Characterization of Symmetric Amenability of Unital Banach Algebras

TL;DR

This paper introduces new concepts of $p$-amenability and symmetric diagonals in Banach algebras, proving their equivalence in unital cases and expanding the understanding of symmetric amenability.

Contribution

It defines $p$-amenability and symmetric diagonals, and establishes their equivalence in unital Banach algebras, advancing the theory of symmetric amenability.

Findings

$p$-amenability implies existence of symmetric diagonals

In unital Banach algebras, $p$-amenability and symmetric amenability are equivalent

New concepts extend the characterization of symmetric amenability

Abstract

In this paper, we introduce $p$-amenability, bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals for a Banach algebra $\mathfrak{A}$ where $s$ is a non-zero element of algebraic center of $\mathfrak{A}$ that is denoted by $Z(\mathfrak{A})$. We show that if a Banach algebra $\mathfrak{A}$ is $p$-amenable then it has bounded $s$-symmetric approximate and $s$-symmetric virtual diagonals and by this fact we prove that if the Banach algebra $\mathfrak{A}$ is unital then $p$-amenability and symmetric amenability are equivalent.