Notes on non-commutative integration

Nelson, Edward

doi:10.1016/0022-1236(74)90014-7

Cited by 545 publications

(456 citation statements)

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“…The completion of M with respect to the norm · 1 is denoted by L 1 (M, τ ). It is known [20] that the spaces L 1 (M, τ ) and M * are isometrically isomorphic; therefore they can be identified. Further we will use this fact without noting it again.…”

Section: Preliminariesmentioning

confidence: 99%

On mixing and completely mixing properties of positive 𝐿¹-contractions of finite von Neumann algebras

Mukhamedov

Temir

Akın

2005

Proc. Amer. Math. Soc.

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Section: Preliminariesmentioning

confidence: 99%

On mixing and completely mixing properties of positive 𝐿¹-contractions of finite von Neumann algebras

Mukhamedov

Temir

Akın

2005

Proc. Amer. Math. Soc.

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“…, then E is said to be symmetric if / , g € E and g << f imply that \\g\\ E < \\f\\ E . Here g <-< f denotes submajorization in the sense of Hardy-Littlewood-Polya [5] It was shown in [19] that (^,d) is a complete metric, Hausdorff, topological *-algebra.…”

Section: E^ := [X : \X\" E E]mentioning

confidence: 99%

“…By taking a subsequence (if necessary), we will assume that for every n > 1, erms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700003359 [19] Non-commutative spaces 349…”

Section: Claim the Sequence {[X N ]}™ =X Is Right Disjointly Supportedmentioning

confidence: 99%

Embeddings of ℓ_p into Non-commutative Spaces

Randrianantoanina

2003

J. Aust. Math. Soc.

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Let ^jK be a semi-finite von Neumann algebra equipped with a faithful normal trace r. We prove a Kadec-Pelczyiiski type dichotomy principle for subspaces of symmetric space of measurable operators of Rademacher type 2. We study subspace structures of non-commutative Lorentz spaces L PJ) {M', r), extending some results of Carothers and Dilworth to the non-commutative settings. In particular, we show that, under natural conditions on indices, l p cannot be embedded into L pq (^, r). As applications, we prove that for 0 < p < oo with p ^ 2, l p cannot be strongly embedded into L p (M', T). This provides a non-commutative extension of a result of Kalton for 0 < p < 1 and a result of Rosenthal for 1 < p < 2 onZ, p [0,1].2000 Mathematics subject classification: primary 46L52, 46L51, 46E30. Keywords and phrases: Lorentz spaces, von Neumann algebras, non-commutative /.,,-spaces. IntroductionThe study of rearrangement invariant Banach spaces of measurable functions is a classical theme. Several studies have been devoted to characterizations of subspaces of rearrangement invariant spaces. Recently, the theory of rearrangement invariant Banach spaces of measurable operators affiliated with semi-finite von Neumann algebra have emerged as the natural non-commutative generalizations of Kothe functions spaces. This theory, which is based on the theory of non-commutative integration introduced by Segal [24], replaces the classical duality (Loo(/it), ^i 0-0) by the duality between a semi-finite von Neumann algebra and its predual. It provides a unified approach to the study of unitary ideals and rearrangement invariant spaces. Several authors have considered these non-commutative spaces of measurable operators (see for instance, [4, 6,7,8,28] The purpose of the present paper is to examine subspaces of symmetric spaces of measurable operators in which the norm topology and the measure topology coincide, and subspaces generated by disjointly supported basic sequences. Such subspaces are of particular importance as they represent in many cases the extreme structures. Our main method is to exploit the notion of uniform integrability of operators introduced in [21]. One of main results of this paper is a dichotomy type result for subspaces of symmetric spaces of measurable operators. More precisely, we prove that any given subspace of a symmetric space of measurable operators either is isomorphic to a Hilbert space or contains a basic sequence equivalent to a disjointly supported sequence.The [0, oo). Such result, not only is of interest in its own right, but also provides an alternative proof to some non-trivial results on L p -spaces.Motivated by such connections, we examine the subspace structures of non commutative Lorentz spaces L pq (^, r), where (Jt, r) is a semi-finite von Neumann algebra. Making use of our dichotomy result and some other results of general nature, we show that some of the results of [2] and [3] extend to the non-commutative settings. Our approach relies on a disjointification techniques based on th...

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“…Equipped with this topology, S(τ ) is a complete topological * -algebra. These facts and their proofs can be found in the papers [14,20].…”

Section: Notation and Preliminariesmentioning

confidence: 94%