Biamenability of Banach algebras and its applications

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 and the group algebra $L^1(G)$, for each locally compact group $G$, tend to lack biamenability, while they may be amenable and highly non-commutative 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 may be applied for studying the character space of some Banach algebras and also for studying some spansion or density problems.

Keywords

• biderivation
• inner biderivation
• biamenability