Quasi-Locality and Property A

Abstract

Let $X$ be a metric space with bounded geometry, $p\in \{0\}\cup [1,\infty ]$, and let $E$ be a Banach space. The main result of this paper is that either if $X$ has Yu's Property A and $p\in (1,\infty )$, or without any condition on $X$ when $p\in \{0,1,\infty \}$, then quasi-local operators on ${\ell }^p(X,E)$ 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 ${\ell }^p$-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.

arXiv comment21 pages; submitted

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• Fredholm theory for band-dominated and related operators: a survey
• The topology of positive scalar curvature
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• Crossed products of $L^p$ operator algebras and the K-theory of Cuntz algebras on $L^p$ spaces
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Cited by in corpus (3)

• Uniform Roe algebras of uniformly locally finite metric spaces are rigid
• A quasi-local characterisation of $L^p$-Roe algebras
• Structure and $K$-theory of ${\ell }^p$ uniform Roe algebras