Surveys in Mathematics and its Applications

ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 12 (2017), 203 -- 217

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This work is licensed under a Creative Commons Attribution 4.0 International License.


Robert Kantrowitz and Michael M. Neumann

Abstract. The purpose of this article is to survey a class of normed algebras that share many central features of Banach algebras, save for completeness. The likeness of these algebras to Banach algebras derives from the fact that the geometric series test is valid, whereas the lack of completeness points to the failure of the absolute convergence test for series in the algebra. Our main result is a compendium of conditions that are all equivalent to the validity of the geometric series test for commutative unital normed algebras. Several examples in the final section showcase some incomplete normed algebras for which the geometric series test is valid, and still others for which it is not.

2010 Mathematics Subject Classification: 46-02; 46H05; 46H20.
Keywords: Normed algebra, Banach algebra, geometric series.

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Robert Kantrowitz
Hamilton College
198 College Hill Road Clinton, N.Y. 13323, USA.

Michael M. Neumann
Mississippi State University, USA