Topological characterizations of amenability and congeniality of bases

Sergio R. López-Permouth

United States

Ohio University

Department of Mathematics

Benjamin Stanley

United States

Ohio University

Department of Mathematics

Submitted: 2019-03-08

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Accepted: 2019-11-28

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Published: 2020-04-03

DOI: https://doi.org/10.4995/agt.2020.11488
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Keywords:

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

Supporting agencies:

Ohio University

Department of Mathematics

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.

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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

P. Aydogdu, S. R. López-Permouth and R. Muhammad, Infinite-dimensional algebras without simple bases, Linear and Multilinear Algebra, to appear.

J. Díaz Boils, S. R. López-Permouth and R. Muhammad, Amenable and simple bases of tensor products of infinite dimensional algebras, preprint.

R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6 (1989).

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

P. Nielsen, Row and column finite matrices, Proc. Amer. Math. Soc. 135, no. 9 (2007), 2689-2697. https://doi.org/10.1090/S0002-9939-07-08790-4

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

S. Willard, General Topology, Dover Publications (1970).

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