scholarly journals TWISTED ENTIRE CYCLIC COHOMOLOGY, J-L-O COCYCLES AND EQUIVARIANT SPECTRAL TRIPLES

2004 ◽  
Vol 16 (05) ◽  
pp. 583-602 ◽  
Author(s):  
DEBASHISH GOSWAMI
Keyword(s):  
Spectral Data ◽  
Noncommutative Geometry ◽  
Positive Operator ◽  
Chern Character ◽  
Cyclic Cohomology ◽  
Spectral Triple ◽  
Usual Sense ◽  
Spectral Triples ◽  
Definition Of ◽  
Made In

We study the "quantized calculus" corresponding to the algebraic ideas related to "twisted cyclic cohomology" introduced in [12]. With very similar definitions and techniques as those used in [9], we define and study "twisted entire cyclic cohomology" and the "twisted Chern character" associated with an appropriate operator theoretic data called "twisted spectral data", which consists of a spectral triple in the conventional sense of noncommutative geometry [1] and an additional positive operator having some specified properties. Furthermore, it is shown that given a spectral triple (in the conventional sense) which is equivariant under the (co-) action of a compact matrix pseudogroup, it is possible to obtain a canonical twisted spectral data and hence the corresponding (twisted) Chern character, which will be invariant (in the usual sense) under the (co-)action of the pseudogroup, in contrast to the fact that the Chern character coming from the conventional noncommutative geometry need not to be invariant. In the last section, we also try to detail out some remarks made in [3], in the context of a new definition of invariance satisfied by the conventional (untwisted) cyclic cocycles when lifted to an appropriate larger algebra.

ON THE JLO COCYCLE AND ITS TRANSGRESSION IN ENTIRE CYCLIC COHOMOLOGY

2013 ◽  
Vol 10 (07) ◽  
pp. 1350037
Author(s):  
ALAN LAI
Keyword(s):  
Noncommutative Geometry ◽  
Von Neumann Algebras ◽  
Chern Character ◽  
Cyclic Cohomology ◽  
Character Formula ◽  
Type Ii ◽  
Von Neumann ◽  
Definition Of ◽  
Dixmier Trace ◽  
Math Phys

The JLO character formula due to Jaffe–Lesniewski–Osterwalder [Quantum K-theory: the Chern character, Commun. Math. Phys.112 (1988) 75–88] assigns to each Fredholm module a cocycle in entire cyclic cohomology. It descends to define a cohomological Chern character on K-homology. This paper extends the definition of the JLO character formula for Breuer–Fredholm modules, the modules that represent type II noncommutative geometry; and shows that the JLO character formula coincides with the Connes character formula [see M. Benameur and T. Fack, Type II noncommutative geometry. I. Dixmier trace in von Neumann algebras, Adv. Math.199 (2006) 29–87] at the level of entire cyclic cohomology.

scholarly journals Temporal Lorentzian spectral triples

2014 ◽  
Vol 26 (08) ◽  
pp. 1430007 ◽  
Author(s):  
Nicolas Franco
Keyword(s):  
Noncommutative Geometry ◽  
Minkowski Spacetime ◽  
Spectral Triple ◽  
Spectral Triples ◽  
Lorentzian Space ◽  
Pure States ◽  
Global Time ◽  
Distance Formula

We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3 + 1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a temporal Lorentzian spectral triple over a Moyal–Minkowski spacetime. We show that, when time is commutative, the algebra can be extended to unbounded elements. Using such an extension, it is possible to define a Lorentzian distance formula between pure states with a well-defined noncommutative formulation.

scholarly journals Multiplicativity of Connes’ calculus

2019 ◽  
Vol 31 (09) ◽  
pp. 1950033
Author(s):  
Partha Sarathi Chakraborty ◽  
Satyajit Guin
Keyword(s):  
Noncommutative Geometry ◽  
Spectral Triple ◽  
Graded Algebra ◽  
Spectral Triples ◽  
Differential Graded Algebra

In his book on noncommutative geometry, Connes constructed a differential graded algebra out of a spectral triple. Lack of monoidality of this construction is investigated. We identify a suitable monoidal subcategory of the category of spectral triples and show that when restricted to this subcategory the construction of Connes is monoidal. Richness of this subcategory is exhibited by establishing a faithful endofunctor to this subcategory.

scholarly journals Spectral Flow for Nonunital Spectral Triples

2015 ◽  
Vol 67 (4) ◽  
pp. 759-794 ◽  
Author(s):  
A. L. Carey ◽  
V. Gayral ◽  
J. Phillips ◽  
A. Rennie ◽  
F. A. Sukochev
Keyword(s):  
Index Theory ◽  
Spectral Triple ◽  
Index Formula ◽  
Spectral Flow ◽  
Spectral Triples ◽  
Local Index ◽  
C Algebra ◽  
Odd Index ◽  
Definition Of ◽  
Special Case

AbstractWe prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a C*-algebra with invariant trace satisûes the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting, we are able to connect with earlier approaches to the analytic definition of spectral flow

Need of Human Rights as a Better Way of Life

Think India ◽  
2019 ◽  
Vol 22 (3) ◽  
pp. 72-83
Author(s):  
Tushar Kadian
Keyword(s):  
Human Rights ◽  
Basic Needs ◽  
Research Paper ◽  
Human Being ◽  
Way Of Life ◽  
Labour Unions ◽  
The People ◽  
The Subject ◽  
Definition Of ◽  
Made In

Actually, basic needs postulates securing of the elementary conditions of existence to every human being. Despite of the practical and theoretical importance of the subject the greatest irony is non- availability of any universal preliminary definition of the concept of basic needs. Moreover, this becomes the reason for unpredictability of various political programmes aiming at providing basic needs to the people. The shift is necessary for development of this or any other conception. No labour reforms could be made in history till labours were treated as objects. Its only after they were started being treating as subjects, labour unions were allowed to represent themselves in strategy formulations that labour reforms could become a reality. The present research paper highlights the basic needs of Human Rights in life.

