Introduction to the theory of noncommutative integration

Trunov, N. V.; Sherstnev, A. N.

doi:10.1007/bf01103856

Cited by 14 publications

(8 citation statements)

References 27 publications

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“…It is known that [20] for the Banach pair L p 1 (M, τ ) and L p 2 (M, τ ), p 1 < p 2 , any space L p (M, τ ), p ∈ [p 1 ; p 2 ], is intermediate. Therefore…”

Section: Lemma 21 If the Intersection A ∩ B Is Dense In Each Membermentioning

confidence: 99%

Non-commutative Arens algebras and their derivations

Albeverio

Ayupov

Kudaybergenov

2007

Journal of Functional Analysis

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Given a von Neumann algebra M with a faithful normal semi-finite trace τ , we consider the non-commutative Arens algebra L ω (M, τ ) = p 1 L p (M, τ ) and the related algebras L ω 2 (M, τ ) = p 2 L p (M, τ ) and M +L ω 2 (M, τ ) which are proved to be complete metrizable locally convex *-algebras. The main purpose of the present paper is to prove that any derivation of the algebra M + L ω 2 (M, τ ) is inner and all derivations of the algebras L ω (M, τ ) and L ω 2 (M, τ ) are spatial and implemented by elements of M + L ω 2 (M, τ ). In particular we obtain that if the trace τ is finite then any derivation on the noncommutative Arens algebra L ω (M, τ ) is inner.

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“…It is known that [20] for the Banach pair L p 1 (M, τ ) and L p 2 (M, τ ), p 1 < p 2 , any space L p (M, τ ), p ∈ [p 1 ; p 2 ], is intermediate. Therefore…”

Section: Lemma 21 If the Intersection A ∩ B Is Dense In Each Membermentioning

confidence: 99%

Non-commutative Arens algebras and their derivations

Albeverio

Ayupov

Kudaybergenov

2007

Journal of Functional Analysis

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“…We will use facts and the terminology from the general theory of von Neumann algebras (see [5,7,17,19,22]), the general theory of Jordan and real operator algebras (see [2,3,11,16]), and the theory of noncommutative integration (see [20,23,24]).…”

Section: Introductionmentioning

confidence: 99%

On weak convergence of iterates in quantum L_p‐spaces(p ≥ 1)

Grabarnik

Katz

Shwartz

2005

International Journal of Mathematics and Mathematical Sciences

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“…It can be verified trivially that a wedge K in a normed space is filled if and only if it ideally convex in the sense of Lifshits [10[ (see also Section 1.6 in [9]), that is, for any bounded sequence {x n } ⊂ K and any sequence {α n } ⊂ R + such that ∞ n=1 α n = 1, the series ∞ n=1 α n x n converges to an element of K. Exercise 1.14 in [9] implies that any wedge in a finite-dimensional space is filled. Another instructive example can be given in the framework of Sherstnev's approach to the theory of integration with respect to a faithful normal semifinite weight ϕ on a von Neumann algebra M (see [11,12]): in the Banach space L h 1 (ϕ) of Hermitian integrable bilinear forms, the cone of closable positive integrable bilinear forms is not generally closed, but it generates L h 1 (ϕ) and is filled. …”

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confidence: 99%