Quasi-isometric embedding between $*$-algebras

TL;DR

This paper introduces the concept of quasi-isometric embedding maps between $*$-algebras, establishing foundational results and criteria for such embeddings akin to metric space cases.

Contribution

It defines quasi-isometric embeddings for $*$-algebras and provides necessary and sufficient conditions for $*$-homomorphisms to qualify as such.

Findings

Established basic properties of quasi-isometric embeddings in $*$-algebras.

Derived necessary and sufficient conditions for $*$-homomorphisms to be quasi-isometric embeddings.

Abstract

The concept of quasi-isometric embedding maps between $*$-algebras is introduced. We have obtained some basic results related to this notion and similar to quasi-isometric embedding maps on metric spaces, under some conditions, we give a necessary and sufficient condition on a $*$-homomorphism to be a quasi-isometric embedding between $*$-algebras.