scholarly journals Classification of injective factors: The wok of Alain Connes

1983 ◽  
Vol 6 (1) ◽  
pp. 1-39
Author(s):  
Steve Wright
Keyword(s):  
Structure Theory ◽  
Historical Survey ◽  
Isomorphism Classes ◽  
Subsequent Discussion ◽  
Complete Set ◽  
Principal Lines

The fundamental results ofA. Connes which determine a complete set of isomorphism classes for most injective factors are discussed in detail. After some introductory remarks which lay the foundation for the subsequent discussion, an historical survey of some of the principal lines of the investigation in the classification of factors is presented, culminating in the Connes-Takesaki structure theory of typeIIIfactors. After a discussion of injectivity for finite factors, the main result of the paper, the uniqueness of the injectiveII1factor, is deduced, and the structure ofII∞and typeIIIinjective factors is then obtained as corollaries of the main result.

scholarly journals The Classification of 7- and 8-dimensional Naturally Reductive Spaces

2019 ◽  
Vol 72 (5) ◽  
pp. 1246-1274 ◽  
Author(s):  
Reinier Storm
Keyword(s):  
New Method ◽  
Structure Theory ◽  
New Construction ◽  
Naturally Reductive

AbstractA new method for classifying naturally reductive spaces is presented. This method relies on a new construction and the structure theory of naturally reductive spaces recently developed by the author. This method is applied to obtain the classification of all naturally reductive spaces in dimension 7 and 8.

scholarly journals Equivariant Map Queer Lie Superalgebras

2016 ◽  
Vol 68 (2) ◽  
pp. 258-279 ◽  
Author(s):  
Lucas Calixto ◽  
Adriano Moura ◽  
Alistair Savage
Keyword(s):  
Finite Group ◽  
Lie Superalgebra ◽  
Lie Superalgebras ◽  
Equivariant Map ◽  
Isomorphism Classes ◽  
Finite Dimensional ◽  
Regular Maps ◽  
A Finite Group ◽  
Special Case

AbstractAn equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a finite group Γ acting on X and q. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant finitely supported maps from X to the set of isomorphism classes of irreducible finite-dimensional representations of q. In the special case where X is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra.

scholarly journals On the classification of Schreier extensions of monoids with non-abelian kernel

2020 ◽  
Vol 32 (3) ◽  
pp. 607-623
Author(s):  
Nelson Martins-Ferreira ◽  
Andrea Montoli ◽  
Alex Patchkoria ◽  
Manuela Sobral
Keyword(s):  
Abelian Group ◽  
Cohomology Group ◽  
Isomorphism Classes ◽  
The Third ◽  
Abelian Kernel ◽  
Invertible Elements ◽  
Abstract Kernel

AbstractWe show that any regular (right) Schreier extension of a monoid M by a monoid A induces an abstract kernel {\Phi\colon M\to\frac{\operatorname{End}(A)}{\operatorname{Inn}(A)}}. If an abstract kernel factors through {\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}}, where {\operatorname{SEnd}(A)} is the monoid of surjective endomorphisms of A, then we associate to it an obstruction, which is an element of the third cohomology group of M with coefficients in the abelian group {U(Z(A))} of invertible elements of the center {Z(A)} of A, on which M acts via Φ. An abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}) is induced by a regular weakly homogeneous (resp. homogeneous) Schreier extension of M by A if and only if its obstruction is zero. We also show that the set of isomorphism classes of regular weakly homogeneous (resp. homogeneous) Schreier extensions inducing a given abstract kernel {\Phi\colon M\to\frac{\operatorname{SEnd}(A)}{\operatorname{Inn}(A)}} (resp. {\Phi\colon M\to\frac{\operatorname{Aut}(A)}{\operatorname{Inn}(A)}}), when it is not empty, is in bijection with the second cohomology group of M with coefficients in {U(Z(A))}.

scholarly journals Projective Modules Over Quantum Projective Line

2017 ◽  
Vol 28 (03) ◽  
pp. 1750022 ◽  
Author(s):  
Albert Jeu-Liang Sheu
Keyword(s):  
Projective Line ◽  
Complete Classification ◽  
Projective Modules ◽  
C Algebra ◽  
Isomorphism Classes ◽  
Complex Projective Spaces ◽  
Algebra Approach ◽  
Quantum Spheres ◽  
Complex Projective

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces [Formula: see text] constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra [Formula: see text] realized as a concrete groupoid C*-algebra, and find its [Formula: see text]-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra [Formula: see text], we identify those quantum principal [Formula: see text]-bundles introduced by Hajac and collaborators among the projections classified.

