Quasi-isometric embedding maps between $*$-algebras and their relation with an amenability question

Abstract

In this paper, we investigate the following question: under which condition or conditions the amenability of Banach algebras can be transferred back i.e. if $\varphi$ is a continuous homomorphism from a Banach algebra $A$ into a Banach algebra $B$ such that $B$ is amenable, whenever $A$ is amenable? To answer this question, first, we define a new version of quasi-isometric embedding maps between $*$-algebras and obtain some primary results related to this notion.  Moreover, as a main result, we show that if $\varphi$ is an injective quasi-isometry between $*$-Banach algebras $A$ and $B$, then the amenability of $B$ implies the amenability of $A$.