Johnson pseudo-contractibility of certain Banach algebras and their nilpotent ideals

TL;DR

This paper investigates Johnson pseudo-contractibility in Banach algebras, demonstrating that the algebra of a bicyclic semigroup lacks this property and that such algebras cannot have non-zero complemented closed nilpotent ideals.

Contribution

It establishes new results linking Johnson pseudo-contractibility with the structure of Banach algebras and their nilpotent ideals, especially for semigroup algebras.

Findings

ll^{1}(S) is not Johnson pseudo-contractible for a bicyclic semigroup S

Johnson pseudo-contractible Banach algebras have no non-zero complemented closed nilpotent ideals

The paper provides structural insights into the properties of Banach algebras related to Johnson pseudo-contractibility

Abstract

In this paper, we study the notion of Johnson pseudo-contractibility for certain Banach algebras. For a bicyclic semigroup $S$, we show that $\ell^{1}(S)$ is not Johnson pseudo-contractible. Also for a Johnson pseudo-contractible Banach algebra $A$, we show that $A$ has no non-zero complemented closed nilpotent ideal.