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Property A and the operator norm localization property for discrete metric spaces. (English) Zbl 1295.46017
The main result of the paper is that, for discrete metric spaces with bounded geometry, the property A introduced in [G.-L. Yu, Invent. Math. 139, No. 1, 201–240 (2000; Zbl 0956.19004)] is equivalent to the operator norm localization property introduced in [X.-M. Chen et al., Adv. Math. 218, No. 5, 1496–1511 (2008; Zbl 1148.46040)].
This result is a very important contribution to the study of properties of metric spaces related to higher index theory; see [J. Brodzki et al., J. Noncommut. Geom. 7, No. 2, 583–603 (2013; Zbl 1281.46023)] for recent related results and discussion on this matter.
The methods of the paper are operator algebraic.

##### MSC:
 46B85 Embeddings of discrete metric spaces into Banach spaces; applications in topology and computer science 19K56 Index theory 46L89 Other “noncommutative” mathematics based on $$C^*$$-algebra theory
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