On the Banach-isomorphic classification of Lp spaces of hyperfinite von Neumann algebras

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.

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Fubini's Theorem for Ultraproducts of Noncommutative Lp-Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2004-045-1 ◽

2004 ◽

Vol 56 (5) ◽

pp. 983-1021 ◽

Cited By ~ 7

Author(s):

Marius Junge

Keyword(s):

Von Neumann Algebras ◽

Type Result ◽

Von Neumann ◽

Lp Spaces ◽

Fubini’S Theorem ◽

Image Position ◽

Local Reflexivity ◽

Fubini's Theorem

AbstractLet (ℳi)i∈I, be families of von Neumann algebras and be ultrafilters in I, J, respectively. Let 1 ≤ p < ∞ and n ∈ ℕ. Let x1,… ,xn in ΠLp(ℓi ) and y1,… ,yn in be bounded families. We show the following equalityFor p = 1 this Fubini type result is related to the local reflexivity of duals of C*-algebras. This fails for p = ∞.

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Kadec–Pełczyński decomposition for Haagerup Lp-spaces

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004101005370 ◽

2002 ◽

Vol 132 (1) ◽

pp. 137-154 ◽

Cited By ~ 9

Author(s):

NARCISSE RANDRIANANTOANINA

Keyword(s):

Von Neumann Algebra ◽

Von Neumann Algebras ◽

Bounded Sequence ◽

Point Property ◽

Weakly Compact ◽

Von Neumann ◽

Lp Spaces ◽

Reflexive Subspace ◽

Finite Von Neumann Algebras ◽

Bounded Sequences

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

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Classification of Real Von Neumann Algebras (I)

Functional Analysis in China ◽

10.1007/978-94-009-0185-8_28 ◽

1996 ◽

pp. 322-332

Author(s):

Bingren Li

Keyword(s):

Von Neumann Algebras ◽

Von Neumann

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Borel Reductibility and Classification of von Neumann Algebras

Bulletin of Symbolic Logic ◽

10.2178/bsl/1243948485 ◽

2009 ◽

Vol 15 (2) ◽

pp. 169-183 ◽

Cited By ~ 1

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.

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Level bounds for exceptional quantum subgroups in rank two

International Journal of Mathematics ◽

10.1142/s0129167x18500349 ◽

2018 ◽

Vol 29 (05) ◽

pp. 1850034 ◽

Cited By ~ 3

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].

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