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

scholarly journals Missing the point in noncommutative geometry

Synthese ◽  
2021 ◽  
Author(s):  
Nick Huggett ◽  
Fedele Lizzi ◽  
Tushar Menon
Keyword(s):  
Scalar Field ◽  
Noncommutative Geometry ◽  
Smooth Manifold ◽  
Operational Definition ◽  
Spectral Triple ◽  
Fundamental Quantity

AbstractNoncommutative geometries generalize standard smooth geometries, parametrizing the noncommutativity of dimensions with a fundamental quantity with the dimensions of area. The question arises then of whether the concept of a region smaller than the scale—and ultimately the concept of a point—makes sense in such a theory. We argue that it does not, in two interrelated ways. In the context of Connes’ spectral triple approach, we show that arbitrarily small regions are not definable in the formal sense. While in the scalar field Moyal–Weyl approach, we show that they cannot be given an operational definition. We conclude that points do not exist in such geometries. We therefore investigate (a) the metaphysics of such a geometry, and (b) how the appearance of smooth manifold might be recovered as an approximation to a fundamental noncommutative geometry.

scholarly journals Higher spectral flow and an entire bivariant JLO cocycle

2012 ◽  
Vol 11 (1) ◽  
pp. 183-232 ◽  
Author(s):  
Moulay-Tahar Benameur ◽  
Alan L. Carey
Keyword(s):  
Dirac Operator ◽  
Chern Character ◽  
Dirac Operators ◽  
Cyclic Cohomology ◽  
Closed Manifold ◽  
Spectral Flow ◽  
Smooth Functions ◽  
Base Manifold ◽  
K Theory ◽  
Higher Spectral Flow

AbstractFor a single Dirac operator on a closed manifold the cocycle introduced by Jaffe-Lesniewski-Osterwalder [19] (abbreviated here to JLO), is a representative of Connes' Chern character map from the K-theory of the algebra of smooth functions on the manifold to its entire cyclic cohomology. Given a smooth fibration of closed manifolds and a family of generalized Dirac operators along the fibers, we define in this paper an associated bivariant JLO cocycle. We then prove that, for any l ≥ 0, our bivariant JLO cocycle is entire when we endow smoooth functions on the total manifold with the Cl+1 topology and functions on the base manifold with the Cl topology. As a by-product of our theorem, we deduce that the bivariant JLO cocycle is entire for the Fréchet smooth topologies. We then prove that our JLO bivariant cocycle computes the Chern character of the Dai-Zhang higher spectral flow.

Defining hormesis: the necessary tool to clarify experimentally the low dose–response relationship

2002 ◽  
Vol 21 (2) ◽  
pp. 103-104 ◽  
Author(s):  
G Carelli ◽  
I Iavicoli
Keyword(s):  
Dose Response ◽  
Low Dose ◽  
Experimental Models ◽  
Assessment Criteria ◽  
Response Relationship ◽  
Dose Response Relationship ◽  
Specific Context ◽  
Definition Of ◽  
Time Specific ◽  
Made In

The authors comment on Calabrese and Baldwin's paper ‘Defining Hormesis’, which, to date, is the first attempt to provide a definition of hormesis that goes beyond the different interpretations of this phenomenon reported in the literature. While appreciating the effort made in this study to place hormesis in a general and at the same time specific context, the authors believe some clarifications are needed as regards the quantitative features of this phenomenon. In this connection, they speculate on whether Calabrese and Baldwin think it appropriate to include hormesis assessment criteria in the document, referring in particular to those reported in a previous paper. The authors share Calabrese and Baldwin's conclusion that future experimental models designed to study hormetic phenomena must necessarily include the time factor, which not only guarantees this phenomenon will be detected, but is also able to detect the specific type of hormesis.

The construction of units in conversational talk

2000 ◽  
Vol 29 (4) ◽  
pp. 477-517 ◽  
Author(s):  
MARGRET SELTING
Keyword(s):  
Conversation Analysis ◽  
Activity Type ◽  
Seminal Work ◽  
Conversational Context ◽  
Linguistic Unit ◽  
Definition Of ◽  
Made In

The notion of Turn-Constructional Unit (TCU) in Conversation Analysis has become unclear for many researchers. The underlying problems inherent in the definition of this notion are here identified, and a possible solution is suggested. This amounts to separating more clearly the notions of TCU and Transition Relevance Place (TRP). In this view, the TCU is defined as the smallest interactionally relevant complete linguistic unit, in a given context, that is constructed with syntactic and prosodic resources within their semantic, pragmatic, activity-type-specific, and sequential conversational context. It ends in a TRP unless particular linguistic and interactional resources are used to project and postpone the TRP to the end of a larger multi-unit turn. This suggestion tries to spell out some of the assumptions that the seminal work in CA made in principle, but never formulated explicitly.

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