  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 commutative
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.
