scholarly journals Kadec–Pełczyński decomposition for Haagerup Lp-spaces

2002 ◽  
Vol 132 (1) ◽  
pp. 137-154 ◽  
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.

scholarly journals ON EQUALITY CONDITION FOR TRACE JENSEN INEQUALITY IN SEMI-FINITE VON NEUMANN ALGEBRAS

2008 ◽  
Vol 19 (04) ◽  
pp. 481-501 ◽  
Author(s):  
TETSUO HARADA ◽  
HIDEKI KOSAKI
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Strict Convexity ◽  
Jensen Inequality ◽  
Von Neumann ◽  
Convexity Assumption ◽  
Finite Von Neumann Algebra ◽  
Equality Condition ◽  
Finite Von Neumann Algebras

Let τ be a faithful semi-finite normal trace on a semi-finite von Neumann algebra, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality states τ(f(a* xa)) ≤ τ(a* f(x)a) for a contraction a and a self-adjoint operator x. Under certain strict convexity assumption on f(t), we will study when this inequality reduces to the equality.

scholarly journals THE RELATIVE WEAK ASYMPTOTIC HOMOMORPHISM PROPERTY FOR INCLUSIONS OF FINITE VON NEUMANN ALGEBRAS

2011 ◽  
Vol 22 (07) ◽  
pp. 991-1011 ◽  
Author(s):  
JUNSHENG FANG ◽  
MINGCHU GAO ◽  
ROGER R. SMITH
Keyword(s):  
Finite Number ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Unitary Operators ◽  
Von Neumann ◽  
The One ◽  
Finite Von Neumann Algebras

A triple of finite von Neumann algebras B ⊆ N ⊆ M is said to have the relative weak asymptotic homomorphism property if there exists a net of unitary operators {uλ}λ∈Λ in B such that [Formula: see text] for all x,y ∈ M. We prove that a triple of finite von Neumann algebras B ⊆ N ⊆ M has the relative weak asymptotic homomorphism property if and only if N contains the set of all x ∈ M such that [Formula: see text] for a finite number of elements x1, …, xn in M. Such an x is called a one-sided quasi-normalizer of B, and the von Neumann algebra generated by all one-sided quasi-normalizers of B is called the one-sided quasi-normalizer algebra of B. We characterize one-sided quasi-normalizer algebras for inclusions of group von Neumann algebras and use this to show that one-sided quasi-normalizer algebras and quasi-normalizer algebras are not equal in general. We also give some applications to inclusions L(H) ⊆ L(G) arising from containments of groups. For example, when L(H) is a masa we determine the unitary normalizer algebra as the von Neumann algebra generated by the normalizers of H in G.

scholarly journals 2-LOCAL DERIVATIONS ON SEMI-FINITE VON NEUMANN ALGEBRAS

2013 ◽  
Vol 56 (1) ◽  
pp. 9-12 ◽  
Author(s):  
SHAVKAT AYUPOV ◽  
FARKHAD ARZIKULOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Local Derivation ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras

AbstractIn the present paper we prove that every 2-local derivation on a semi-finite von Neumann algebra is a derivation.

scholarly journals Remarks on complemented subspaces of von Neumann algebras

1992 ◽  
Vol 121 (1-2) ◽  
pp. 1-4 ◽  
Author(s):  
G. Pisier
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
C Algebra ◽  
Know How ◽  
Complemented Subspaces ◽  
Reflexive Subspace

SynopsisIn this note we include two remarks about bounded (not necessarily contractive) linear projections on a von Neumann algebra. We show that if M is a von Neumann subalgebra of B(H) which is complemented in B(H) and isomorphic to M⊗M, then M is injective (or equivalently M is contractively complemented). We do not know how to get rid of the second assumption on M. In the second part, we show that any complemented reflexive subspace of a C*-algebra is necessarily linearly isomorphic to a Hilbert space.

BOUNDED COCYCLES ON FINITE VON NEUMANN ALGEBRAS

2001 ◽  
Vol 12 (06) ◽  
pp. 743-750 ◽  
Author(s):  
TERESA BATES ◽  
THIERRY GIORDANO
Keyword(s):  
Discrete Group ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
Countable Discrete Group ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras ◽  
Uniformly Bounded

In this note we prove that if G is a countable discrete group, then every uniformly bounded cocycle from a standard Borel G-space into a finite Von Neumann algebra is cohomologous to a unitary cocycle. This generalizes results of both F. H. Vasilescu and L. Zsidó and R. J. Zimmer.

TRACE JENSEN INEQUALITY FOR SELF-ADJOINT OPERATORS IN SEMI-FINITE VON NEUMANN ALGEBRAS

2013 ◽  
Vol 24 (09) ◽  
pp. 1350075
Author(s):  
HIDEKI KOSAKI
Keyword(s):  
Convex Function ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Jensen Inequality ◽  
Measurable Operator ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators ◽  
Adjoint Operators ◽  
Finite Von Neumann Algebras

Let [Formula: see text] be a semi-finite von Neumann algebra equipped with a faithful semi-finite normal trace τ, and f(t) be a convex function with f(0) = 0. The trace Jensen inequality in our previous work states τ(f(a*xa)) ≤ τ(a*f(x)a) (as long as the both sides are well-defined) for a contraction [Formula: see text] and a semi-bounded τ-measurable operator x. Validity of this inequality for (not necessarily semi-bounded) self-adjoint τ-measurable operators is investigated.

scholarly journals Tensor Products of Noncommutative Lp-Spaces

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-9
Author(s):  
Somlak Utudee
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebras ◽  
Tensor Products ◽  
Von Neumann ◽  
Lp Spaces ◽  
Finite Von Neumann Algebras

We consider the notion of tensor product of noncommutative Lp spaces associated with finite von Neumann algebras and define the notion of tensor product of Haagerup noncommutative Lp spaces associated with σ-finite von Neumann algebras.

STRUCTURE RESULTS FOR NORMALIZERS OF II1 FACTORS

2011 ◽  
Vol 22 (07) ◽  
pp. 947-979 ◽  
Author(s):  
JAN M. CAMERON
Keyword(s):  
Tensor Product ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Type Theorem ◽  
Discrete Structure ◽  
Von Neumann ◽  
Product Group ◽  
Countable Subgroup ◽  
Finite Von Neumann Algebras ◽  
Group Von Neumann Algebra

For an inclusion N ⊆ M of II1 factors, we study the group of normalizers [Formula: see text] and the von Neumann algebra it generates. We first show that [Formula: see text] imposes a certain "discrete" structure on the generated von Neumann algebra. By analyzing the bimodule structure of certain subalgebras of [Formula: see text], this leads to a "Galois-type" theorem for normalizers, in which we find a description of the subalgebras of [Formula: see text] in terms of a unique countable subgroup of [Formula: see text]. Implications for inclusions B ⊆ M arising from the crossed product, group von Neumann algebra, and tensor product constructions will also be addressed. Our work leads to a construction of new examples of norming subalgebras in finite von Neumann algebras: If N ⊆ M is a regular inclusion of II1 factors, then N norms M.

scholarly journals Haagerup approximation property via bimodules

2017 ◽  
Vol 121 (1) ◽  
pp. 75 ◽  
Author(s):  
Rui Okayasu ◽  
Narutaka Ozawa ◽  
Reiji Tomatsu
Keyword(s):  
Quantum Group ◽  
Von Neumann Algebra ◽  
Approximation Property ◽  
Von Neumann Algebras ◽  
Locally Compact ◽  
Haagerup Property ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Finite Von Neumann Algebras

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it was recently generalized to arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.

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