Classification of Real Von Neumann Algebras (I)

1996 ◽  
pp. 322-332
Author(s):  
Bingren Li
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

1971 ◽  
Vol 23 (4) ◽  
pp. 598-607 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Direct Integral ◽  
Von Neumann ◽  
Ratio Set ◽  
Almost All ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

scholarly journals Borel Reductibility and Classification of von Neumann Algebras

2009 ◽  
Vol 15 (2) ◽  
pp. 169-183 ◽  
Author(s):  
Román Sasyk ◽  
Asger Törnquist
Keyword(s):  
Von Neumann Algebras ◽  
Classification Problem ◽  
Von Neumann ◽  
Borel Reducibility

AbstractWe announce some new results regarding the classification problem for separable von Neumann algebras. Our results are obtained by applying the notion of Borel reducibility and Hjorth's theory of turbulence to the isomorphism relation for separable von Neumann algebras.

scholarly journals Level bounds for exceptional quantum subgroups in rank two

2018 ◽  
Vol 29 (05) ◽  
pp. 1850034 ◽  
Author(s):  
Andrew Schopieray
Keyword(s):  
Von Neumann Algebras ◽  
Witt Group ◽  
Von Neumann ◽  
Conformal Field ◽  
Fusion Categories ◽  
Finite Dimensional ◽  
Modular Invariants ◽  
Rank 2 ◽  
Modular Tensor Categories

There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when [Formula: see text] has rank 2, contributing proofs and explicit bounds when [Formula: see text] is of type [Formula: see text] or [Formula: see text], adding to the previously known positive results for types [Formula: see text] and [Formula: see text].

On classification of von Neumann algebras in a space with conjugation

2014 ◽  
Vol 58 (4) ◽  
pp. 1-6
Author(s):  
E. V. Vladova ◽  
M. S. Matveichuk ◽  
Yu. G. Ogai
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann

scholarly journals Classification of actions of compact abelian groups on subfactors with index less than 4

2015 ◽  
Vol 26 (07) ◽  
pp. 1550044
Author(s):  
Koichi Shimada
Keyword(s):  
Von Neumann Algebras ◽  
Abelian Groups ◽  
Type Ii ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Cocycle Conjugacy ◽  
Compact Abelian Groups

We classify actions of discrete abelian groups on some inclusions of von Neumann algebras, up to cocycle conjugacy. As an application, we classify actions of compact abelian groups on the inclusions of approximately finite dimensional (AFD) factors of type II1 with index less than 4, up to stable conjugacy.

Factoriality and Type Classification of k-Graph von Neumann Algebras

2016 ◽  
Vol 60 (2) ◽  
pp. 499-518 ◽  
Author(s):  
Dilian Yang
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Single Vertex ◽  
Tracial State ◽  
Graph Von Neumann Algebras ◽  
Fixed Point Algebra ◽  
Type Classification

AbstractLet be a single vertex k-graph and let be the von Neumann algebra induced from the Gelfand–Naimark–Segal (GNS) representation of a distinguished state ω of its k-graph C*-algebra . In this paper we prove the factoriality of , and furthermore determine its type when either has the little pullback property, or the intrinsic group of has rank 0. The key step to achieving this is to show that the fixed-point algebra of the modular action corresponding to ω has a unique tracial state.

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