Extending positive definite linear forms

Abstract:

In the classical literature two properties, called here symmetry and bounded variation, of a positive definite linear form $a’$ on an involutive Banach algebra $\mathcal {A}$ are given. Together they form a necessary and sufficient condition that $a’$ admit a positive definite linear extension to the involutive algebra obtained from $\mathcal {A}$ by adjoining an identity. In this note we show that bounded variation alone suffices, in that it already implies symmetry.