A generalization of cyclic amenability of Banach algebras

TL;DR

This paper extends the understanding of approximate cyclic amenability in Banach algebras, demonstrating its preservation under homomorphic images, matrix algebra equivalence, and conditions relating to second duals.

Contribution

It proves that approximate cyclic amenability is preserved under homomorphic images and matrix algebra equivalence, and explores conditions linking the property of a Banach algebra to its second dual.

Findings

Homomorphic images of approximately cyclic amenable algebras are approximately cyclic amenable.

Approximate cyclic amenability of a Banach algebra is equivalent to that of its matrix algebra $M_{n}( ext{A})$.

Under certain conditions, the approximate cyclic amenability of the second dual implies that of the original algebra.

Abstract

This paper continues the investigation of Esslamzadeh and the first author which was begun in [7]. It is shown that homomorphic image of an approximately cyclic amenable Banach algebra is again approximately cyclic amenable. Equivalence of approximate cyclic amenability of a Banach algebra $\mathcal{A}$ and approximate cyclic amenability of $M_{n}(\mathcal{A})$ is proved. It is shown that under certain conditions the approximate cyclic amenability of second dual $\mathcal{A}^{**}$ implies the approximate cyclic amenability of $\mathcal{A}$.