On the paper: Numerical radius preserving linear maps on Banach algebras

TL;DR

This paper provides a counterexample in Banach algebra theory showing that certain normalized states are not spectral or multiplicative, challenging previous assumptions and results in the field.

Contribution

It constructs a specific example of a Banach algebra with normalized states that are not spectral or multiplicative, disproving earlier claims.

Findings

Existence of a normalized state that is not spectral

Existence of an extreme normalized state that is not multiplicative

Disproof of two previous results by Golfarshchi and Khalilzadeh

Abstract

We give an example of a unital commutative complex Banach algebra having a normalized state which is not a spectral state and admitting an extreme normalized state which is not multiplicative. This disproves two results by Golfarshchi and Khalilzadeh.