Journal of Noncommutative Geometry

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Volume 7, Issue 2, 2013, pp. 583–603
DOI: 10.4171/JNCG/128

Published online: 2013-05-06

Uniform local amenability

Jacek Brodzki[1], Graham A. Niblo[2], Ján Špakula[3], Rufus Willett[4] and Nick J. Wright[5]

(1) University of Southampton, UK
(2) University of Southampton, UK
(3) University of Münster, Germany
(4) University of Hawaii at Manoa, USA
(5) University of Southampton, UK

The main results of this paper show that various coarse (‘large scale’) geometric properties are closely related. In particular, we show that property A implies the operator norm localisation property, and thus that norms of operators associated to a very large class of metric spaces can be effectively estimated.

The main tool is a new property called uniform local amenability. This property is easy to negate, which we use to study some `bad' spaces: specifically, expanders and graphs with large girth. We also generalise and reprove a theorem of Nowak relating amenability and asymptotic dimension in the quantitative setting.

Keywords: Coarse embedding in Hilbert space, property A, operator norm localization, metric sparsification, expander, graphs with large girth

Brodzki Jacek, Niblo Graham, Špakula Ján, Willett Rufus, Wright Nick: Uniform local amenability. J. Noncommut. Geom. 7 (2013), 583-603. doi: 10.4171/JNCG/128