scholarly journals ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

2016 ◽  
Vol 17 (2) ◽  
pp. 453-488 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Category Theory ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Adjoint Functors ◽  
Left Action ◽  
Parabolic Induction ◽  
Left And Right ◽  
Restriction Functor ◽  
Tempered Representation ◽  
Standard Concept

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 Clareet al.[Parabolic induction and restriction via$C^{\ast }$-algebras and Hilbert$C^{\ast }$-modules,Compos. Math.FirstView(2016), 1–33, 2].

scholarly journals Parabolic induction and restriction via -algebras and Hilbert -modules

2016 ◽  
Vol 152 (6) ◽  
pp. 1286-1318 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Hilbert Space ◽  
Representation Theory ◽  
Inner Product ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Parabolic Induction ◽  
Full Treatment ◽  
Reduced Group ◽  
The Right ◽  
Tempered Representation

This paper is about the reduced group $C^{\ast }$-algebras of real reductive groups, and about Hilbert $C^{\ast }$-modules over these $C^{\ast }$-algebras. We shall do three things. First, we shall apply theorems from the tempered representation theory of reductive groups to determine the structure of the reduced $C^{\ast }$-algebra (the result has been known for some time, but it is difficult to assemble a full treatment from the existing literature). Second, we shall use the structure of the reduced $C^{\ast }$-algebra to determine the structure of the Hilbert $C^{\ast }$-bimodule that represents the functor of parabolic induction. Third, we shall prove that the parabolic induction bimodule admits a secondary inner product, using which we can define a functor of parabolic restriction in tempered representation theory. We shall prove in a sequel to this paper that parabolic restriction is adjoint, on both the left and the right, to parabolic induction in the context of tempered unitary Hilbert space representations.

scholarly journals On Reducibility and Unitarizability for Classical p-Adic Groups, Some General Results

2009 ◽  
Vol 61 (2) ◽  
pp. 427-450 ◽  
Author(s):  
Marko Tadić
Keyword(s):  
Reductive Groups ◽  
Parabolic Induction

Abstract. The aim of this paper is to prove two general results on parabolic induction of classical p-adic groups (actually, one of them holds also in the archimedean case), and to obtain from them some consequences about irreducible unitarizable representations. One of these consequences is a reduction of the unitarizability problem for these groups. This reduction is similar to the reduction of the unitarizability problem to the case of real infinitesimal character for real reductive groups.

Groupoid Enriched Categories and Homotopy Theory

1983 ◽  
Vol 35 (3) ◽  
pp. 385-416 ◽  
Author(s):  
P. H. H. Fantham ◽  
E. J. Moore
Keyword(s):  
Algebraic Topology ◽  
Category Theory ◽  
Homotopy Theory ◽  
Adjoint Functors ◽  
Homotopy Classes ◽  
Enriched Categories ◽  
Special Cases ◽  
Hopf Invariants

We are concerned in this paper with category-theoretic aspects of homotopy theory. Originally, category theory developed as a simplifying language in the context of algebraic topology and yet one primary example: the category Π of spaces and homotopy classes of maps admits only limited use of the language owing to the very sparse occurrence of limits. Of course, full use has been made of them nevertheless: limits and colimits exist in the case of products and coproducts, and in almost no other case; yet, from this we obtain the theory of Samelson products, Whitehead products, and Hopf invariants which can all be expressed in Π see [8]. In addition, there are hosts of adjoint functors and yet the outcome is disappointing because the language applies only to special cases rather than to the situation as a whole.

scholarly journals Category Algebras and States on Categories

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1172
Author(s):  
Hayato Saigo
Keyword(s):  
Hilbert Spaces ◽  
Category Theory ◽  
Linear Functional ◽  
Quantum Probability ◽  
Probability Measures ◽  
Mathematical Framework ◽  
Hilbert Modules ◽  
Matrix Algebras ◽  
Generalized Matrix ◽  
Category Algebras

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.

Triangulated Categories

2007 ◽  
Author(s):  
D. Huybrechts
Keyword(s):  
Triangulated Categories ◽  
Adjoint Functors ◽  
Abelian Categories ◽  
Orthogonal Decompositions ◽  
Left And Right ◽  
Serre Functors

Reviewing the basic notions of additive and abelian categories, left and right adjoint functors, and Serre functors, this chapter is mainly devoted to triangulated categories. In particular, criteria are established which decide when a given functor is fully-faithful or an equivalence. This is formulated in terms of spanning classes. The last section discusses exceptional objects in triangulated categories which lead naturally to the notion of orthogonal decompositions of categories.

