Convergence in Measure and τ-Compactness of τ-Measurable Operators, Affiliated with a Semifinite von Neumann Algebra

2020 ◽  
Vol 64 (5) ◽  
pp. 79-82
Author(s):  
A. M. Bikchentaev
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann ◽  
Convergence In Measure ◽  
Measurable Operators

scholarly journals LOCAL DERIVATIONS ON ALGEBRAS OF MEASURABLE OPERATORS

2011 ◽  
Vol 13 (04) ◽  
pp. 643-657 ◽  
Author(s):  
S. ALBEVERIO ◽  
SH. A. AYUPOV ◽  
K. K. KUDAYBERGENOV ◽  
B. O. NURJANOV
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Continuity Condition ◽  
Type I ◽  
Atomic Lattice ◽  
Von Neumann ◽  
Measurable Operators ◽  
Finite Trace ◽  
Necessary And Sufficient

The paper is devoted to local derivations on the algebra [Formula: see text] of τ-measurable operators affiliated with a von Neumann algebra [Formula: see text] and a faithful normal semi-finite trace τ. We prove that every local derivation on [Formula: see text] which is continuous in the measure topology, is in fact a derivation. In the particular case of type I von Neumann algebras, they all are inner derivations. It is proved that for type I finite von Neumann algebras without an abelian direct summand, and also for von Neumann algebras with the atomic lattice of projections, the continuity condition on local derivations in the above results is redundant. Finally we give necessary and sufficient conditions on a commutative von Neumann algebra [Formula: see text] for the algebra [Formula: see text] to admit local derivations which are not derivations.

scholarly journals On the uniform Kadec-Klee property with respect to convergence in measure

1995 ◽  
Vol 59 (3) ◽  
pp. 343-352 ◽  
Author(s):  
F. A. Sukochev
Keyword(s):  
Function Space ◽  
Symmetric Function ◽  
Lorentz Space ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Finite Measure ◽  
Operator Space ◽  
Von Neumann ◽  
Convergence In Measure ◽  
Uniform Monotonicity

AbstractLet E(0, ∞) be a separable symmetric function space, let M be a semifinite von Neumann algebra with normal faithful semifinite trace μ, and let E(M, μ) be the symmetric operator space associated with E(0, ∞). If E(0, ∞) has the uniform Kadec-Klee property with respect to convergence in measure then E(M, μ) also has this property. In particular, if LΦ(0, ∞) (ϕ(0, ∞)) is a separable Orlicz (Lorentz) space then LΦ(M, μ) (Λϕ (M, μ)) has the uniform Kadec-Klee property with respect to convergence in measure on sets of finite measure if and only if the norm of E(0, ∞) satisfies G. Birkhoff's condition of uniform monotonicity.

scholarly journals Embeddings of ℓp into Non-commutative Spaces

2003 ◽  
Vol 74 (3) ◽  
pp. 331-350 ◽  
Author(s):  
Narcisse Randrianantoanina
Keyword(s):  
Symmetric Space ◽  
Von Neumann Algebra ◽  
Lorentz Spaces ◽  
Natural Conditions ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators

AbstractLet ℳ be a semi-finite von Neumann algebra equipped with a faithful normal trace τ. We prove a Kadec-Pelczyński type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces Lp, q, (ℳ, τ), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, ℓp cannot be embedded into Lp, q (ℳ, τ). As applications, we prove that for 0 < p < ∞ with p ≠ 2, ℓp cannot be strongly embedded into Lp(ℳ, τ). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 ≦ p < 2 on Lp [0, 1].

scholarly journals Derivations on the algebra of measurable operators affiliated with a type I von Neumann algebra

2008 ◽  
Vol 18 (2) ◽  
pp. 86-94 ◽  
Author(s):  
S. Albeverio ◽  
Sh. A. Ayupov ◽  
K. K. Kudaybergenov
Keyword(s):  
Von Neumann Algebra ◽  
Type I ◽  
Von Neumann ◽  
Measurable Operators

Weakly compact subsets of symmetric operator spaces

1991 ◽  
Vol 110 (1) ◽  
pp. 169-182 ◽  
Author(s):  
Peter G. Dodds ◽  
Theresa K. Dodds ◽  
Ben De Pagter
Keyword(s):  
Banach Space ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Operator Spaces ◽  
Natural Conditions ◽  
Weakly Compact ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Measurable Operators ◽  
Symmetric Operator Spaces

AbstractUnder natural conditions it is shown that the rearrangement invariant hull of a weakly compact subset of a properly symmetric Banach space of measurable operators affiliated with a semi-finite von Neumann algebra is again relatively weakly compact.

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