Commutator estimates in $W^{*}$-factors

Ber, A. F.; Sukochev, Fedor

doi:10.1090/s0002-9947-2012-05568-1

Cited by 13 publications

(15 citation statements)

References 34 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Derivations with values in ideals of a von Neumann algebra have important applications in the study of hyperreflexivity of operator spaces [50], commutants mod normed ideals [65], automorphisms and epimorphisms of operator algebras [12,13,39,40], etc. In Sect.…”

Section: ])mentioning

confidence: 99%

“…Kadison [38] and Sakai [58] gave an affirmative answer to the special case when the A-bimodule J coincides with the algebra A itself. Further, it was proved that every derivation from a von Neumann algebra into its arbitrary ideal is automatically inner [12,13]. However, when one considers more general A-bimodules J (see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…[33] and [36,Section 10.11]) and a number of mathematicians have been working on this problem since the 1960s (see e.g. [10,12,13,16,32,33,37,38,44,54,58]). Even though the special case when E = M is still open (e.g.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Derivations with Values in the Ideal of $$\tau $$-Compact Operators Affiliated with a Semifinite von Neumann Algebra

Ber

Huang

Levitina

et al. 2022

Commun. Math. Phys.

View full text Add to dashboard Cite

Let $${{\mathcal {M}}}$$ M be a semifinite von Neumann algebra with a faithful normal semifinite trace $$\tau $$ τ and let $${{\mathcal {A}}}$$ A be an arbitrary von Neumann subalgebra of $${{\mathcal {M}}}$$ M . We characterize the class of symmetric ideals $${{\mathcal {E}}}$$ E in $${{\mathcal {M}}}$$ M such that derivations $$\delta :{{\mathcal {A}}}\rightarrow {{\mathcal {E}}}$$ δ : A → E are necessarily inner, which is a unification and far-reaching extension of the results due to Johnson and Parrott (J Funct Anal 11:39–61, 1972), due to Kaftal and Weiss (J Funct Anal 62:202–220, 1985), and due to Popa (J Funct Anal 71:393–408, 1987). In particular, we show that every derivation from $${{\mathcal {A}}}$$ A into the ideal $${{\mathcal {C}}}_0({{\mathcal {M}}},\tau )$$ C 0 ( M , τ ) of all $$\tau $$ τ -compact operators is inner, establishing a semifinite version of the Johnson–Parrott–Popa Theorem which is different from Popa and Rădulescu (Duke Math J 57(2):485–518, 1988, Theorem 1.1) and contrasts to the example of a non-inner derivation established in Popa and Rădulescu (1988, Theorem 1.2).

show abstract

Section: ])mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Derivations with Values in the Ideal of $$\tau $$-Compact Operators Affiliated with a Semifinite von Neumann Algebra

Ber

Huang

Levitina

et al. 2022

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…It concerns derivations from a C * -subalgebra (or von Neumann subalgebra) into ideals of a larger von Neumann algebra and it has been widely studied during the last decades (see e.g. [8,9,10,11,12,24,26,29,38,39]). However, very few results are available in the setting when the derivation in question takes values in a non-commutative operator space (of possibly unbounded operators) (see e.g.…”

Section: Introductionmentioning

confidence: 99%

M-embedded symmetric operator spaces and the derivation problem

Huang¹,

Levitina²,

Sukochev³

2019

Math. Proc. Camb. Phil. Soc.

View full text Add to dashboard Cite

Let ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

show abstract

“…There exists a self-adjoint element a ∈ M such that the inequality |[a, u]| |a − λ1| fails for every λ ∈ C and every unitary u ∈ M [1]. Hence the factor 1 − ε in Theorem 1(1) cannot be omitted.…”

mentioning

confidence: 99%

Commutator estimates in von Neumann algebras

Ber

Sukochev

2013

Funct Anal Its Appl

Self Cite

View full text Add to dashboard Cite

Let M be a von Neumann algebra. For every self-adjoint locally measurable operator a, there exists a central self-adjoint locally measurable operator c 0 such that, given any ε > 0, |[a, u ε ]| (1 − ε)|a − c 0 | for some unitary operator u ε ∈ M . In particular, every derivation δ : M → I (where I is an ideal in M ) is inner, and δ = δ a for a ∈ I .Let M be a von Neumann algebra with center Z(M ). We denote the inner derivation on M generated by an element a ∈ M by δ a = [a, · ]. Clearly, δ a is a continuous linear operator on (M , · M ) (here, · M denotes the C * -norm on M ). The operator norm of δ a is estimated from below as δ a a − c M for some c ∈ Z(M ) [4]. It is natural to ask whether the positive operator |a − c| with some c ∈ Z(M ) is majorized, with respect to some natural partial order on M h = {x ∈ M : x = x * }, by the operator δ a (y) for some self-adjoint y ∈ M such that y M 1. To state our main result, we need the notion of an operator (locally) measurable with respect to a von Neumann algebra. A closed densely defined operator a affiliated with a von Neumann algebra M is said to be measurable with respect to M if the spectral projection e |a| (λ, +∞) of |a| is finite for some λ > 0. The set S(M ) of all operators measurable with respect to M is a * -algebra under the strong sum and strong multiplication operations [5]. A closed densely defined operator a affiliated with a von Neumann algebra M is said to be locally measurable with respect to M if there exists a sequence of projections {z n } ⊂ Z(M ) such that z n ↑ 1 and az n ∈ S(M ) for every n. The set LS(M ) of all operators locally measurable with respect to M is a * -algebra under the strong sum and strong multiplication operations; moreover, S(M ) ⊂ LS(M ) [5].Below, by Z(A ) we denote the center of an algebra A . Theorem 1. Let M be a von Neumann algebra, and let a = a * ∈ LS(M ). Then the following assertions hold.(1) There exists an operator c 0 = c * 0 ∈ Z(LS(M )) such that, for every ε > 0, there exists a unitary self-adjoint operator(2) If M is a finite or purely infinite σ-finite von Neumann algebra, then there exists an operator c 0 = c * 0 ∈ Z(LS(M )) and a self-adjoint unitary operatorLet M be a σ-finite factor of type I ∞ or II ∞ . There exists a self-adjoint element a ∈ M such that the inequality |[a, u]| |a − λ1| fails for every λ ∈ C and every unitary u ∈ M [1]. Hence the factor 1 − ε in Theorem 1(1) cannot be omitted.The main technique in the proof of this theorem consists in thoroughly comparing spectral projections of the self-adjoint operator a ∈ LS(M ) and "gluing toghether" the required unitary operators by partial isometries between these projections. This technique was first used (in the case of factors) in [1].The inequality in Theorem 1(1) strengthens classical results going back to Calkin's celebrated paper [2]. Interestingly, our methods are completely different from those of Calkin and have wider range of application.Corollary 2. Let M be a von Neumann algebra, and let I be an ideal in M . Then the following asse...

show abstract

Commutator estimates in $W^{*}$-factors

Cited by 13 publications

References 34 publications

Derivations with Values in the Ideal of $$\tau $$-Compact Operators Affiliated with a Semifinite von Neumann Algebra

Derivations with Values in the Ideal of $$\tau $$-Compact Operators Affiliated with a Semifinite von Neumann Algebra

M-embedded symmetric operator spaces and the derivation problem

Commutator estimates in von Neumann algebras

Contact Info

Product

Resources

About