Extreme Cases of Limit Operator Theory on Metric Spaces

Zhang, Jiawen

doi:10.1007/s00020-018-2498-7

Cited by 6 publications

(9 citation statements)

References 19 publications

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“…More generally, we have the following result. The proof is similar to that of Theorem 5.1, "(i) ⇔ (ii)" in [37]. Second, in the case of p ∈ (1, ∞) or especially when p = 2, is the assumption of Property A really necessary for all the quasi-local operators to be approximable by operators with finite propagation?…”

mentioning

confidence: 72%

“…, which is the band-dominated operator algebra (see [30,37]). And it is clear that K(X, B) = K p E (X) defined thereby, which is the set of all P-compact operators on ℓ p E (X).…”

Section: Definition 29 ([20]) Let (X D) Be a Discrete Metric Spacementioning

confidence: 99%

“…Note that the above definition assumes p ∈ [1, ∞). For the case that p ∈ {0, ∞}, we need a few more notions and a lemma from [37].…”

Section: Property a And Partition Of Unitymentioning

confidence: 99%

“…The following technical lemma is taken from [37], using partition of unity and the associated dual family to construct operators via blocks. To simplify the statement, we declare that a metric p-partition of unity always means a metric 1-partition of unity when p ∈ {0, 1, ∞}.…”

Section: Definition 215 ([37]mentioning

confidence: 99%

“…The main result of this paper provides an affirmative answer to this question for any metric space X with bounded geometry in the following cases: without any further assumption on X for p ∈ {0, 1, ∞}; or when X has Property A and p ∈ (1, ∞). For the former case, our proof is inspired by the recent work on limit operator theory [37]. For the latter case, we utilise the key technical trick from [17,29] (cf.…”

Section: Introductionmentioning

confidence: 99%

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Quasi-locality and Property A

Spakula

Zhang

2020

Journal of Functional Analysis

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Let X be a metric space with bounded geometry, p ∈ {0} ∪ [1, ∞], and let E be a Banach space. The main result of this paper is that either if X has Yu's Property A and p ∈ (1, ∞), or without any condition on X when p ∈ {0, 1, ∞}, then quasi-local operators on ℓ p (X, E) belong to (the appropriate variant of) Roe algebra of X. This generalises the existing results of this type by Lange and Rabinovich, Engel, Tikuisis and the first author, and Li, Wang and the second author. As consequences, we obtain that uniform ℓ p-Roe algebras (of spaces with Property A) are closed under taking inverses, and another condition characterising Property A, akin to operator norm localisation for quasi-local operators.

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mentioning

confidence: 72%

“…, which is the band-dominated operator algebra (see [30,37]). And it is clear that K(X, B) = K p E (X) defined thereby, which is the set of all P-compact operators on ℓ p E (X).…”

Section: Definition 29 ([20]) Let (X D) Be a Discrete Metric Spacementioning

confidence: 99%

“…Note that the above definition assumes p ∈ [1, ∞). For the case that p ∈ {0, ∞}, we need a few more notions and a lemma from [37].…”

Section: Property a And Partition Of Unitymentioning

confidence: 99%

Section: Definition 215 ([37]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Quasi-locality and Property A

Spakula

Zhang

2020

Journal of Functional Analysis

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A quasi-local characterisation of L-Roe algebras

Wang

Zhang

2019

Journal of Mathematical Analysis and Applications

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Very recently,Špakula and Tikuisis provide a new characterisation of (uniform) Roe algebras via quasi-locality when the underlying metric spaces have straight finite decomposition complexity. In this paper, we improve their method to deal with the L p -version of (uniform) Roe algebras for any p ∈ [1, ∞). Due to the lack of reflexivity on L 1 -spaces, some extra work is required for the case of p = 1.Mathematics Subject Classification (2010): 20F65, 46H35, 47L10. Introduction(Uniform) Roe algebras are C * -algebras associated to metric spaces, which reflect coarse properties of the underlying metric spaces. These algebras have been well-studied and have fruitful applications, among which the most important ones would be the (uniform) coarse Baum-Connes conjecture and the Novikov conjecture (e.g., [32,33,40,41,42,43]). Meanwhile, they also provide a link between coarse geometry of metric spaces and the theory of C * -algebras (e.g., [1,11,15,16,19,20,23,29,30,32,37,39]), and turn out to be useful in the study of topological phases of matter (e.g., [17,10]) as well as the theory of limit operators in the study of Fredholmness of band-dominated operators (e.g., [14,21,31,35]).By definition, the (uniform) Roe algebra of a proper metric space X is the norm closure of all bounded locally compact operators T with finite propagation in the sense that there exists R > 0 such that for any f, g ∈ C b (X) acting on L 2 (X) by pointwise multiplication, we have f Tg = 0 provided their supports are Rseparated (i.e., the distance between the supports of f and g is at least R). Since general elements in (uniform) Roe algebras may not have finite propagation, it is usually difficult to tell what operators exactly belong to them. On the other hand, Roe [27] defined an asymptotic version of finite propagation as follows: An operator T on L 2 (X) has finite ε-propagation for ε > 0, if there is R > 0 such that for any f, g ∈ C b (X), we have f Tg ≤ ε f · g provided their supports Key words and phrases. Quasi-local operators, L p -Roe algebras, straight finite decomposition complexity.

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Limit operators techniques on general metric measure spaces of bounded geometry

Hagger

Seifert

2020

Journal of Mathematical Analysis and Applications

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We study band-dominated operators on (subspaces of) Lp-spaces over metric measure spaces of bounded geometry satisfying an additional property. We single out core assumptions to obtain, in an abstract setting, definitions of limit operators, characterizations of compactness and Fredholmness using limit operators; and thus also spectral consequences. In this way, we recover and unify the classical and recent results on limit operator techniques, but also gain new insights and are able to treat further applications.MSC2010: Primary: 47A53; Secondary: 47B07, 47B35-38, 47L10

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Extreme Cases of Limit Operator Theory on Metric Spaces

Cited by 6 publications

References 19 publications

Quasi-locality and Property A

Quasi-locality and Property A

A quasi-local characterisation of L-Roe algebras

Limit operators techniques on general metric measure spaces of bounded geometry

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