Property A and the operator norm localization property for discrete metric spaces

Sako, Hiroki

doi:10.1515/crelle-2012-0065

Cited by 60 publications

(47 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So, Ψ is a compact preserving strongly continuous linear map, hence, by Proposition 3.3, Ψ is coarse-like. Since X has property A, X has the operator norm localization [21,Theorem 4.1], so [4,Lemma 7.2] gives r > 0 such that for all A ⊆ X and all B ⊆ Y with…”

Section: Embeddings Onto Hereditary Subalgebrasmentioning

confidence: 99%

Embeddings of Uniform Roe Algebras

Braga

Farah

Vignati

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

In this paper, we study embeddings of uniform Roe algebras. Generally speaking, given metric spaces X and Y , we are interested in which large scale geometric properties are stable under embedding of the uniform Roe algebra of X into the uniform Roe algebra of Y . Contents 1. Introduction 1 2. Preliminaries 5 3. Coarse-like maps and quasi-locality 8 4. Making embeddings strongly continuous 13 5. Embeddings and geometry preservation 16 6. Embeddings onto hereditary subalgebras 22 7. Digression on our geometric property 27 8. Open problems 30 References 32

show abstract

Section: Embeddings Onto Hereditary Subalgebrasmentioning

confidence: 99%

Embeddings of Uniform Roe Algebras

Braga

Farah

Vignati

2019

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…This localizability of the operator norm is no longer a property of a particular operator but rather of the space X. There is recent work by X. Chen, R. Tessera, X. Wang, G. Yu and H. Sako (see [32] and references therein) on metric spaces M with a certain measure such that X = l 2 (M ) has the operator norm localization property (ONL). Sako proves in [32] that in case of a discrete metric space M with sup m∈M |{n ∈ M : d(m, n) ≤ R}| < ∞ for all radii R > 0 (which clearly holds in our case, M = Z N ), property (ONL) is equivalent to the so-called Property A that was introduced by G. Yu and is connected with amenability.…”

Section: P-essential Norm Of Band-dominated Operatorsmentioning

confidence: 99%

Essential pseudospectra and essential norms of band-dominated operators

Hagger

Lindner

Seidel

2016

Journal of Mathematical Analysis and Applications

View full text Add to dashboard Cite

An operator A on an l p -space is called band-dominated if it can be approximated, in the operator norm, by operators with a banded matrix representation. The coset of A in the Calkin algebra determines, for example, the Fredholmness of A, the Fredholm index, the essential spectrum, the essential norm and the so-called essential pseudospectrum of A. This coset can be identified with the collection of all so-called limit operators of A. It is known that this identification preserves invertibility (hence spectra). We now show that it also preserves norms and in particular resolvent norms (hence pseudospectra). In fact we work with a generalization of the ideal of compact operators, so-called P-compact operators, allowing for a more flexible framework that naturally extends to l p -spaces with p ∈ {1, ∞} and/or vector-valued l p -spaces.

show abstract

“…Proposition 7.6 fails for any space X that does not have the metric sparsification property. By [19] (see also [3]), this is equivalent to not having the operator norm localisation property, and so [18,Lemma 4.2] provides us with r ą 0, κ ă 1, a sequence of disjoint finite subsets X n of X, a sequence of positive, norm one operators A n P Lpℓ 2 X n q with propagation at most r and an increasing sequence of positive reals s n tending to infinity, such that for any v P ℓ 2 X n of norm one, with support of diameter at most s n , one has }A n v} ď κ. Furthermore, it is argued in [18, Proof of Theorem 1.3] that there are eigenvectors of A n with eigenvalue 1.…”

Section: Consequently We Getmentioning

confidence: 99%

“…The gist of the property is that one can choose big sets (in a given measure) that split into well separated uniformly bounded sets. It does not seem to be obvious that the metric sparsification property is equivalent to property A: this follows on combining results from [3] and [19].…”

Section: Metric Sparsification and Uniform Boundednessmentioning

confidence: 99%

See 1 more Smart Citation

A metric approach to limit operators

Spakula

Willett

2016

Trans. Amer. Math. Soc.

106

View full text Add to dashboard Cite

Abstract. We extend the limit operator machinery of Rabinovich, Roch, and Silbermann from Z N to (bounded geometry, strongly) discrete metric spaces. We do not assume the presence of any group structure or action on our metric spaces. Using this machinery and recent ideas of Lindner and Seidel, we show that if a metric space X has Yu's property A, then a band-dominated operator on X is Fredholm if and only if all of its limit operators are invertible. We also show that this always fails for metric spaces without property A.

show abstract

Property A and the operator norm localization property for discrete metric spaces

Cited by 60 publications

References 16 publications

Embeddings of Uniform Roe Algebras

Embeddings of Uniform Roe Algebras

Essential pseudospectra and essential norms of band-dominated operators

A metric approach to limit operators

Contact Info

Product

Resources

About