Topological characterizations of amenability and congeniality of bases

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© AGT, UPV, 2020. We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of infinite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed. A basis B over an infinite dimensional F-algebra A is called amenable if FB, the direct product indexed by B of copies of the field F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules. (Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimor- phism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.