English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In [3] S. J. Bhatt and H. V. Dedania exposed certain classes of Banachalgebras in which every element is a topological divisor of zero. We identify anew (large) class of Banach algebras with the aforementioned property, namely,the class of non-unital Banach algebras which admits either an approximateidentity or approximate units. This also leads to improvements of results by R.J. Loy and J. Wichmann, respectively. If we observe that every single examplethat 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 thisproperty, 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 showsomewhat elusive counterexamples. Furthermore, we investigate the rolecompleteness plays in the results and show, by giving a suitable example, thatthe assumptions are not superfluous. The ideas considered here also yields apleasing characterization: The socle of a semisimple Banach algebra isinfinite-dimensional if and only if every socle-element is a topologicaldivisor of zero in the socle.

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