Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Albeverio, Sergio; Ayupov, Sh. A.; Kudaybergenov, Karimbergen

doi:10.1016/j.jfa.2008.11.003

Cited by 59 publications

(84 citation statements)

References 17 publications

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“…we obtain a linear operator D δ on M n (S(Z, τ Z )), which is a derivation on the algebra M n (S(Z, τ Z )) (see [2]). Now if M is an arbitrary finite von Neumann algebra of type I with the center Z, then there exists a family {z n } n∈F , F ⊆ N, of orthogonal central projections in M such that sup n∈F z n = 1 and M is * -isomorphic to the C * -product of homogeneous von Neumann algebras z n M of type I n respectively, n ∈ F, i.e.…”

Section: ) Is a Derivation On S(m τ) In Particular Any T τ -Continmentioning

confidence: 99%

“…We have conjectured in [1,2] that the existence of such "exotic" examples of derivations is closely connected with the commutative nature of these algebras. This was confirmed for the particular case of type I von Neumann algebras in [1,2], moreover we have investigated and completely described derivations on the algebra LS(M) of all locally measurable operators affiliated with a type I von Neumann algebra M and on its various subalgebras. Recently the above conjecture was also confirmed for the type I case in the paper [6] by a representation of measurable operators as operator valued functions.…”

Section: Introductionmentioning

confidence: 99%

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Local Derivations on Algebras of Measurable Operators

Albeverio

Ayupov

Kudaybergenov

et al. 2011

Commun. Contemp. Math.

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The paper is devoted to local derivations on the algebra S(M, τ) of τ -measurable operators affiliated with a von Neumann algebra M and a faithful normal semi-finite trace τ. We prove that every local derivation on S(M, τ) 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 M for the algebra S(M, τ) to admit local derivations which are not derivations.Lemma 3.1. Given any element a ∈ A there exists an idempotent e ∈ ∇ such that (i) ea is integral with respect to K c (∇), moreover in this case ea ∈ K c (∇); (ii) if e = 1, then e ⊥ a is weakly transcendental with respect to K c (∇).Proof. Denote by ∇ int the set of all idempotents e ∈ ∇ such that ea is integral with respect to K c (∇). By [5, Proposition 3.8] each integral element with respect to K c (∇) in fact belongs to K c (∇). Therefore ∇ int = {e ∈ ∇ : ea ∈ K c (∇)}. We set e = sup ∇ int . Since ∇ is a complete Boolean algebra of countable type (see [5, Proposition 2.7]), there exists a countable family of mutually disjoint elements {e k } k≥1 in ∇ with sup k≥1 e k = e, such that given any e ∈ ∇ int there exists k ≥ 1, for which e k ≤ e . It is clear that from e k ≤ e and e ∈ ∇ int it follows that e k ∈ ∇ int , and thus e k a ∈ K c (∇). Therefore ea = k≥1 e k a ∈ K c (∇). Further since s(a) ⊥ a = 0 ∈ K c (∇) we have that s(a) ⊥ ≤ e, i.e. e ⊥ ≤ s(a) and hence s(e ⊥ a) = e ⊥ . Now let us show that if e = 1 then e ⊥ a is weakly transcendental with respect to K c (∇). Suppose the opposite, i.e. there exists a non zero idempotent q ≤ e ⊥ = s(e ⊥ a) such that qa is integral with respect to K c (∇). This means that q ∈ ∇ int , i.e. q ≤ e. This is a contradiction with 0 = q ≤ e ⊥ . Therefore e ⊥ a is weakly transcendental with respect to K c (∇). The proof is complete.

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Section: ) Is a Derivation On S(m τ) In Particular Any T τ -Continmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local Derivations on Algebras of Measurable Operators

Albeverio

Ayupov

Kudaybergenov

et al. 2011

Commun. Contemp. Math.

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“…Recently, some properties of cyclically compact operators on Hilbert-Kaplansky modules have been investigated in [7,8]. Certain natural applications of Hilbert-Kaplansky modules appeared in [1]. Namely, it has been shown that the algebra of all locally measurable operators with respect to a type I von Neumann algebra can be represented as an algebra of all bounded module-linear operators acting on a Hilbert-Kaplansky module over the ring of measurable function on a measure space.…”

Section: Introductionmentioning

confidence: 99%

“…Namely, it has been shown that the algebra of all locally measurable operators with respect to a type I von Neumann algebra can be represented as an algebra of all bounded module-linear operators acting on a Hilbert-Kaplansky module over the ring of measurable function on a measure space. This result played a crucial role in the description of derivations on algebras of locally measurable operators with respect to type I von Neumann algebras and their subalgebras (see for example, [1][2][3][4]). …”

Section: Introductionmentioning

confidence: 99%

Self-Adjoint Cyclically Compact Operators and Its Application

Kudaybergenov¹,

Mukhamedov²

2017

Bulletin of the Korean Mathematical Society

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Abstract. The present paper is devoted to self-adjoint cyclically compact operators on Hilbert-Kaplansky module over a ring of bounded measurable functions. The spectral theorem for such a class of operators is given. We use more simple and constructive method, which allowed to apply this result to compact operators relative to von Neumann algebras. Namely, a general form of compact operators relative to a type I von Neumann algebra is given.

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“…В случае, когда ℳ коммутативная * -алгебра, критерием существования ненулевых дифференцирований в (ℳ) служит отсутствие свойства -дистрибутивности у булевой алгебры проекторов из ℳ [5]. Для алгебры фон Ней-мана ℳ типа задачи описания дифференцирований в алгебрах (ℳ) всех изме-римых, (ℳ, ) всех -измеримых и (ℳ) всех локально измеримых операторов решены в работах [6] и [7]. В частности, для алгебр фон Неймана ℳ типа ∞ уста-новлено, что любое дифференцирование на алгебрах (ℳ), (ℳ, ), (ℳ) явля-ется внутренним.…”

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Непрерывные дифференцирования на $*$-алгебрах $\tau$-измеримых операторов - внутренние

Бер¹

2013

Matematicheskie Zametki

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Structure of derivations on various algebras of measurable operators for type I von Neumann algebras

Cited by 59 publications

References 17 publications

Local Derivations on Algebras of Measurable Operators

Local Derivations on Algebras of Measurable Operators

Self-Adjoint Cyclically Compact Operators and Its Application

Непрерывные дифференцирования на $*$-алгебрах $\tau$-измеримых операторов - внутренние

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