A non-unital algebra has UUNP iff its unitization has UUNP

TL;DR

This paper proves that a non-unital algebra has the unique uniform norm property if and only if its unitization has this property, extending previous results to all non-unital algebras.

Contribution

The paper generalizes the equivalence of UUNP between a non-unital algebra and its unitization to all non-unital algebras.

Findings

A non-unital algebra has UUNP iff its unitization has UUNP.

Extends previous results from Banach algebras to all non-unital algebras.

Provides a complete characterization of UUNP in relation to algebra unitization.

Abstract

Let $A$ be a non-unital Banach algebra, S. J. Bhatt and H. V. Dedania showed that $A$ has the unique uniform norm property (UUNP) if and only if its unitization has UUNP. Here we prove this result for any non-unital algebra.