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 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 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 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.

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.

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.

A Note on Uniformly Bounded Cocycles into Finite Von Neumann Algebras

2018 ◽  
Vol 61 (2) ◽  
pp. 236-239
Author(s):  
Remi Boutonnet ◽  
Jean Roydor
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Short Proof ◽  
Positive Curvature ◽  
Positive Cone ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras ◽  
Uniformly Bounded

AbstractWe give a short proof of a result of T. Bates and T. Giordano stating that any uniformly bounded Borel cocycle into a finite von Neumann algebra is cohomologous to a unitary cocycle. We also point out a separability issue in their proof. Our approach is based on the existence of a non-positive curvature metric on the positive cone of a finite von Neumann algebra.

scholarly journals Linear maps on von Neumann algebras preserving zero products on tr-rank

2002 ◽  
Vol 65 (1) ◽  
pp. 79-91 ◽  
Author(s):  
Cui Jianlian ◽  
Hou Jinchuan
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Linear Subspace ◽  
Invertible Element ◽  
Bounded Linear ◽  
Von Neumann ◽  
Linear Map ◽  
Linear Maps ◽  
Finite Von Neumann Algebra ◽  
Finite Von Neumann Algebras

In this paper, we give some characterisations of homomorphisms on von Neumann algebras by linear preservers. We prove that a bounded linear surjective map from a von Neumann algebra onto another is zero-product preserving if and only if it is a homomorphism multiplied by an invertible element in the centre of the image algebra. By introducing the notion of tr-rank of the elements in finite von Neumann algebras, we show that a unital linear map from a linear subspace ℳ of a finite von Neumann algebra ℛ into ℛ can be extended to an algebraic homomorphism from the subalgebra generated by ℳ into ℛ; and a unital self-adjoint linear map from a finite von Neumann algebra onto itself is completely tr-rank preserving if and only if it is a spatial *-automorphism.

Best approximation in von Neumann algebras

1977 ◽  
Vol 81 (2) ◽  
pp. 233-236 ◽  
Author(s):  
A. Guyan Robertson
Keyword(s):  
Best Approximation ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Identity Operator ◽  
Chebyshev Subspace ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Continuous Complex ◽  
The One

We investigate here the question of uniqueness of best approximation to operators in von Neumann algebras by elements of certain linear subspaces. Recall that a linear subspace V of a Banach space X is called a Chebyshev subspace if each vector in X has a unique best approximation by vectors in V. Our first main result characterizes the one-dimensional Chebyshev subspaces of a von Neumann algebra. This may be regarded as a generalization of a result of Stampfli [(4), theorem 2, corollary] which states that the scalar multiples of the identity operator form a Chebyshev subspace. Alternatively it may be regarded as a generalization of the commutative situation in which a continuous complex-valued function f on a compact Hausdorff space X spans a Chebyshev subspace of C(X) if and only if f does not vanish on X [(3), p. 215]. Our second main result is that a finite dimensional * subalgebra, of dimension > 1, of an infinite dimensional von Neumann algebra cannot be a Chebyshev subspace. This imposes limits to further generalization of Stampfli's result.

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