Topological characterizations of amenability and congeniality of bases

Sergio R. López-Permouth, Benjamin Stanley

Abstract

We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite 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 innite dimensional F-algebra A is called amenable if FB, the direct product indexed by B of copies of the eld 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 epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.


Keywords

uniform topologies; linear vector spaces; amenable bases; congeniality of bases; Schauder bases; infinite-dimensional modules and algebras

Subject classification

16K40; 16D99; 15B99.

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References

L. M. Al-Essa, S. R. López-Permouth and N. M. Muthana, Modules over infinite-dimensional algebras, Linear and Multilinear Algebra 66 (2018), 488-496. https://doi.org/10.1080/03081087.2017.1301365

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S. R. López-Permouth and B. Stanley, On the amenability profile of an infinite dimensional module over an algebra, preprint.

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B. Stanley, Perspectives on amenability and congeniality of bases, Ph. Dissertation, Ohio University, February 2019.

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Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt