scholarly journals Non-classification of Cartan subalgebras for a class of von Neumann algebras

2018 ◽  
Vol 332 ◽  
pp. 510-552 ◽  
Author(s):  
Pieter Spaas
Keyword(s):  
Von Neumann Algebras ◽  
Von Neumann ◽  
Cartan Subalgebras

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 Cartan subalgebras of regular extensions of von Neumann algebras

1988 ◽  
Vol 45 (2) ◽  
pp. 249-274
Author(s):  
Colin E. Sutherland
Keyword(s):  
Cartan Subalgebra ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Regular Representation ◽  
Projective Representation ◽  
Von Neumann ◽  
Left Regular ◽  
Cartan Subalgebras ◽  
Inner Automorphisms ◽  
The Given

AbstractWe analyse the structure of a regular extension ℳ ⋊ γ, υQ of a von Neumann algebra ℳ by an action (modulo inner automorphisms) γ of a discrete group Q, and a nonabelian 2-cycle υ for γ, under the assumption that the “action” γ of Q is cocycle conjugate to an “action”, α which leaves globally invariant a cartan subalgebra of ℳ. we show that ℳ ⋊ γ, υQ is isomorphic with the algebra of the left regular projective representation of a certain discrete, non-principal groupoid ℜ V Q determined by the action of Q on the given cartan subalgebrs, where ℜ is the Takesaki relation associated to the pair (ℳ, ) we apply this description to give a decomposition of the regular representation of a group G into irreducibles, where G is a split extension of a type I group K by an abelian group Q, and work out the details of the author's earlier abstract plancherel theorem in the case when K is abelian.

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
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