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Amenability of coarse spaces and \(\mathbb {K}\)-algebras. (English) Zbl 1427.16020

Summary: In this article we analyze the notions of amenability and paradoxical decomposition from an algebraic perspective. We consider this dichotomy for locally finite extended metric spaces and for general algebras over fields. In the context of algebras we also study the relation of amenability with proper infiniteness. We apply our general analysis to two important classes of algebras: the unital Leavitt path algebras and the translation algebras on locally finite extended metric spaces. In particular, we show that the amenability of a metric space is equivalent to the algebraic amenability of the corresponding translation algebra.

MSC:

16S99 Associative rings and algebras arising under various constructions
16S88 Leavitt path algebras
16P90 Growth rate, Gelfand-Kirillov dimension
20F65 Geometric group theory
37A15 General groups of measure-preserving transformations and dynamical systems
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