The three-dimensional structure of frozen-hydrated left- and right-handed forms of bacterial flagellar filaments from an identical serotype

1986 ◽  
Vol 44 ◽  
pp. 298-299
Author(s):  
S. Trachtenberg ◽  
D. J. DeRosier
Keyword(s):  
Ionic Strength ◽  
Basal Body ◽  
Three Dimensional ◽  
Dimensional Structure ◽  
Protein Species ◽  
Liquid Environment ◽  
Bacterial Flagellum ◽  
Left And Right ◽  
Rotary Motor ◽  
Helical Parameters

The bacterial cell is propelled through the liquid environment by means of one or more rotating flagella. The bacterial flagellum is composed of a basal body (rotary motor), hook (universal coupler), and filament (propellor). The filament is a rigid helical assembly of only one protein species — flagellin. The filament can adopt different morphologies and change, reversibly, its helical parameters (pitch and hand) as a function of mechanical stress and chemical changes (pH, ionic strength) in the environment.

Advantages of Complementary Stereo Images as Viewed with the Low-Temperature Field-Emission SEM

1992 ◽  
Vol 50 (2) ◽  
pp. 1314-1315
Author(s):  
William P. Wergin ◽  
Eric F. Erbe
Keyword(s):  
Fold Increase ◽  
Horizontal Displacement ◽  
Three Dimensions ◽  
Stereo Image ◽  
Stereo Pair ◽  
Sem Micrograph ◽  
Left And Right ◽  
The Common ◽  
The Brain ◽  
Pair Analysis

The eye-brain complex allows those of us with normal vision to perceive and evaluate our surroundings in three-dimensions (3-D). The principle factor that makes this possible is parallax - the horizontal displacement of objects that results from the independent views that the left and right eyes detect and simultaneously transmit to the brain for superimposition. The common SEM micrograph is a 2-D representation of a 3-D specimen. Depriving the brain of the 3-D view can lead to erroneous conclusions about the relative sizes, positions and convergence of structures within a specimen. In addition, Walter has suggested that the stereo image contains information equivalent to a two-fold increase in magnification over that found in a 2-D image. Because of these factors, stereo pair analysis should be routinely employed when studying specimens.Imaging complementary faces of a fractured specimen is a second method by which the topography of a specimen can be more accurately evaluated.

Histological evaluation of cochlear hair cell damage from noise-induced hearing loss in chinchillas

1990 ◽  
Vol 48 (3) ◽  
pp. 332-333
Author(s):  
R.V. Harrison ◽  
R.J. Mount ◽  
P. White ◽  
N. Fukushima
Keyword(s):  
Hair Cell ◽  
Evoked Potential ◽  
Cell Damage ◽  
Acoustic Trauma ◽  
Histological Evaluation ◽  
Cell Integrity ◽  
Functional Changes ◽  
Cochlear Hair Cell ◽  
Left And Right ◽  
Contralateral Ear

In studies which attempt to define the influence of various factors on recovery of hair cell integrity after acoustic trauma, an experimental and a control ear which initially have equal degrees of damage are required. With in a group of animals receiving an identical level of acoustic trauma there is more symmetry between the ears of each individual, in respect to function, than between animals. Figure 1 illustrates this, left and right cochlear evoked potential (CAP) audiograms are shown for two chinchillas receiving identical trauma. For this reason the contralateral ear is used as control.To compliment such functional evaluations we have devised a scoring system, based on the condition of hair cell stereocilia as revealed by scanning electron microscopy, which permits total stereociliar damage to be expressed numerically. This quantification permits correlation of the degree of structural pathology with functional changes. In this paper wereport experiments to verify the symmetry of stereociliar integrity between two ears, both for normal (non-exposed) animals and chinchillas in which each ear has received identical noise trauma.

The crystallography characteristic of (Mn,Fe)S single crystal

1990 ◽  
Vol 48 (4) ◽  
pp. 898-899
Author(s):  
Jiang Xishan
Keyword(s):  
Single Crystal ◽  
Cast Steel ◽  
Point Group ◽  
Central Area ◽  
Hexagonal Crystal ◽  
Growth Step ◽  
Left And Right ◽  
Relationship Of ◽  
Fcc Structure ◽  
New Discovery

This paper reports the growth step pattern and morphology at equilibrium and growth states of (Mn,Fe)S single crystal on the wall of micro-voids in ZG25 cast steel by using scanning electron microscope. Seldom report was presented on the growth morphology and steppattern of (Mn,Fe)S single crystal.Fig.1 shows the front half of the polyhedron of(Mn,Fe)S single crystal,its central area being the square crystal plane,the two pairs of hexagons symmetrically located in the high and low, the left and right with a certain, angle to the square crystal plane.According to the symmetrical relationship of crystal, it was defined that the (Mn,Fe)S single crystal at equilibrium state is tetrakaidecahedron consisted of eight hexagonal crystal planes and six square crystal planes. The macroscopic symmetry elements of the tetrakaidecahedron correpond to Oh—n3m symmetry class of fcc structure,in which the hexagonal crystal planes are the { 111 } crystal planes group,square crystal plaits are the { 100 } crystal planes group. This new discovery of the (Mn,Fe)S single crystal provides a typical example of the point group of Oh—n3m.

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