On decidable varieties of Heyting algebras

1992 ◽  
Vol 57 (3) ◽  
pp. 988-991 ◽  
Author(s):  
Devdatt P. Dubhashi
Keyword(s):  
Order Theory ◽  
Boolean Algebras ◽  
Semantic Interpretation ◽  
Structure Theory ◽  
Heyting Algebras ◽  
Quantitative Classification ◽  
First Order ◽  
First Order Theory ◽  
Famous Result

In this paper we present a new proof of a decidability result for the firstorder theories of certain subvarieties of Heyting algebras. By a famous result of Grzegorczyk, the full first-order theory of Heyting algebras is undecidable. In contrast, the first-order theory of Boolean algebras and of many interesting subvarieties of Boolean algebras is decidable by a result of Tarski [8]. In fact, Kozen [6] gives a comprehensive quantitative classification of the complexities of the first-order theories of various subclasses of Boolean algebras (including the full variety).This stark contrast may be reconciled from the standpoint of universal algebra as arising out of the byplay between structure and decidability: A good structure theory entails positive decidability results. Boolean algebras have a well-developed structure theory [5], while the corresponding theory for Heyting algebras is quite meagre. Viewed in this way, we may hope to obtain decidability results if we focus attention on subclasses of Heyting algebras with good structural properties.K. Idziak and P. M. Idziak [4] have considered an interesting subvariety of Heyting algebras, , which is the variety generated by all linearly-ordered Heyting algebras. This variety is shown to be the largest subvariety of Heyting algebras with a decidable theory of its finite members. However their proof is rather indirect, proceeding via semantic interpretation into the monadic second order theory of trees. The latter is a powerful theory—it interprets many other theories—but is computationally highly infeasible. In fact, by a celebrated theorem of Rabin, its complexity is not bounded by any elementary recursive function. Consequently, the proof of [4], besides being indirect, also gives no information on the quantitative computational complexity of the theory of .Here we pursue the theme of structure and decidability. We isolate the indecomposable algebras in and use this to prove a theorem on the structure of if -algebras. This theorem relates the -algebras structurally to Boolean algebras. This enables us to bootstrap the known decidability results for Boolean algebras to the variety if .

A propensity score analysis for comparison of T-3b and VATET in myasthenia gravis

Neurology ◽  
2017 ◽  
Vol 89 (2) ◽  
pp. 189-195 ◽  
Author(s):  
Greta Brenna ◽  
Carlo Antozzi ◽  
Cristina Montomoli ◽  
Fulvio Baggi ◽  
Renato Mantegazza ◽  
...  
Keyword(s):  
Propensity Score ◽  
Myasthenia Gravis ◽  
Clinical Data ◽  
Propensity Score Analysis ◽  
Statistical Tests ◽  
Surgical Approaches ◽  
Extended Thymectomy ◽  
Score Analysis ◽  
Complete Set

Objective:We performed propensity score (PS) models to compare the outcome of patients with myasthenia gravis (MG) submitted to 2 different surgical approaches: extended transsternal (T-3b) or thoracoscopic extended thymectomy (VATET).Methods:Patients' clinical data were retrieved from the MG database of the C. Besta Neurologic Institute Foundation. In the PS analysis, a matching ratio of 1:1 of the main clinical variables was obtained for the 2 groups of patients and treatment effect was estimated by comparing their outcome.Results:A total of 210 patients met the inclusion criteria, by having a complete set of clinical data, and were included in the PS model; a matched dataset of 122 participants (61 per group) showed an adequate balance of all the covariates. Our analysis demonstrated that 68.9% of patients who had thymectomy by the VATET technique reached the pharmacologic remission/remission status at 2 years from thymectomy compared to 34.4% of those operated on by the T-3b technique (p < 0.001), had a lower INCB-MG score (p < 0.001), and had less muscle fatigability (p = 0.004). Similar results were found considering only nonthymomatous patients with MG. Results were also confirmed by paired statistical tests.Conclusions:Our PS matching analysis showed that VATET is a reliable and effective surgical approach alternative to T-3b in patients with MG who are candidates for thymectomy.Classification of evidence:This study provides Class IV evidence that for patients with MG, VATET is more effective than T-3b thymectomy.

CLASSIFICATION OF REAL ALGEBRAIC VECTOR BUNDLES OVER THE REAL ANISOTROPIC CONIC

2005 ◽  
Vol 16 (10) ◽  
pp. 1207-1220 ◽  
Author(s):  
INDRANIL BISWAS ◽  
D. S. NAGARAJ
Keyword(s):  
Vector Bundles ◽  
Complete Classification ◽  
Real Numbers ◽  
The Real ◽  
Isomorphism Classes ◽  
Algebraic Vector

We give a complete classification of isomorphism classes of real algebraic vector bundles over the scheme defined by a nondegenerate anisotropic conic defined over the field of real numbers.

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