Weak* continuous states on Banach algebras

TL;DR

This paper proves that in unital Banach algebras that are dual spaces, weak* continuous states are dense among all states and linearly span the dual space, revealing structural properties of states.

Contribution

It establishes the density and spanning properties of weak* continuous states in dual Banach algebras, a novel insight into their structure.

Findings

Weak* continuous states are weak* dense in all states.

Weak* continuous states linearly span the dual space.

Structural insight into states on dual Banach algebras.

Abstract

We prove that if a unital Banach algebra $A$ is the dual of a Banach space $\pd{A}$, then the set of weak* continuous states is weak* dense in the set of all states on $A$. Further, weak* continuous states linearly span $\pd{A}$.