A remark on continuity of positive linear functionals on separable Banach *-algebras

TL;DR

This paper provides a new proof that positive linear functionals on certain separable Banach *-algebras are continuous, extending the understanding of functional behavior in these algebraic structures.

Contribution

It introduces a novel proof technique based on a variation of the Murphy-Varopoulos Theorem for positive linear functionals.

Findings

Positive linear functionals are continuous under specified conditions.

The proof extends the R.J.Loy Theorem to broader classes of Banach *-algebras.

The approach simplifies understanding of functional continuity in algebraic contexts.

Abstract

Using a variation of the Murphy-Varopoulos Theorem, we give a new proof of the following R.J.Loy Theorem: Let A be a separable Banach*-algebra with center Z such that ZA has at most countable codimension, then every positive linear functional on A is continuous.