Interpolation between non-commutative BMO and non-commutative Lp-spaces

Musat, Magdalena

doi:10.1016/s0022-1236(03)00099-5

Cited by 57 publications

(76 citation statements)

References 18 publications

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“…For more information on noncommutative martingale BM O-spaces, we refer to [27,17,24,16]. It was shown in [27] that the classical Feffermann duality is still valid in the noncommutative settings.…”

Section: Now We Apply Lemma 23(i) To Get That Since For Everymentioning

confidence: 99%

Noncommutative fractional integrals

Randrianantoanina

2016

Studia Math.

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Abstract. Let M be a hyperfinite finite von Nemann algebra and (M k ) k≥1 be an increasing filtration of finite dimensional von Neumann subalgebras of M. We investigate abstract fractional integrals associated to the filtration (M k ) k≥1 . For a finite noncommutative martingale x = (x k ) 1≤k≤n ⊆ L1(M) adapted to (M k ) k≥1 and 0 < α < 1, the fractional integral of x of order α is defined by setting:

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Section: Now We Apply Lemma 23(i) To Get That Since For Everymentioning

confidence: 99%

Noncommutative fractional integrals

Randrianantoanina

2016

Studia Math.

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“…The following lemma, which can be proved in the same way as in the tracial case (see [26]), will enable us to use interpolation for these norms.…”

Section: Interpolation Resultsmentioning

confidence: 99%

“…In this paper we will make crucial use of Kosaki's interpolation results (see [23]) , which will enable us to extend the results in [26] to the nontracial setting.…”

Section: Interpolation Resultsmentioning

confidence: 99%

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A noncommutative version of the John-Nirenberg theorem

Junge

Musat

2006

Trans. Amer. Math. Soc.

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“…For more information on non-commutative martingale BMO-spaces, we refer to the articles [20,10,15,9]. Special attention will be given to the subspace called vanishing mean oscillation, denoted by VMO(M), and defined as the closure (for the BMO-norm) of the linear subspace of those x ∈ BMO(M) for which E n (x) = x for some n ∈ N.…”

Section: Notation and Preliminary Definitionsmentioning

confidence: 99%

Non-commutative martingale VMO-spaces

Randrianantoanina

2009

Studia Math.

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Abstract. We study Banach space properties of non-commutative martingale VMOspaces associated with general von Neumann algebras. More precisely, we obtain a version of the classical Kadets-Pełczyński dichotomy theorem for subspaces of non-commutative martingale VMO-spaces. As application we prove that if M is hyperfinite then the noncommutative martingale VMO-space associated with a filtration of finite-dimensional von Neumannn subalgebras of M has property (u).1. Introduction. The space of functions of bounded mean oscillation generally referred to as BMO-space has been instrumental in several aspects of analysis. Its martingale version plays an equally important role in probability.In this paper, we analyze subspaces of BMO-spaces related to non-commutative martingales. Our main motivation comes primarily from a paper by Müller and Schechtman [14] who studied structural properties of closed subspaces of dyadic martingale VMO (vanishing mean oscillation) as Banach spaces. More precisely, they provided, among other things, a version of the classical Kadets-Pełczyński dichotomy theorem for closed subspaces of dyadic martingale VMO-spaces. In order to explain the details, we first recall the celebrated Kadets-Pełczyński dichotomy theorem, which states that every closed subspace of L p (0, 1), 2 < p < ∞, either is isomorphic to a Hilbert space or contains a subspace which is isomorphic to l p . This dichotomy plays a crucial role in the development of L p -spaces and the theory of function spaces in general. Non-commutative analogues of the Kadets-Pełczyński dichotomy has been considered by several authors with the most general result obtained by Raynaud and Xu (see [21]) in the context of Haagerup L p -spaces when 2 ≤ p < ∞. Clearly, the dichotomy does not extend to closed subspaces of L ∞ (0, 1) or any C(K)-spaces in general. As a substitute, the following result was obtained by Müller and Schechtman:

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Interpolation between non-commutative BMO and non-commutative Lp-spaces

Cited by 57 publications

References 18 publications

Noncommutative fractional integrals

Noncommutative fractional integrals

A noncommutative version of the John-Nirenberg theorem

Non-commutative martingale VMO-spaces

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