An article titled “Normed Algebras and the Geometric Series Test” by Professor of Mathematics Robert Kantrowitz ’82 and Michael M. Neumann of Mississippi State University was recently published in the journal Surveys in Mathematics and its Applications. The paper highlights a class of normed algebras that share remarkably many features with Banach algebras.
The defining property of the algebras under consideration is that the geometric series test is valid, whereas Banach algebras, named after the 20th-century Polish mathematician Stefan Banach, are characterized by the more stringent requirement that the absolute convergence test for series holds. Both the geometric series test and the test for absolute convergence migrate over to the realm of normed algebras from the traditional calculus setting.
The main result of the paper is an inventory of conditions that are all equivalent to the validity of the geometric series test for commutative unital normed algebras.