Identities, Approximate Identities and Topological Divisors of Zero in Banach Algebras

TL;DR

This paper identifies a broad class of non-unital Banach algebras with approximate identities where every element is a topological divisor of zero, extending previous results and exploring the role of completeness.

Contribution

It introduces a new large class of Banach algebras with all elements as topological divisors of zero, improving earlier results and providing characterizations related to the socle.

Findings

Many classical examples belong to the identified class.

Counterexamples show the class inclusion is not universal.

Completeness affects the properties of these algebras.

Abstract

In [3] S. J. Bhatt and H. V. Dedania exposed certain classes of Banach algebras in which every element is a topological divisor of zero. We identify a new (large) class of Banach algebras with the aforementioned property, namely, the class of non-unital Banach algebras which admits either an approximate identity or approximate units. This also leads to improvements of results by R. J. Loy and J. Wichmann, respectively. If we observe that every single example that appears in [3] belongs to the class identified in the current paper, and, moreover, that many of them are classical examples of Banach algebras with this property, then it is tempting to conjecture that the classes exposed in [3] must be contained in the class that we have identified here. However, we show somewhat elusive counterexamples. Furthermore, we investigate the role completeness plays in the results and show, by giving a suitable example, that the assumptions are not superfluous. The ideas considered here also yields a pleasing characterization: The socle of a semisimple Banach algebra is infinite-dimensional if and only if every socle-element is a topological divisor of zero in the socle.