10.1017/S1474748016000074

}

*Journal of the Institute of Mathematics of Jussieu*, vol. 17, no. 2, pp. 453-488. https://doi.org/10.1017/S1474748016000074

**ADJOINT FUNCTORS between CATEGORIES of HILBERT C∗-MODULES.** / Clare, Pierre; Crisp, Tyrone; Higson, Nigel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - ADJOINT FUNCTORS between CATEGORIES of HILBERT C∗-MODULES

AU - Clare, Pierre

AU - Crisp, Tyrone

AU - Higson, Nigel

PY - 2018/4/1

Y1 - 2018/4/1

N2 - 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].

AB - 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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U2 - 10.1017/S1474748016000074

DO - 10.1017/S1474748016000074

M3 - Article

VL - 17

SP - 453

EP - 488

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

SN - 1474-7480

IS - 2

ER -