Jones index theory for Hilbert C-bimodules and its equivalence with conjugation theory

Kajiwara, Tsuyoshi; Pinzari, Claudia; Watatani, Yasuo

doi:10.1016/j.jfa.2003.09.008

Cited by 48 publications

(93 citation statements)

References 19 publications

(6 reference statements)

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“…The full relationship of our results to those of [KPW04] will be discussed in detail in Section 3.7. …”

Section: Adjunctionsmentioning

confidence: 83%

“…The purpose of this section is to explicate the relationship between our results and those of [KPW04]. Throughout this section, F denotes a correspondence from A to B, and we assume that F * has been equipped with an A-valued inner product making it into a correspondence from B to A.…”

Section: Left and Right Indexesmentioning

confidence: 96%

“…Kajiwara, Pinzari and Watatani [KPW04] relate the existence of such an inner product on F * to the existence of adjoint functors, as follows. (a) The conjugate B-A-bimodule F * carries an A-valued inner product making it into a correspondence from B to A.…”

Section: Adjunctionsmentioning

confidence: 99%

“…The operator space X * is usually called the adjoint of X, but once again we shall try to avoid over-using this word in this paper. However we warn the reader that the term conjugate as it is used in [KPW04] refers to adjoint functors.…”

Section: Hilbert Modules As Operator Spacesmentioning

confidence: 99%

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Adjoint Functors Between Categories of Hilbert -Modules

Clare

Crisp

Higson³

2016

J. Inst. Math. Jussieu

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Let E be a (right) Hilbert module over a C * -algebra A. If E is equipped with a left action of a second C * -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 [CCH16].

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“…The full relationship of our results to those of [KPW04] will be discussed in detail in Section 3.7. …”

Section: Adjunctionsmentioning

confidence: 83%

Section: Left and Right Indexesmentioning

confidence: 96%

Section: Adjunctionsmentioning

confidence: 99%

Section: Hilbert Modules As Operator Spacesmentioning

confidence: 99%

See 2 more Smart Citations

Adjoint Functors Between Categories of Hilbert -Modules

Clare

Crisp

Higson³

2016

J. Inst. Math. Jussieu

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“…In this paper, we shall use the phrase 'Hilbert C * -bimodule' in the sense of Kajiwara and Watatani [13].…”

Section: Three Sets and Their Equivalence Relationsmentioning

confidence: 99%

A CHARACTERIZATION OF SATURATED C^*-ALGEBRAIC BUNDLES OVER FINITE GROUPS

Kodaka

Teruya

2010

J. Aust. Math. Soc.

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Let A be a unital C * -algebra. Let (B, E) be a pair consisting of a unital C * -algebra B containing A as a C * -subalgebra with a unit that is also the unit of B, and a conditional expectation E from B onto A that is of index-finite type and of depth 2. Let B 1 be the C * -basic construction induced by (B, E). In this paper, we shall show that any such pair (B, E) satisfying the conditions that A ∩ B = C1 and that A ∩ B 1 is commutative is constructed by a saturated C * -algebraic bundle over a finite group. Furthermore, we shall give a necessary and sufficient condition for B to be described as a twisted crossed product of A by its twisted action of a finite group under the condition that A ∩ B 1 is commutative.2000 Mathematics subject classification: primary 46L08; secondary 46L40.

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