On biamenability of Banach algebras

TL;DR

This paper introduces the concept of biamenability in Banach algebras, highlighting its differences from amenability and exploring its implications for various algebra classes and their character spaces.

Contribution

It defines biamenability, compares it with amenability, and investigates its properties and applications in characterizing Banach algebras.

Findings

Commutative Banach algebras like R and C are not biamenable.

Noncommutative Banach algebras such as B(H) can be biamenable despite not being amenable.

Unconditional unitization preserves amenability but not necessarily biamenability.

Abstract

In this paper, we introduce the concept of biamenability of Banach algebras and we show that despite the apparent similarities between amenability and biamenability of Banach algebras, they lead to very different, and somewhat opposed, theories. In this regard, we show that commutative Banach algebras such as R and C tend to lack biamenability, while they may be amenable and highly noncommutative Banach algebras such as B(H) for an infinite dimensional Hilbert space H tend to be biamenable, while they are not amenable. Also, we show that although the unconditional unitization of an amenable Banach algebra is amenable but in general unconditional unitization of a Banach algebra is not biamenable. This concept is used for finding the character space of some Banach algebras.